playground.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import torch\n",
    "import warnings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def read_datasets(data_pq_file):\n",
    "    df = pd.read_parquet(data_pq_file)\n",
    "    captions, img_names, img_bytes = df['formula'], df['filename'], df['image']\n",
    "    # assert len(captions) == len(img_names) == len(img_bytes), 'dataset parquet got errors'\n",
    "    if not len(captions) == len(img_names) == len(img_bytes):\n",
    "        warnings.warn('Warning! Dataset may got errors.')\n",
    "    return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def combine_pred_gt(predict_res, test_df):\n",
    "    pairs = []\n",
    "    \n",
    "    # 读取 predictions 和 label\n",
    "    with open(predict_res, 'r', encoding='utf-8') as f:\n",
    "        for line_num, line in enumerate(f, 1):\n",
    "            line = line.strip()\n",
    "            if not line:\n",
    "                continue  # 跳过空行\n",
    "            if ':' not in line:\n",
    "                warnings.warn(f\"Line {line_num} in prediction file does not contain ':'. Skipping.\")\n",
    "                continue\n",
    "            file, pred = line.split(':', 1)  # 只对第一个 : 分割\n",
    "            file = file.strip()\n",
    "            pred = pred.strip()\n",
    "            if not file or not pred:\n",
    "                warnings.warn(f\"Line {line_num} in prediction file has empty filename or prediction. Skipping.\")\n",
    "                continue\n",
    "            # 从 test_df 中获取对应的 label\n",
    "            label_series = test_df.loc[test_df[\"filename\"] == file, \"formula\"]\n",
    "            if label_series.empty:\n",
    "                warnings.warn(f\"Filename '{file}' not found in test_df. Skipping.\")\n",
    "                continue\n",
    "            label_formula = label_series.values[0]\n",
    "            pairs.append((pred, label_formula))\n",
    "    return pairs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "from Levenshtein import distance  # 需要安装 python-Levenshtein 库：pip install python-Levenshtein\n",
    "\n",
    "# 计算 ExpRate 和 ≤1、≤2 的准确率\n",
    "def compute_exprate(pairs, test_data_name=None):\n",
    "    print('------------------------------ Calculating the Exp Rate ------------------------------')\n",
    "    if not pairs:\n",
    "        raise ValueError(\"No valid prediction-label pairs found. Please check the prediction file and test_df.\")\n",
    "\n",
    "    length = len(pairs)\n",
    "    correct_count = 0\n",
    "    correct_within_1 = 0\n",
    "    correct_within_2 = 0\n",
    "    \n",
    "    # 遍历预测-标签对\n",
    "    for pred, gt in pairs:\n",
    "        # 将标签字符串分解成单字符\n",
    "        gt = ' '.join(c for st in gt.split() for c in list(st))\n",
    "        \n",
    "        # 计算完全匹配\n",
    "        if pred == gt:\n",
    "            correct_count += 1\n",
    "        else:\n",
    "            # 计算编辑距离\n",
    "            dist = distance(pred, gt)\n",
    "            if dist <= 1:\n",
    "                correct_within_1 += 1\n",
    "            if dist <= 2:\n",
    "                correct_within_2 += 1\n",
    "\n",
    "    # 计算各项指标\n",
    "    exprate = (correct_count / length) * 100\n",
    "    exprate_within_1 = ((correct_count + correct_within_1) / length) * 100\n",
    "    exprate_within_2 = ((correct_count + correct_within_2) / length) * 100\n",
    "\n",
    "    print(f\"ExpRate: {exprate:.2f}%\")\n",
    "    print(f\"ExpRate (≤1 error): {exprate_within_1:.2f}%\")\n",
    "    print(f\"ExpRate (≤2 errors): {exprate_within_2:.2f}%\")\n",
    "\n",
    "    return exprate, exprate_within_1, exprate_within_2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>formula</th>\n",
       "      <th>filename</th>\n",
       "      <th>image</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>\\frac { 5 } { 1 4 } = \\frac { 1 } { x }\\n</td>\n",
       "      <td>test_2.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>- 2 x ^ { 4 } - x ^ { 2 } + 5 = ( - x ^ { 2 } ...</td>\n",
       "      <td>test_3.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>S _ { \\Delta D B F } = \\frac { 3 } { 8 } S _ {...</td>\n",
       "      <td>test_5.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>: x _ { 1 } = - \\frac { 3 } { 5 }\\n</td>\n",
       "      <td>test_6.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7 5 \\div ( 7 5 + 2 5 ) = 7 5 \\div 1 0 0 = 7 5 ...</td>\n",
       "      <td>test_7.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>y = 3 . y = - 1\\n</td>\n",
       "      <td>test_8.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>x _ { 1 } = - 6 x _ { 2 } = 4\\n</td>\n",
       "      <td>test_9.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>2 3 . 5 + \\frac { 1 } { 2 } x = 1 3 + x\\n</td>\n",
       "      <td>test_10.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>\\frac { 9 } { 1 0 } \\div \\frac { 3 } { 5 } = \\...</td>\n",
       "      <td>test_13.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>= \\frac { ( 7 + 1 8 ) + ( 9 - 1 4 ) i } { 5 } ...</td>\n",
       "      <td>test_14.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                             formula     filename  \\\n",
       "0          \\frac { 5 } { 1 4 } = \\frac { 1 } { x }\\n   test_2.png   \n",
       "1  - 2 x ^ { 4 } - x ^ { 2 } + 5 = ( - x ^ { 2 } ...   test_3.png   \n",
       "2  S _ { \\Delta D B F } = \\frac { 3 } { 8 } S _ {...   test_5.png   \n",
       "3                : x _ { 1 } = - \\frac { 3 } { 5 }\\n   test_6.png   \n",
       "4  7 5 \\div ( 7 5 + 2 5 ) = 7 5 \\div 1 0 0 = 7 5 ...   test_7.png   \n",
       "5                                  y = 3 . y = - 1\\n   test_8.png   \n",
       "6                    x _ { 1 } = - 6 x _ { 2 } = 4\\n   test_9.png   \n",
       "7          2 3 . 5 + \\frac { 1 } { 2 } x = 1 3 + x\\n  test_10.png   \n",
       "8  \\frac { 9 } { 1 0 } \\div \\frac { 3 } { 5 } = \\...  test_13.png   \n",
       "9  = \\frac { ( 7 + 1 8 ) + ( 9 - 1 4 ) i } { 5 } ...  test_14.png   \n",
       "\n",
       "                                               image  \n",
       "0  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "1  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "2  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "3  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "4  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "5  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "6  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "7  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "8  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "9  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_dataset_df = read_datasets('dataset/test_parquets/hmer_test.parquet')\n",
    "test_dataset_df.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('\\\\ f r a c { 5 } { 1 4 } = \\\\ f r a c { 1 } { x } i c i r c f r a c { 1 } { x }',\n",
       "  '\\\\frac { 5 } { 1 4 } = \\\\frac { 1 } { x }\\n'),\n",
       " ('\\\\ t h e r e f o r e S _ { \\\\ D e l t a } P H _ { 2 } = ( - \\\\ s q r t { 1 + k ^ { 2 } } ) ( x + 1 ) + \\\\ s q r t { 5 } x ^ { 2 } + ( 1 - \\\\ s q r t { x } ) ^ { 2 } - ( x - \\\\ s q r t { 5 } ) x + 1',\n",
       "  '- 2 x ^ { 4 } - x ^ { 2 } + 5 = ( - x ^ { 2 } + 1 ) ( x + a ) + b = x ^ { 2 } + a x + 3 x - 3 a - b\\n'),\n",
       " ('S _ { \\\\ D e l t a A B F } = \\\\ f r a c { 3 } { 8 } S _ { \\\\ D e l t a A B C } = \\\\ f r a c { 3 } { 8 } \\\\ t i m e s \\\\ f r a c { 1 5 } { 2 } = \\\\ f r a c { 4 5 } { 1 6 }',\n",
       "  'S _ { \\\\Delta D B F } = \\\\frac { 3 } { 8 } S _ { \\\\Delta A B C } = \\\\frac { 3 } { 8 } \\\\times \\\\frac { 1 5 } { 2 } = \\\\frac { 4 5 } { 1 6 }\\n'),\n",
       " ('\\\\ t h e r e f o r e x _ { 1 } = - \\\\ f r a c { 3 } { 5 }',\n",
       "  ': x _ { 1 } = - \\\\frac { 3 } { 5 }\\n'),\n",
       " ('7 5 \\\\ d i v ( 7 5 + 2 5 ) = 7 5 \\\\ d i v 1 0 0 = 7 5 \\\\ %',\n",
       "  '7 5 \\\\div ( 7 5 + 2 5 ) = 7 5 \\\\div 1 0 0 = 7 5 \\\\%\\n'),\n",
       " ('y = 3 . y = - 1', 'y = 3 . y = - 1\\n'),\n",
       " ('x _ { 1 } = - 6 x _ { 2 } = 4 \\\\ c d o t s = 4 \\\\ t i m e s 2 5 = 4 \\\\ t i m e s 2 5 = 4 \\\\ t i m e s 2 5 = 4 \\\\ t i m e s 2 5 = 4 \\\\ t i m e s 2',\n",
       "  'x _ { 1 } = - 6 x _ { 2 } = 4\\n'),\n",
       " ('2 3 . 5 + \\\\ f r a c { 1 } { 2 } x = 1 3 + x',\n",
       "  '2 3 . 5 + \\\\frac { 1 } { 2 } x = 1 3 + x\\n'),\n",
       " ('\\\\ f r a c { 9 } { 1 0 0 } \\\\ d i v \\\\ f r a c { 3 } { 5 } = \\\\ f r a c { ( 9 ) } { ( 1 0 ) } \\\\ t i m e s \\\\ f r a c { ( 5 ) } { ( 6 ) } = ( \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { 9 } { 1 0 } \\\\div \\\\frac { 3 } { 5 } = \\\\frac { ( 9 ) } { ( 1 0 ) } \\\\textcircled { \\\\times } \\\\frac { ( 5 ) } { ( 3 ) } = ( \\\\frac { 2 } { 3 } )\\n'),\n",
       " ('= \\\\ f r a c { ( 7 + 1 8 ) + ( 9 - 4 ) } { 5 } = \\\\ f r a c { 2 5 - 5 } { 5 } = 5 - 1',\n",
       "  '= \\\\frac { ( 7 + 1 8 ) + ( 9 - 1 4 ) i } { 5 } = \\\\frac { 2 5 - 5 i } { 5 } = 5 - i\\n'),\n",
       " ('\\\\ a n g l e C P Q = \\\\ a n g l e B = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle C P Q = \\\\angle B = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('a - 4 b = - 1', 'a - 4 b = - 1\\n'),\n",
       " ('a \\\\ n e q \\\\ f r a c { 5 } { 2 }', 'a \\\\neq \\\\frac { 3 } { 2 }\\n'),\n",
       " ('a \\\\ v e r t b \\\\ v e r t', 'a \\\\bot b\\n'),\n",
       " ('\\\\ f r a c { O E } { A C } = \\\\ f r a c { O B } { A B }',\n",
       "  '\\\\frac { O E } { A C } = \\\\frac { O B } { A B }\\n'),\n",
       " ('\\\\ f r a c { 3 } { 2 } \\\\ t i m e s \\\\ f r a c { 7 } { 9 } - \\\\ f r a c { 3 } { 2 } \\\\ t i m e s \\\\ f r a c { 7 } { 9 } + \\\\ f r a c { 7 } { 9 } \\\\ t i m e s \\\\ f r a c { 4 } { 5 } \\\\ t i m e s \\\\ f r a c { 7 }',\n",
       "  '\\\\frac { \\\\frac { 8 } { 3 } } { 2 \\\\times \\\\frac { 4 } { 9 } } = - \\\\frac { - \\\\frac { 8 } { 3 } } { \\\\frac { 8 } { 9 } } = \\\\frac { 8 } { 3 } \\\\times \\\\frac { 9 } { 8 } = 3\\n'),\n",
       " ('y = \\\\ f r a c { 7 } { 3 }', 'y = \\\\frac { 7 } { 3 }\\n'),\n",
       " ('( - 1 + 0 ) ^ { 2 } = ( - 1 ) ^ { 2 } = 1',\n",
       "  '( - 1 + 0 ) ^ { 2 } = ( - 1 ) ^ { 2 } = 1\\n'),\n",
       " ('x _ { 3 } = \\\\ f r a c { - 5 + \\\\ s q r t { 4 1 } } { 4 }',\n",
       "  'x _ { 3 } = \\\\frac { - 5 + \\\\sqrt { 4 1 } } { 4 }\\n'),\n",
       " ('b ^ { 2 } \\\\ d i v a ^ { 2 } + c ^ { 2 } - 2 a c \\\\ c d o t \\\\ c o s b',\n",
       "  'b ^ { 2 } \\\\dot { = } a ^ { 2 } + c ^ { 2 } - 2 a c \\\\cdot \\\\cos B\\n'),\n",
       " ('1 0 \\\\ s q r t { 1 0 x - x ^ { 2 } } = 1 4',\n",
       "  '1 0 \\\\sqrt { 1 0 x - x ^ { 2 } } = 1 4\\n'),\n",
       " ('5 . 5 - 2 . 5 ) x + 4 8 \\\\ t i m e s 5 . 5 = 4 8 0',\n",
       "  '5 . 5 - 2 . 5 ) x + 4 8 \\\\times 5 . 5 = 4 8 0\\n'),\n",
       " ('4 . 5 8 5 > 4 . 6 5 > 4 . 5 6 > 4 . 5 0 6',\n",
       "  '4 . 5 8 5 > 4 . 6 5 > 4 . 5 6 > 4 . 5 0 6\\n'),\n",
       " ('\\\\ a n g l e D D E = \\\\ a n g l e D B E = \\\\ a n g l e A E B = 7 5 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle D O E = \\\\angle D B E = \\\\angle A E B = 7 5 ^ { \\\\circ }\\n'),\n",
       " ('x = \\\\ f r a c { 1 7 } { 5 }', 'x = \\\\frac { 1 7 } { 5 }\\n'),\n",
       " ('9 \\\\ t i m e s \\\\ f r a c { x } { 4 } = \\\\ f r a c { 1 } { 2 } = 1 2 . ( c m )',\n",
       "  '\\\\frac { 1 } { 3 } \\\\times \\\\frac { 1 } { 4 } = \\\\frac { 1 } { 1 2 } = 1 2 ( c m )\\n'),\n",
       " ('\\\\ f r a c { 2 x - 6 } { 2 } = 6', '\\\\frac { 2 x - 6 } { 2 } = 6\\n'),\n",
       " ('y = \\\\ f r a c { x ( x - 2 ) } { ( x - 1 ) ^ { 2 } }',\n",
       "  'y = \\\\frac { x ( x - 2 ) } { ( x - 1 ) ^ { 2 } }\\n'),\n",
       " ('x _ { 1 } = \\\\ f r a c { 1 } { 9 }', 'x _ { 1 } = \\\\frac { 1 } { 9 }\\n'),\n",
       " ('y _ { 1 } = 0 . y _ { 2 } = \\\\ f r a c { 8 . } { 3 } = \\\\ f r a c { 8 . } { 3 }',\n",
       "  '\"y _ { 1 } = 0 , y _ { 2 } = \\\\frac { 8 } { 3 } \\\\cdot\"\\n'),\n",
       " ('( 2 ) \\\\ v e r t A B \\\\ v e r t = \\\\ v e r t - 1 - 5 \\\\ v e r t = 6',\n",
       "  '( 2 ) \\\\vert A B \\\\vert = \\\\vert - 1 - 5 \\\\vert = 6\\n'),\n",
       " ('\\\\ f r a c { 4 x } { 3 } + \\\\ f r a c { 3 } { 2 } > - \\\\ f r a c { x } { 6 }',\n",
       "  '\\\\frac { 4 x } { 3 } + \\\\frac { 3 } { 2 } > - \\\\frac { x } { 6 }\\n'),\n",
       " ('\\\\ f r a c { ( 1 2 + 4 ) } { 8 } + \\\\ l e f t ( 3 0 2 1 > 2 0 1 \\\\ r i g h t )',\n",
       "  '( \\\\frac { ( 2 \\\\times 4 ) } { 8 } + 1 3 = 2 1 > 2 0 )\\n'),\n",
       " ('n = \\\\ f r a c { 1 } { 2 } \\\\ d i v \\\\ f r a c { 3 } { 2 }',\n",
       "  'n = \\\\frac { 1 } { 2 } \\\\div \\\\frac { 3 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } \\\\ t i m e s 3 \\\\ t i m e s 6 + \\\\ f r a c { 1 } { 2 } \\\\ t i m e s 5 \\\\ t i m e s 6 + 5 \\\\ t i m e s 2 \\\\ t i m e s 1 + 5 \\\\ t i m e s 2 \\\\ t i m e s 6 = 9 4',\n",
       "  '\\\\frac { 1 } { 2 } \\\\times 3 \\\\times 6 + \\\\frac { 1 } { 2 } \\\\times ( 6 + 8 ) \\\\times 1 1 + \\\\frac { 1 } { 2 } \\\\times 2 \\\\times 6 = 9 4\\n'),\n",
       " ('x = 2 ( \\\\ f r a c { 5 x - 1 7 } { 4 } )',\n",
       "  'x = 2 ( \\\\frac { 5 x - 1 7 } { 4 } )\\n'),\n",
       " ('P ( x = 0 ) = \\\\ f r a c { C _ { 4 } ^ { 3 } } { C _ { 7 } ^ { 3 } } = \\\\ f r a c { 4 } { 3 5 }',\n",
       "  'P ( x = 0 ) : \\\\frac { C _ { 4 } ^ { 3 } } { C _ { 7 } ^ { 3 } } = \\\\frac { 4 } { 3 5 }\\n'),\n",
       " ('B Q = A B \\\\ t a n A = 2 0 \\\\ t a n 6 0 ^ { \\\\ c i r c } = 2 0 \\\\ s q r t { 2 }',\n",
       "  'B Q = A B \\\\tan A = 2 0 \\\\tan 6 0 ^ { \\\\circ } = 2 0 \\\\sqrt { 2 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } \\\\ t i m e s \\\\ f r a c { 1 } { 3 } = \\\\ f r a c { 1 } { 9 }',\n",
       "  '\\\\frac { 1 } { 3 } \\\\times \\\\frac { 1 } { 3 } = \\\\frac { 1 } { 9 }\\n'),\n",
       " ('1 9 . 2 6 \\\\ t i m e s \\\\ f r a c { 3 } { 4 } = 1 4 7 . 1 9 5 ( m l )',\n",
       "  '1 9 6 . 2 6 \\\\times \\\\frac { 3 } { 4 } = 1 4 7 . 1 9 5 ( m l )\\n'),\n",
       " ('\\\\ s i n \\\\ a n g l e 3 = \\\\ f r a c { A D } { A C } = \\\\ f r a c { A O } { O C } = \\\\ f r a c { \\\\ s q r t { 5 } } { 5 }',\n",
       "  '\\\\sin \\\\angle 3 = \\\\frac { A D } { A C } = \\\\frac { A O } { O C } = \\\\frac { 5 } { 5 \\\\sqrt { 5 } }\\n'),\n",
       " ('5 \\\\ f r a c { 1 } { 2 } - 3 x = a + \\\\ f r a c { 1 } { 4 } \\\\ g e q',\n",
       "  '5 \\\\frac { 1 } { 2 } - 3 x = a + \\\\frac { 1 } { 4 } x\\n'),\n",
       " ('\\\\ v e r t \\\\ o v e r r i g h t a r r o w { b } \\\\ v e r t = \\\\ s q r t { \\\\ o v e r r i g h t a r r o w { b } } \\\\ v e r t ^ { 2 } = \\\\ s q r t { \\\\ o v e r r i g h t a r r o w { b } } ^ { 2 } \\\\ c d o t \\\\ o',\n",
       "  '\\\\vert \\\\overrightarrow { b } \\\\vert = \\\\sqrt { \\\\vert \\\\overrightarrow { b } \\\\vert ^ { 2 } } = \\\\sqrt { \\\\overrightarrow { b } \\\\cdot \\\\overrightarrow { b } } = \\\\sqrt { x _ { 2 } ^ { 2 } + y _ { 2 } ^ { 2 } }\\n'),\n",
       " ('\" 1 + a ^ { 4 } b ^ { 2 } - a ^ { 2 } - b ^ { 2 } > 0 \" \" \" \" \" \" \" \" \" \" \" \" \" , \" , \" , \" , \" , \" , 1 , 1 , 3 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ,',\n",
       "  '1 + a ^ { 2 } b ^ { 2 } - a ^ { 2 } - b ^ { 2 } > 0\\n'),\n",
       " ('x = \\\\ f r a c { 2 2 5 } { 2 } \\\\ t i m e s \\\\ f r a c { 1 0 } { 1 2 }',\n",
       "  'x = \\\\frac { 2 2 5 } { 2 } \\\\times \\\\frac { 1 0 } { 1 2 }\\n'),\n",
       " ('S _ { \\\\ D e l t a A O N } = \\\\ f r a c { 1 } { 2 } \\\\ t i m e s 4 \\\\ t i m e s 6 = 1 2',\n",
       "  'S _ { \\\\Delta A O V } = \\\\frac { 1 } { 2 } \\\\times 4 \\\\times 6 = 1 2\\n'),\n",
       " ('A H = 3 k B H = 4 k', 'A H = 3 k B H = 4 k\\n'),\n",
       " ('\\\\ f r a c { x ^ { 2 } } { 4 } + \\\\ f r a c { y ^ { 2 } } { 2 } = 1 ( 1 x + 2 )',\n",
       "  '\\\\frac { x ^ { 2 } } { 4 } + \\\\frac { y ^ { 2 } } { 2 } = 1 ( \\\\vert x \\\\vert \\\\neq 2 )\\n'),\n",
       " ('\\\\ f r a c { 2 x } { 3 } = \\\\ f r a c { 4 x - 8 } { 9 } - 5',\n",
       "  '\\\\frac { 2 x } { 3 } = \\\\frac { 4 x - 8 } { 9 } - 5\\n'),\n",
       " ('x = - \\\\ f r a c { 6 4 } { 3 }', 'x = - \\\\frac { 6 4 } { 3 }\\n'),\n",
       " ('( x + 1 ) ^ { 2 } - 4 = ( x + 1 ) ( x - 1 )',\n",
       "  '( x + 1 ) ^ { 2 } - 4 = ( x + 1 ) ( x - 1 )\\n'),\n",
       " ('x ^ { 3 } + 4 x ^ { 2 } + 4 x ^ { 2 } - 2 x ^ { 2 } - 2 x ^ { 2 } - 4 x - 0',\n",
       "  'x ^ { 2 } + a x ^ { 2 } + a ^ { 2 } x - a x ^ { 2 } - a ^ { 2 } x - a\\n'),\n",
       " ('3 6 0 0 0 c m ^ { 3 } = 3 6 0 d m ^ { 3 }',\n",
       "  '3 6 0 0 0 c m ^ { 3 } = 3 6 d m ^ { 3 }\\n'),\n",
       " ('x - y _ { 0 } = - \\\\ f r a c { 1 1 } { 1 2 }',\n",
       "  'x - y = - \\\\frac { 1 1 } { 1 2 }\\n'),\n",
       " ('0 . 5 < 0 . 5 0 1 < 0 . 5 1 < 0 . 5 1 1 1 1 1 1 1 1 1 5 1 1 5 1 1 1 1 1 1 ) 1 1 \\\\ d o t { 5 0 1 1 } \\\\ d o t { 5 0 1 1 } \\\\ d o t { 5 0 1 1 } \\\\ d o',\n",
       "  '0 . 5 < 0 . 5 0 1 < 0 . 5 1 < 0 . 5 1 1\\n'),\n",
       " ('\\\\ f r a c { 8 } { 8 } - \\\\ f r a c { 7 } { 8 } = \\\\ f r a c { 1 } { 8 }',\n",
       "  '\\\\frac { 8 } { 8 } - \\\\frac { 7 } { 8 } = \\\\frac { 1 } { 8 }\\n'),\n",
       " ('\\\\ f r a c { 2 8 } { 4 2 } = \\\\ f r a c { 1 4 \\\\ t i m e s 2 } { 1 4 \\\\ t i m e s 3 } = \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { 2 8 } { 4 2 } = \\\\frac { 1 4 \\\\times 2 } { 1 4 \\\\times 3 } = \\\\frac { 2 } { 3 }\\n'),\n",
       " ('\" 9 x - 5 , 2 = 2 1 . 8 \" \" \" \" \" \" \" \" \" \" \" \" \" \" , - 2 , 2 1 , 8 , 8 , \" \" \" \" \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , 3 , , 1 , 6 , 8 , 6 , 8 ,',\n",
       "  '\"9 x - 5 , 2 = 2 1 . 8\"\\n'),\n",
       " ('W = ( 1 2 0 - a ) \\\\ t i m e s 1 0 0 + 1 0 0 ( 3 0 0 0 - m ) 0 . 1 m ) 0 . 1 m ) 0 . 1 m ) 0 . 1 m ) 0 . 1 m ( 0 . 1 m ) 0 . 1 m ) 0 . 1 m ) 0 . 1 m > 0 . 0 . 0 5 0 )',\n",
       "  'W = ( 1 2 0 - a ) m + 1 0 0 ( 3 0 0 - m )\\n'),\n",
       " ('x _ { 1 } = \\\\ s q r t { 2 } x _ { 2 } = - \\\\ s q r t { 2 }',\n",
       "  'x _ { 1 } = \\\\sqrt { 2 } x _ { 2 } = - \\\\sqrt { 2 }\\n'),\n",
       " ('\\\\ t h e r e f o r e F 1 8 0', '\\\\therefore F 1 8 0\\n'),\n",
       " ('1 0 0 = 2 0 0 k + b', '1 0 0 = 2 0 0 k + b\\n'),\n",
       " ('x _ { 3 } = \\\\ f r a c { 1 } { 4 }', 'x _ { 3 } = \\\\frac { 1 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 5 } \\\\ t i m e s 1 0 0 0 = 2 0 0 ( g ) )',\n",
       "  '\\\\frac { 1 } { 5 } \\\\times 1 0 0 0 = 2 0 0 ( g )\\n'),\n",
       " ('\\\\ a n g l e A E D = \\\\ a n g l e D A E = \\\\ f r a c { 1 } { 2 } ( 1 8 0 ^ { \\\\ c i r c } -',\n",
       "  '\\\\angle A E D = \\\\angle D A E = \\\\frac { 1 } { 2 } ( 1 8 0 ^ { \\\\circ } -\\n'),\n",
       " ('\\\\ f r a c { x ^ { 2 } + 1 } { x + 1 } - \\\\ f r a c { 2 } { x + 1 }',\n",
       "  '\\\\frac { x ^ { 2 } + 1 } { x + 1 } - \\\\frac { 2 } { x + 1 }\\n'),\n",
       " ('3 0 0 - 3 9 7 ( c m )', '3 0 - 3 = 2 7 ( c m )\\n'),\n",
       " ('3 y - 2 = 9 - 6 \\\\ s q r t { y + 3 } + y + 3',\n",
       "  '3 y - 2 = 9 - 6 \\\\sqrt { y + 3 } + y + 3\\n'),\n",
       " ('\" 3 , 1 4 \\\\ t i m e s 5 \\\\ t i m e s 8 = 1 2 5 . 6 ( m ^ { 2 } ) \" ) \"',\n",
       "  '3 . 1 4 \\\\times 5 \\\\times 8 = 1 2 5 . 6 ( m ^ { 2 } )\\n'),\n",
       " ('V = - \\\\ f r a c { 2 } { 2 }', 'v = \\\\frac { 2 } { 3 }\\n'),\n",
       " ('F e + H _ { 2 } S O 4 = F e S O 4 + H _ { 2 } \\\\ u p a r r o w',\n",
       "  'F e + H _ { 2 } S O 4 = F e _ { 5 } S O _ { 4 } + H 2 \\\\uparrow\\n'),\n",
       " ('\\\\ a n g l e 2 + \\\\ a n g l e 3 = \\\\ a n g l e E B F = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle 2 + \\\\angle 3 = \\\\angle E B F = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('5 \\\\ t i m e s 3 \\\\ t i m e s 4 = 6 0 d m ^ { 3 }',\n",
       "  '5 \\\\times 3 \\\\times 4 = 6 0 ( d m ) ^ { 3 }\\n'),\n",
       " ('a m + b n > a n + b m > 0', 'a m + b n > a n + b m > 0\\n'),\n",
       " ('2 x = 8 3 . 2', '2 x = 8\\n'),\n",
       " ('x = \\\\ f r a c { 5 } { 8 } i g h t a r r o w \\\\ f r a c { 5 } { 8 } i g h t a r r o w \\\\ f r a c { 5 } { 8 }',\n",
       "  'x = \\\\frac { 5 } { 8 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 4 } + \\\\ f r a c { 3 } { 2 } \\\\ d i v \\\\ f r a c { 1 } { 6 }',\n",
       "  '\\\\frac { 1 } { 4 } \\\\div \\\\frac { 3 } { 2 } = \\\\frac { 1 } { 6 }\\n'),\n",
       " ('3 x _ { ( g ) } + y _ { ( g ) } \\\\ r i g h t l e f t h a r p o o n s 2 z ( g )',\n",
       "  '3 x _ { ( g ) } + y _ { ( g ) } \\\\rightleftharpoons 2 z ( g )\\n'),\n",
       " ('x + 1 = \\\\ p m \\\\ f r a c { 4 } { 5 }',\n",
       "  'x + 1 = \\\\pm \\\\frac { 4 } { 5 }\\n'),\n",
       " ('\\\\ f r a c { 2 x } { x - 2 } = \\\\ f r a c { 2 - x } { 2 - x } - \\\\ f r a c { 1 } { 2 - x }',\n",
       "  '\\\\frac { 2 x } { x - 2 } = \\\\frac { 2 - x } { 2 - x } - \\\\frac { 1 } { 2 - x }\\n'),\n",
       " ('4 . 5 0 6 < 4 . 5 6 < 4 . 5 6 5 < 5 . 6 5',\n",
       "  '4 . 5 0 6 < 4 . 5 6 < 4 . 5 6 < 4 . 5 6 5 < 5 . 6 5\\n'),\n",
       " ('= 1 0 - \\\\ f r a c { 1 } { 2 }', '- 1 0 - \\\\frac { 1 } { 5 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x = \\\\ f r a c { 2 } { 3 } \\\\ t i m e s 6',\n",
       "  '\\\\frac { 1 } { 2 } x = \\\\frac { 2 } { 3 } \\\\times 6\\n'),\n",
       " ('1 x + 7 5 0 - 1 5 5 a \\\\ g e q', '1 0 a + 1 5 0 - 1 5 a \\\\geq\\n'),\n",
       " ('\\\\ t e x t c i r c l e d { 5 } \\\\ t e x t c i r c l e d { 2 } ^ { 2 } = x \\\\ t e x t c i r c l e d { 7 } + x ^ { 2 }',\n",
       "  'B D ^ { 2 } - A B ^ { 2 } - A D ^ { 2 }\\n'),\n",
       " ('\" ( 1 ) a _ { 2 } ^ { x } = a _ { 1 } + ( 2 - 1 ) d \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" , 1 , 2 , 1 , 0 , 1 , 2 , 1 , 0 , 1 , 0 , 1 \" \" \" \" \" , 1 , 0 , 1 , 0',\n",
       "  '( 1 ) a ^ { x } _ { 2 } = a _ { 1 } + ( 2 - 1 ) d\\n'),\n",
       " ('2 ^ { 2 } + b ^ { 2 } = a ^ { 2 } + b ^ { 2 }',\n",
       "  '2 ^ { 2 } + b ^ { 2 } = a ^ { 2 } + b ^ { 2 }\\n'),\n",
       " ('2 N a O H + C u S O _ { 4 } = C u ( O H ) _ { 2 } \\\\ d o w n a r r o w + N _ { 2 } S O _ { 4 }',\n",
       "  '2 N a O H + C u S O _ { 4 } = C u ( O H ) _ { 2 } \\\\downarrow + N a _ { 2 } S O _ { 4 }\\n'),\n",
       " ('\\\\ f r a c { x } { 3 } + \\\\ f r a c { a + x } { 2 } = 1',\n",
       "  '\\\\frac { x } { 3 } + \\\\frac { a + x } { 2 } = 1\\n'),\n",
       " ('a = - 7 - ( 2 . 5 )', '\"a = - 7 - ( 2 , s )\"\\n'),\n",
       " ('\\\\ a n g l e B O C = 1 2 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B O C = 1 2 0 ^ { \\\\circ }\\n'),\n",
       " ('a _ { 2 } = 3 b _ { 2 } = 6', 'a _ { 2 } = 3 b _ { 2 } = 6\\n'),\n",
       " ('1 5 0 0 \\\\ t i m e s \\\\ f r a c { 1 } { 5 } = \\\\ f r a c { 1 5 0 0 \\\\ t i m e s 1 } { 5 } = \\\\ f r a c { 1 5 0 0 } { 5 } = \\\\ f r a c { 3 0 0 } { 5 } =',\n",
       "  '1 5 0 0 \\\\times \\\\frac { 1 } { 5 } = \\\\frac { 1 5 0 0 \\\\times 1 } { 5 } = \\\\frac { 1 5 0 0 } { 5 } = \\\\frac { 3 0 0 } { 5 } =\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } + \\\\ f r a c { 1 } { 4 } = \\\\ f r a c { 2 } { a + a }',\n",
       "  '\\\\frac { 1 } { 2 c } + \\\\frac { 1 } { 2 a } \\\\geq \\\\frac { 2 } { c + a }\\n'),\n",
       " ('c = R + R \\\\ s i n 3 0 ^ { \\\\ c i r c }',\n",
       "  'l = R + R \\\\sin 3 0 ^ { \\\\circ }\\n'),\n",
       " ('x - 5 - 2 x + 7 - 4 \\\\ s q r t { 2 x } ) - 4 = 0',\n",
       "  'x - 5 - 2 x + 7 - 4 \\\\sqrt { 2 x - 7 } - 4 = 0\\n'),\n",
       " ('a n c < 0', 'a n c = 0\\n'),\n",
       " ('\" ( x - \\\\ f r a c { \\\\ p i } { 2 } ( - \\\\ i n f t y , - \\\\ f r a c { 1 } { 2 } ) \\\\ c u p ( \\\\ f r a c { 1 } { 2 } , + \\\\ i n f t y ) \\\\ u p a r r o w \" \" \" \" \" \" \" \" \" \" \" \" \" \"',\n",
       "  '\"k > 0 E ( - \\\\infty , - \\\\frac { 1 } { k } ) \\\\downarrow ( - \\\\frac { 1 } { k } , - \\\\infty ) \\\\uparrow\"\\n'),\n",
       " ('\\\\ a l p h a = \\\\ f r a c { B E } { A B } = \\\\ f r a c { 1 } { 4 }',\n",
       "  'x = \\\\frac { B E } { A B } = \\\\frac { 1 } { 4 }\\n'),\n",
       " ('2 C a ( O H ) _ { 2 } - 2 C O _ { 2 } = C a C O _ { 3 } \\\\ d o w n a r r o w + 2 C O _ { 2 } \\\\ u p a r r o w 2 C O _ { 2 } 2 C O _ { 3 }',\n",
       "  '2 c m : 6 0 0 m = 2 c m : 6 0 0 0 0 c m = 1 : 3 0 0 0 0\\n'),\n",
       " ('\\\\ f r a c { m + 2 } { m ^ { 2 } - 1 }',\n",
       "  '\\\\frac { m ^ { 2 } + 4 } { m ^ { 2 } - 1 }\\n'),\n",
       " ('+ a _ { 6 } + ( a _ { 3 } + a _ { 8 } ) = 5 + 5 + 5 + 5 + 5 = 2 5 = 2 5',\n",
       "  '+ a _ { 6 } + ( a _ { 3 } + a _ { 8 } ) = 5 + 5 + 5 + 5 + 5 = 2 5\\n'),\n",
       " ('\" x _ { 1 } = v + 2 , x _ { 2 } = 2 ( n + 1 ) \" ) \" 1 \" 1 \" ) \" ) \" \" \" \" 1 , \" 1 , \" 1 , 2 , 1 1 , 1 1 , 3 , 1 1 , 1 \" ) \" \" , \" ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1',\n",
       "  '\"x _ { 1 } = n + 2 , x _ { 2 } = 2 ( n + 1 )\"\\n'),\n",
       " ('\" x = m + 1 , y = - m - 3 \" \" \" \" \" \" \" \" , \" , m = - 3 , \" \" \" \" - 3 , \" , \" - 1 , 1 , 1 , - 1 , - 1 , - 1 , - 1 . \" \" \" \" \" , \" , 1 , 1 , 1 , 1 , \\\\ t h e r e f',\n",
       "  '\"- x = m + 1 , y = - m - 3\"\\n'),\n",
       " ('\\\\ f r a c { 4 \\\\ t i m e s 1 2 } { 4 8 } + \\\\ f r a c { 1 8 } { 4 } \\\\ t i m e s 3 = 6 0',\n",
       "  '\\\\frac { 4 \\\\times 1 2 } { 4 8 } + \\\\frac { ( 8 - 4 ) } { 4 } \\\\times 3 = 6 0\\n'),\n",
       " ('\\\\ f r a c { 3 } { 2 } = \\\\ f r a c { 2 } { x } i c i r c h c i r c h } { 2 1 }',\n",
       "  '\\\\frac { 3 } { 2 } = \\\\frac { 2 } { x }\\n'),\n",
       " ('x = \\\\ f r a c { 2 4 } { 5 }', 'x = \\\\frac { 2 4 } { 5 }\\n'),\n",
       " ('x = - \\\\ f r a c { 2 4 } { 5 }', 'x = \\\\frac { 2 4 } { 5 }\\n'),\n",
       " ('2 L i O H + C O _ { 2 } = L i _ { 2 } C O _ { 3 } + H _ { 2 } O',\n",
       "  '2 L i O H + C O _ { 2 } = L i _ { 2 } C O _ { 3 } + H _ { 2 } O\\n'),\n",
       " ('= - D P = \\\\ f r a c { - 3 + \\\\ s q r t { 6 7 } } { 2 }',\n",
       "  '\\\\therefore D P = \\\\frac { - 3 + \\\\sqrt { 6 9 } } { 2 }\\n'),\n",
       " ('3 ^ { \\\\ c i r c } \\\\ c d o t \\\\ a l p h a - \\\\ a l p h a ^ { \\\\ c i r c } \\\\ c d o t \\\\ a l p h a = \\\\ a l p h a',\n",
       "  '\\\\sin \\\\alpha x = \\\\alpha\\n'),\n",
       " ('3 x - \\\\ f r a c { 3 } { 4 } = \\\\ f r a c { 1 } { 4 }',\n",
       "  '3 x - \\\\frac { 3 } { 4 } = \\\\frac { 1 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 2 7 } { 2 } - 6 + 3 \\\\ s q r t { 3 } + \\\\ f r a c { 9 } { 2 }',\n",
       "  '\\\\frac { 2 7 } { 2 } - 6 + 3 \\\\sqrt { 3 } + \\\\frac { 9 } { 2 }\\n'),\n",
       " ('y ( 2 y + x ) = 0 \\\\ c d o t s y = y',\n",
       "  'y ( 2 y + x ) = 0 \\\\cdot \\\\therefore y\\n'),\n",
       " ('1 0 0 y \\\\ t i m e s 2 \\\\ t i m e s 2 = 1 0 0 x \\\\ t i m e s 1 \\\\ t i m e s 1 1 1 1 1',\n",
       "  '1 0 0 y \\\\times 2 \\\\times 2 = 1 0 0 x \\\\times 1 \\\\times 1\\n'),\n",
       " ('y = \\\\ f r a c { 3 } { 2 } = \\\\ f r a c { 3 } { 2 } = \\\\ f r a c { 3 } { 2 } g h = \\\\ f r a c { 3 } { 2 }',\n",
       "  'y = \\\\frac { 3 } { 2 }\\n'),\n",
       " ('A \\\\ t i m e s 0 . k \\\\ g e q \\\\ s q r t { 2 }',\n",
       "  'k \\\\geq 0 . k \\\\geq \\\\sqrt { 2 }\\n'),\n",
       " ('A D \\\\ b o t B C', 'A D \\\\bot B C\\n'),\n",
       " ('2 t - \\\\ f r a c { 2 5 } { 2 4 } )', '2 t - \\\\frac { 2 5 } { 2 4 } )\\n'),\n",
       " (': I _ { 1 } : I _ { 2 } = 3 : 1', 'I _ { 1 } : I _ { 2 } = 3 : 1\\n'),\n",
       " ('t ( 7 = \\\\ f r a c { 1 } { 7 }', '\\\\cdot E G = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('F e + H _ { 2 } S O _ { 4 } = F e S O _ { 4 } + H _ { 2 } \\\\ u p a r r o w',\n",
       "  'F e + H _ { 2 } S O _ { 4 } = F e S O _ { 4 } + H _ { 2 } \\\\uparrow\\n'),\n",
       " ('x = \\\\ f r a c { 3 } { 4 }', 'x = \\\\frac { 3 } { 4 }\\n'),\n",
       " ('( y - 2 ) ( y + 1 ) = 0', '( y - 2 ) ( y + 1 ) = 0\\n'),\n",
       " ('4 - 2 b = \\\\ s q r t { b ^ { 2 } + r ^ { 2 } } + \\\\ t e x t c i r c l e d { 1 }',\n",
       "  '4 - 2 h = \\\\sqrt { h ^ { 2 } + r ^ { 2 } } . \\\\textcircled { 1 }\\n'),\n",
       " ('\\\\ s q r t { x + 4 5 } \\\\ r i g h t a r r o w x + 1 = 0',\n",
       "  '\\\\sqrt { x + 4 5 } - 2 x + 1 = 0\\n'),\n",
       " ('\\\\ f r a c { 1 } { 4 } \\\\ t i m e s + \\\\ f r a c { 3 } { 1 0 } y = 1',\n",
       "  '\\\\frac { 1 } { 4 } x + \\\\frac { 3 } { 1 0 } y = 1\\n'),\n",
       " ('x = \\\\ f r a c { 2 } { 8 }', 'x = \\\\frac { 3 } { 8 }\\n'),\n",
       " ('\\\\ t e x t c i r c l e d { < } 9 9 9 9 < \\\\ t e x t c i r c l e d { < } 9 8 9 9 < 9 9 9 9 5',\n",
       "  '( 8 9 9 9 ) < ( 9 0 0 1 ) < ( 9 8 9 9 ) < ( 9 9 9 9 )\\n'),\n",
       " ('\\\\ a n g l e P F E = 1 8 0 ^ { \\\\ c i r c } - \\\\ a n g l e 1 - 9 0 ^ { \\\\ c i r c } ^ { \\\\ c i r c } ^ { \\\\ c i r c } ^ { \\\\ c i r c } ^ { \\\\ c i r c } ^ { \\\\ c i r c } ^ { \\\\ c i r c }',\n",
       "  '\\\\angle P F E = 1 8 0 ^ { \\\\circ } - \\\\angle 1 - 9 0 ^ { \\\\circ }\\n'),\n",
       " ('S _ { \\\\ D e l t a } A F C = \\\\ f r a c { 1 } { 2 } A F \\\\ c d o t B C = 1 0',\n",
       "  'S _ { \\\\Delta } A F C = \\\\frac { 1 } { 2 } A F \\\\cdot B C = 1 0\\n'),\n",
       " ('6 0 \\\\ t i m e s \\\\ f r a c { 3 } { 4 } = \\\\ f r a c { 6 0 \\\\ t i m e s 3 } { 4 } = \\\\ f r a c { 1 8 0 } { 4 } = 4 5',\n",
       "  '6 0 \\\\times \\\\frac { 3 } { 4 } = \\\\frac { 6 0 \\\\times 3 } { 4 } = \\\\frac { 1 8 0 } { 4 } = 4 5\\n'),\n",
       " ('1 4 0 ^ { \\\\ c i r c } - 5 m + 5 m - 5 0 ^ { \\\\ c i r c } = 9 0',\n",
       "  '1 4 0 ^ { \\\\circ } - 5 m + 5 m - 5 0 ^ { \\\\circ } = 9 0\\n'),\n",
       " ('x > - \\\\ f r a c { 5 } { 2 }', 'x > - \\\\frac { 5 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { x + 2 } { 3 } = 1', '\\\\frac { x + 2 } { 3 } = 1\\n'),\n",
       " ('F H = \\\\ s q r t { x ^ { 2 } - x ^ { 2 } + 5 x ^ { 2 } - \\\\ f r a c { 2 5 } { 4 } } =',\n",
       "  'E H = \\\\sqrt { x ^ { 2 } - x ^ { 2 } + 5 x - \\\\frac { 2 5 } { 4 } } =\\n'),\n",
       " ('I = \\\\ f r a c { Q } { t } = \\\\ f r a c { 2 } { 4 } = 0 . 5',\n",
       "  'I = \\\\frac { Q } { t } = \\\\frac { 2 } { 4 } = 0 . 5\\n'),\n",
       " ('\\\\ a n g l e A O F = \\\\ a n g l e 3 = 4 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A O F = \\\\angle 3 = 4 0 ^ { \\\\circ }\\n'),\n",
       " ('. ( 4 2 \\\\ d i v 2 ) + 2 = 2 5 ( m )', '( 4 2 \\\\div 2 ) + 2 = 2 3 ( m )\\n'),\n",
       " ('2 N a O H + C O _ { 2 } = 2 N a _ { 2 } C O _ { 3 } + H _ { 2 } O',\n",
       "  '2 N a O H + C O _ { 2 } = O N a _ { 2 } C O _ { 3 } + H _ { 2 } O\\n'),\n",
       " ('n ( n + 2 ) + 1 = ( n + 1 ) ^ { \\\\ c d o t }',\n",
       "  'n ( n + 2 ) + 1 = ( n + 1 ) ^ { 2 }\\n'),\n",
       " ('\\\\ a n g l e P A C = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e B A C = 4 8 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle P A C = \\\\frac { 1 } { 2 } \\\\angle B A C = 4 8\\n'),\n",
       " ('\\\\ a n g l e B O D + \\\\ a n g l e C O D = 6 0 ^ { \\\\ c i r c } + 3 0 ^ { \\\\ c i r c } = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B O D + \\\\angle C O D = 6 0 ^ { \\\\circ } + 3 0 ^ { \\\\circ } = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('x = \\\\ f r a c { 1 0 8 } { 1 3 }', 'x = \\\\frac { 1 0 8 } { 1 3 }\\n'),\n",
       " ('= 1 - 1 . 5', 'x = 3 . 5\\n'),\n",
       " ('3 0 h = 2 7 \\\\ D e l t a h - 8 \\\\ D e l t a h',\n",
       "  '3 0 h = 2 7 \\\\Delta h - 8 \\\\Delta h\\n'),\n",
       " ('3 a _ { 1 } + 4 d = 1 0', '3 a _ { 1 } + 9 d = 1 0\\n'),\n",
       " ('1 \\\\ s i m 5 : B B D C B B', '1 \\\\sim 5 : B B D C B\\n'),\n",
       " ('\\\\ a n g l e B = \\\\ a n g l e C ^ { \\\\ p r i m e } m g h t r i g h t a r r o w 0 . 0 5 m / h 0 . 0 5 5 m < 0 . 0 5 5 \\\\ t e x t c i r c l e d { 2',\n",
       "  '\\\\angle B = \\\\angle C ^ { \\\\prime }\\n'),\n",
       " ('d = \\\\ s q r t { a ^ { 2 } - 1 - \\\\ f r a c { a ^ { 2 } - 1 } { 4 } } = \\\\ f r a c { \\\\ v e r t a ^ { 2 } - 1 \\\\ v e r t } { \\\\ s q r t { a + 1 } } = \\\\ s q r t { a + 1 }',\n",
       "  'd = \\\\sqrt { a ^ { 2 } - 1 - \\\\frac { a ^ { 2 } - 1 } { 4 } } = \\\\frac { \\\\vert a ^ { 2 } - 1 \\\\vert } { \\\\sqrt { a - 1 } }\\n'),\n",
       " ('\" B D = D M , D H \\\\ b o t B M \"', '\"B D = D M , D H \\\\bot B M\"\\n'),\n",
       " ('( y + 1 ) ^ { 2 } + ( y + 1 ) y + y ^ { 2 } = 1',\n",
       "  '( y + 1 ) ^ { 2 } + ( y + 1 ) y + y ^ { 2 } = 1\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } + 1 1 y = 1 0 y - 7',\n",
       "  '\\\\frac { 1 } { 3 } + 1 1 y = 1 0 y - 7\\n'),\n",
       " ('\\\\ s q r t { 1 3 ^ { 3 } - 5 ^ { 2 } } - \\\\ s q r t { ( \\\\ s q r t { 2 } - 1 ) ^ { 2 } }',\n",
       "  '\\\\sqrt { 1 3 ^ { 3 } - 5 ^ { 2 } } - \\\\sqrt { ( \\\\sqrt { 2 } - 1 ) ^ { 2 } }\\n'),\n",
       " ('\\\\ f r a c { 6 + a - 5 } { 5 } = \\\\ f r a c { 1 0 } { 5 }',\n",
       "  '\\\\frac { 6 + a - 5 } { 5 } = \\\\frac { 1 0 } { 6 }\\n'),\n",
       " ('x = \\\\ f r a c { 6 4 0 } { 3 }', 'x = \\\\frac { 6 4 0 } { 3 }\\n'),\n",
       " ('\\\\ f r a c { 1 7 } { 2 0 } > \\\\ f r a c { 1 6 } { 2 0 }',\n",
       "  '\\\\frac { 1 7 } { 2 0 } > \\\\frac { 1 6 } { 2 0 }\\n'),\n",
       " ('1 = s h = 1 . 0 \\\\ t i m e s 1 0 ^ { - 3 } m ^ { 2 } \\\\ t i m e s 0 . 2 m = 2 \\\\ t i m e s 1 0 ^ { - 4 } m ^ { 3 } m ^ { 3 }',\n",
       "  '= s h = 1 . 0 \\\\times 1 0 ^ { - 3 } m ^ { 2 } \\\\times 0 . 2 m = 2 \\\\times 1 0 ^ { - 4 } m ^ { 3 }\\n'),\n",
       " ('4 x ^ { 2 } - y ^ { 2 } = 4', '4 x ^ { 2 } - y ^ { 2 } = 4\\n'),\n",
       " ('I = \\\\ f r a c { u } { R _ { 1 } } = \\\\ f r a c { 1 2 V } { 3 0 \\\\ O m e g a } = 0 . 4 A',\n",
       "  'I = \\\\frac { U } { R } = \\\\frac { 1 2 V } { 3 0 \\\\Omega } = 0 . 4 A\\n'),\n",
       " ('+ \\\\ f r a c { 7 } { 4 3 } \\\\ t i m e s 3 6 0 ^ { \\\\ c i r c } = 2 1 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\frac { 7 } { + 7 + 3 } \\\\times 3 6 0 ^ { \\\\circ } = 2 6 0 ^ { \\\\circ }\\n'),\n",
       " ('1 . 5 < 0 . 5 0 1 < 0 . 5 1 < 0 . 5 1 1',\n",
       "  '0 . 5 < 0 . 5 0 1 < 0 . 5 1 < 0 . 5 1 1\\n'),\n",
       " ('\\\\ a n g l e O B A = \\\\ a n g l e A B C = 3 6 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle O B A = \\\\angle A B C = 6 0\\n'),\n",
       " ('6 0 \\\\ t i m e s 6 0 \\\\ t i m e s 1 0 = 3 6 0 0 0 ( c m ^ { 2 } )',\n",
       "  '6 0 \\\\times 6 0 \\\\times 1 0 = 3 6 0 0 0 ( c m ^ { 2 } )\\n'),\n",
       " ('x ^ { 2 } + 1 2 x - 1 1 ^ { 2 } + 1 3 ( 2 x - 1 ) + x + 1 = 9',\n",
       "  'x ^ { 2 } + ( 2 x - 1 ) ^ { 2 } + 3 ( 2 x - 1 ) + x + 1 = 0\\n'),\n",
       " ('F _ { 1 } = k _ { 0 } 0 . 0 5', 'F _ { 1 } = 1 0 0 0 N\\n'),\n",
       " ('\\\\ f r a c { 3 . 6 } { 4 . 5 } = \\\\ f r a c { 4 } { x } C u x =',\n",
       "  '\\\\frac { 3 . 6 } { 4 . 5 } = \\\\frac { 4 } { x }\\n'),\n",
       " ('\\\\ a n g l e 1 = 1 8 0 ^ { \\\\ c i r c } - \\\\ a n g l e B A E = 1 8 0 ^ { \\\\ c i r c } - 4 6 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle 1 = 1 8 0 ^ { \\\\circ } - \\\\angle B A E = 1 8 0 ^ { \\\\circ } - 4 6 ^ { \\\\circ }\\n'),\n",
       " ('x = \\\\ f r a c { 1 } { 3 }', 'x = \\\\frac { 1 } { 3 }\\n'),\n",
       " ('a = \\\\ f r a c { 9 5 } { 8 6 1 }', 'a = \\\\frac { 9 5 } { 8 6 1 }\\n'),\n",
       " ('2 4 0 \\\\ t i m e s 3 = 7 2 0 2 4 0 \\\\ t i m e s 4 = 9 6 0',\n",
       "  '2 4 0 \\\\times 3 = 7 2 0 2 4 0 \\\\times 4 = 9 6 0\\n'),\n",
       " ('( x - 2 ) ( 3 x + 2 ) - 3 x ( x + 3 ) \\\\ l e q x',\n",
       "  '( x - ) ( 3 x + 2 ) - 3 x ( x + 3 ) + 1 0 x\\n'),\n",
       " ('\\\\ f r a c { 1 } { 6 } < \\\\ f r a c { 4 } { 9 }',\n",
       "  '\\\\frac { 1 } { 6 } < \\\\frac { 4 } { 9 }\\n'),\n",
       " ('y = \\\\ p m \\\\ f r a c { \\\\ s q r t { b ^ { 2 } + 1 } } { b ^ { 2 } + 1 }',\n",
       "  'y = \\\\pm \\\\frac { \\\\sqrt { b ^ { 2 } + 1 } } { b ^ { 2 } + 1 }\\n'),\n",
       " ('\\\\ f r a c { 2 } { 2 } - \\\\ f r a c { 6 } { 2 } + 2 = 1 - 3 + 2 = 0',\n",
       "  '\\\\frac { 2 } { 2 } - \\\\frac { 6 } { 2 } + 2 = 1 - 3 + 2 = 0\\n'),\n",
       " ('Z n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\ u p a r r o w',\n",
       "  'Z n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\uparrow\\n'),\n",
       " ('x = - \\\\ f r a c { 5 } { 1 8 }', 'x = - \\\\frac { 5 } { 1 8 }\\n'),\n",
       " ('. a _ { 1 } = - \\\\ f r a c { 1 } { 4 } = \\\\ f r a c { 1 } { 4 0 } g',\n",
       "  '. a _ { 1 } = - \\\\frac { 1 } { 4 }\\n'),\n",
       " ('x _ { 2 } = \\\\ f r a c { 3 \\\\ s q r t { 6 } } { 4 }',\n",
       "  'x _ { 2 } = \\\\frac { 3 \\\\sqrt { 6 } } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 5 } { 1 4 } - \\\\ f r a c { 1 } { 6 } = \\\\ f r a c { 2 } { 2 1 }',\n",
       "  '\\\\frac { 5 } { 1 4 } - \\\\frac { 1 } { 6 } = \\\\frac { 2 } { 2 1 }\\n'),\n",
       " ('\\\\ f r a c { 1 6 } { 3 } x = \\\\ f r a c { 3 2 } { 3 5 }',\n",
       "  '\\\\frac { 1 6 } { 3 } x = \\\\frac { 3 2 } { 3 5 }\\n'),\n",
       " ('2 N O + O _ { 2 } \\\\ r i g h t a r r o w 2 N O _ { 2 }',\n",
       "  '2 N O + O _ { 2 } \\\\rightarrow 2 N O _ { 2 }\\n'),\n",
       " ('0 . 6 6 > 0 . 6 4 > 0 . 6 0 6 > 0 . 6 0 4',\n",
       "  '0 . 6 6 > 0 . 6 4 > 0 . 6 0 6 > 0 . 6 0 4\\n'),\n",
       " ('\\\\ t a n : y = \\\\ f r a c { 3 } { 2 } x - 1 2',\n",
       "  '\\\\rho _ { N A } : y = \\\\frac { 3 } { 2 } x - 1 2\\n'),\n",
       " ('x + y - \\\\ s q r t { y + y } = 2', 'x + y - \\\\sqrt { x + y } = 2\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } ( 3 0 9 6 - 2 - 4 ) = 4 2 8',\n",
       "  '\\\\frac { 1 } { 2 } ( 3 0 \\\\% x - 4 ) = 2 0 \\\\%\\n'),\n",
       " ('\\\\ c o s \\\\ b e t a = \\\\ f r a c { 3 } { 2 } \\\\ c o s \\\\ a l p h a',\n",
       "  '\\\\cos \\\\beta = \\\\frac { 3 } { 2 } \\\\cos \\\\alpha\\n'),\n",
       " ('C Q = B C - B Q = 6 0 - 2 0 \\\\ s q r t { 3 } ( m',\n",
       "  'C Q = B C - B Q = 6 0 - 2 0 \\\\sqrt { 3 } ( m\\n'),\n",
       " ('5 0 0 \\\\ a p p r o x 4 8 0', '5 0 0 \\\\approx 4 8 0\\n'),\n",
       " ('3 - 1 - 2 \\\\ t i m e s \\\\ f r a c { 1 } { 2 } = 3 - 1 - 1 = 1',\n",
       "  '3 - 1 - 2 \\\\times \\\\frac { 1 } { 2 } = 3 - 1 - 1 = 1\\n'),\n",
       " ('N a _ { 2 } C O _ { 3 } + H _ { 2 } S O 4',\n",
       "  'N a _ { 2 } C O _ { 3 } + H _ { 2 } S O _ { 4 }\\n'),\n",
       " ('( x - 1 ) ( 3 a x ^ { 2 } - 3 a x - 1 ) > 0',\n",
       "  '( x - 1 ) ( 3 a x ^ { 2 } - 3 a x - 1 ) \\\\geqslant 0\\n'),\n",
       " ('\\\\ f r a c { 2 n } { 2 1 + n } = \\\\ f r a c { 1 7 } { 3 5 }',\n",
       "  '\\\\frac { 2 n } { 2 n + 1 } = \\\\frac { 1 7 } { 3 5 }\\n'),\n",
       " ('2 s m ( n - s ) = 1', '2 \\\\sin ( A - \\\\frac { \\\\pi } { 6 } ) = 1\\n'),\n",
       " ('y _ { 1 } = - \\\\ f r a c { 3 } { 2 }',\n",
       "  'y _ { 1 } = - \\\\frac { 3 } { 2 }\\n'),\n",
       " ('( t ^ { 2 } - \\\\ s q r t { 2 } ) ( t ^ { 2 } + \\\\ s q r t { 2 } t + 1 ) = 0',\n",
       "  '( t ^ { 2 } - \\\\sqrt { 2 } ) ( t ^ { 2 } + \\\\sqrt { 2 } t + 1 ) = 0\\n'),\n",
       " ('\\\\ a n g l e A D C = 1 8 0 ^ { \\\\ c i r c } - 6 0 ^ { \\\\ c i r c } = 1 2 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A D C = 1 8 0 ^ { \\\\circ } - 6 0 ^ { \\\\circ } = 1 2 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ f r a c { x } { x - 3 } - 2 m = \\\\ f r a c { 3 m - 1 } { x - 3 }',\n",
       "  '\\\\frac { x } { x - 3 } - 2 m = \\\\frac { 3 m - 1 } { x - 3 }\\n'),\n",
       " ('\\\\ a n g l e B O S + \\\\ a n g l e S = 1 8 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B O S + \\\\angle S = 1 8 0 ^ { \\\\circ }\\n'),\n",
       " ('( - 4 ) + 7 + ( - 9 ) + 8 + 6 + ( - 4 ) + ( - 3 ) = 1 k m',\n",
       "  '. ( - 4 ) + 7 + ( - 9 ) + 8 + 6 + ( - 4 ) + ( - 3 ) = 1 k m\\n'),\n",
       " ('\\\\ s q r t { 2 - 1 + 1 } = \\\\ f r a c { 1 } { \\\\ s q r t { 2 } } = \\\\ f r a c { \\\\ s q r t { 2 } } { 2 }',\n",
       "  '\\\\frac { 1 } { \\\\sqrt { 2 } - 1 + 1 } = \\\\frac { 1 } { \\\\sqrt { 2 } } = \\\\frac { \\\\sqrt { 2 } } { 2 }\\n'),\n",
       " ('2 8 0 0 \\\\ d i v 5 0 = 6 \\\\ c d o t s 3 0',\n",
       "  '2 8 0 \\\\div 5 0 = 6 \\\\cdots 3 0\\n'),\n",
       " ('x \\\\ a n g l e 0 . 5 = \\\\ f r a c { 1 } { 1 + \\\\ s q r t { 1 0 } }',\n",
       "  'x - a ^ { 5 } = - \\\\frac { 1 } { b \\\\sqrt { 1 0 } }\\n'),\n",
       " ('\\\\ f r a c { 1 } { a + b } = \\\\ f r a c { 1 } { 2 - 1 } = 1 = 1',\n",
       "  '\\\\frac { 1 } { a + b } = - \\\\frac { 1 } { 2 - 1 } = 1\\n'),\n",
       " ('\\\\ c o s \\\\ a n g l e A B C = \\\\ f r a c { 5 } { 1 3 } = \\\\ f r a c { B D } { A B }',\n",
       "  '\\\\cos \\\\angle A B C = \\\\frac { 5 } { 1 3 } = \\\\frac { B D } { A B }\\n'),\n",
       " ('x - \\\\ f r a c { x - 1 } { 4 } = 2 \\\\ t i m e s \\\\ f r a c { x - 2 } { 2 }',\n",
       "  'x - \\\\frac { x - 1 } { 4 } = 2 \\\\times \\\\frac { x - 2 } { 3 }\\n'),\n",
       " ('\\\\ f r a c { 1 1 7 } { 4 4 } = \\\\ f r a c { y } { 8 . 8 g }',\n",
       "  '\\\\frac { 1 1 7 } { 4 9 } = \\\\frac { y } { 8 . 8 y }\\n'),\n",
       " ('= 2 \\\\ t i m e s \\\\ f r a c { 3 } { 7 } \\\\ t i m e s \\\\ f r a c { 2 } { 3 } )',\n",
       "  '\\\\frac { 5 } { 6 } \\\\times \\\\frac { 6 } { 5 } \\\\textcircled { = } \\\\frac { 3 } { 7 } \\\\times \\\\frac { 7 } { 3 }\\n'),\n",
       " ('y = \\\\ f r a c { 1 } { 3 } x ^ { 2 } + b x + c',\n",
       "  'y = \\\\frac { 1 } { 3 } x ^ { 2 } + b x + c\\n'),\n",
       " ('6 x - 1 0 - 7 x - 6 x + 2 x + 2 = 3 x - 6',\n",
       "  '6 x - 1 0 - 7 x - 6 x + 2 x + 2 = 3 x - 6\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x - 1 1 1 + x = 8 3 4',\n",
       "  '\\\\frac { 1 } { 2 } x - 1 1 1 + x = 8 3 4\\n'),\n",
       " ('6 2 9 \\\\ d i v 1 3 = 4 8 \\\\ c d o t s 5',\n",
       "  '6 2 9 \\\\div 1 3 = 4 8 \\\\cdots 5\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } [ \\\\ f r a c { 1 } { 4 } ( \\\\ f r a c { 1 } { 5 } x - 2 ) - 8 ] = \\\\ f r a c { 1 } { 7 } 4',\n",
       "  '\\\\frac { 1 } { 3 } [ \\\\frac { 1 } { 4 } ( \\\\frac { 1 } { 5 } x - 2 ) - 8 ] = - 4\\n'),\n",
       " ('3 . 1 4 m ^ { 2 } = 5 2 . 5 d m ^ { 3 }',\n",
       "  '3 . 1 4 \\\\times 4 0 = 1 2 5 . 6 ( c m ^ { 2 } )\\n'),\n",
       " ('a \\\\ g e q - \\\\ f r a c { 1 } { 5 }', 'a \\\\geq - \\\\frac { 1 } { 5 }\\n'),\n",
       " ('v = \\\\ f r a c { m } { \\\\ r h o } = \\\\ f r a c { 1 0 ^ { 2 7 } } { 7 . 9 g / c m ^ { 3 } } = 1 3 0 0 0 0 0 c m ^ { 3 } m ^ { 3 } m ^ { 3 }',\n",
       "  'v = \\\\frac { m } { \\\\rho } = \\\\frac { 1 0 2 7 k g } { 7 . 9 g / c m ^ { 3 } } = 1 3 0 0 0 0 c m ^ { 3 }\\n'),\n",
       " ('8 0 \\\\ % x + 1 5 0 = 5 0 \\\\ t i m e s 6 0 \\\\ % + 3 0 \\\\ %',\n",
       "  '8 0 \\\\% x + 1 5 0 = 5 0 0 \\\\times 6 0 \\\\% + 3 0 \\\\%\\n'),\n",
       " ('\" x _ { 1 } = \\\\ s q r t { 5 } + 1 , x _ { 2 } = 1 - \\\\ s q r t { 5 } \" \" _ { 3 } \" \" \" \" \" \" \" , \" \" , \" , x _ { 3 } \" \" , \" , \" x _ { 3 } , \" , \" x _ { 3 } \" , \" , x',\n",
       "  '\"x _ { 1 } = \\\\sqrt { 5 } + 1 , x _ { 2 } = 1 - \\\\sqrt { 5 }\"\\n'),\n",
       " ('D E F C = D C = 3 . 5 c m', 'D E + E C = D C = 3 . 5 c m\\n'),\n",
       " ('1 \\\\ d i v - 1 . 7 < 2 \\\\ f r a c { 5 } { 2 } = \\\\ f r a c { 5 \\\\ c d o t } { 3 }',\n",
       "  '\"1 = - 1 , x _ { 2 } = \\\\frac { 5 \\\\cdot } { 3 }\"\\n'),\n",
       " ('\\\\ t h e r e f o r e B E = D F = y', '\\\\therefore B E = D F = y\\n'),\n",
       " ('= \\\\ f r a c { 1 0 0 } { 2 0 g } = \\\\ f r a c { 1 1 7 } { y }',\n",
       "  '\\\\frac { 1 0 6 } { x } = \\\\frac { \\\\cdot 1 0 0 } { 2 0 3 } = \\\\frac { 1 1 7 } { y }\\n'),\n",
       " ('1 5 \\\\ t i m e s 4 = 6 0 ( k g )', '1 5 \\\\times 4 = 6 0 ( k g )\\n'),\n",
       " ('8 ( x - 1 ) - 1 2 ( 5 x + 1 ) = 4 8 - 6 ( 3 x + 2 )',\n",
       "  '8 ( 7 x - 1 ) - 1 2 ( 5 x + 1 ) = 4 8 - 6 ( 3 x + 2 )\\n'),\n",
       " ('\\\\ f r a c { \\\\ a n g l e B = \\\\ a n g l e C = 1 8 0 ^ { \\\\ c i r c } - \\\\ a n g l e B A C = \\\\ f r a c { 8 0 ^ { \\\\ c i r c } } { 2 } = 4 0 ^ { \\\\ c i r c } = 4 0 ^ { \\\\ c i r c } ^ { \\\\ c i',\n",
       "  '\\\\cdot \\\\angle B = ^ { \\\\prime } \\\\angle C = \\\\frac { 1 8 0 ^ { \\\\circ } - \\\\angle B A C } { 2 } = \\\\frac { 8 0 ^ { \\\\circ } } { 2 } = 4 0 ^ { \\\\circ }\\n'),\n",
       " ('C D = 4 d \\\\ s q r t { 5 }', 'C D = 4 \\\\sqrt { 5 }\\n'),\n",
       " ('N a O H + H _ { 2 } O + C l _ { 2 } = 2 N a C l + H C O _ { 2 }',\n",
       "  'N a O H + H _ { 2 } O + C l _ { 1 } = Z N a C l + H C l O\\n'),\n",
       " ('\\\\ f r a c { 3 6 } { 4 } = \\\\ f r a c { x } { 2 y }',\n",
       "  '\\\\frac { 3 6 } { 4 } = \\\\frac { x } { 2 9 }\\n'),\n",
       " ('\\\\ s i n ( - \\\\ f r a c { 5 } { 4 } \\\\ p i ) = \\\\ s i n',\n",
       "  '\\\\sin ( - \\\\frac { 8 7 } { 4 } \\\\pi ) = s i\\n'),\n",
       " ('1 6 \\\\ t i m e s 6 = 9 6 c m ^ { 2 }', '1 6 \\\\times 6 = 9 6 c m ^ { 2 }\\n'),\n",
       " ('y = \\\\ f r a c { 9 } { 5 }', 'y = \\\\frac { 9 } { 5 }\\n'),\n",
       " ('7 5 0 + 1 5 0 ( x - 1 ) = \\\\ f r a c { 3 0 0 x } { 2 } + \\\\ f r a c { 1 0 0 x } { 2 }',\n",
       "  '7 5 0 + 1 5 0 ( x - 1 ) = \\\\frac { 3 0 0 x } { 2 } + \\\\frac { 1 0 0 x } { 2 }\\n'),\n",
       " ('\\\\ f r a c { 7 } { 8 } = \\\\ f r a c { 4 2 } { 4 8 }',\n",
       "  '\\\\frac { 7 } { 8 } = \\\\frac { 4 2 } { 4 8 }\\n'),\n",
       " ('\\\\ f r a c { 1 2 } { 1 6 } = \\\\ f r a c { 9 } { 1 2 } 、 \\\\ f r a c { 4 } { 1 2 } > \\\\ f r a c { 5 } { 2 0 }',\n",
       "  '\\\\frac { 1 2 } { 1 6 } = \\\\frac { 9 } { 1 2 } 、 \\\\frac { 4 } { 1 2 } > \\\\frac { 5 } { 2 0 }\\n'),\n",
       " ('\\\\ f r a c { B D } { 6 0 } = 1', '\\\\frac { B D } { C P } = 1\\n'),\n",
       " ('5 x = \\\\ f r a c { 1 } { 2 } x + 3', '5 x = \\\\frac { 1 } { 2 } x + 3\\n'),\n",
       " ('C O _ { 2 } + C a ( O H ) _ { 2 } = C a C O _ { 3 } \\\\ d o w n a r r o w + H _ { 2 } O',\n",
       "  'C O _ { 2 } + C a ( O H ) _ { 2 } = C a C O _ { 3 } \\\\downarrow + H _ { 2 } O\\n'),\n",
       " ('x = \\\\ f r a c { 1 6 } { 4 }', 'x = \\\\frac { 1 6 } { 4 }\\n'),\n",
       " ('\\\\ b e g i n { m a t r i x } x + 5 x = 2 + 1 - 1 \\\\ \\\\ 1 x = 7 + 1 - 1 \\\\ e n d { m a t r i x } = 7',\n",
       "  'x + 3 x = 2 + 1 - 1\\n'),\n",
       " ('P = \\\\ r h o g h = 1 \\\\ t i m e s 1 0 ^ { 3 } k g / m ^ { 3 } \\\\ t i m e s 0 . 1 m \\\\ t i m e s 9 . 8 N / k g = 9 8 0 P a 0 . 0 3 m 0 . 0 3 m 0 . 0 3 m 0 . 0 3 m 0 . 0 3 m 0 . 0 3 m 0 .',\n",
       "  'P = e g h = 1 \\\\times 1 0 ^ { 3 } k g / m ^ { 3 } \\\\times 0 . 1 m \\\\times 9 . 8 N / k g = 9 8 0 P a\\n'),\n",
       " ('\\\\ f r a c { O E } { O B } = \\\\ f r a c { O F } { O C }',\n",
       "  '\\\\frac { O E } { O B } = \\\\frac { O F } { O C }\\n'),\n",
       " ('x ^ { 2 } + x + 1 = ( x + \\\\ f r a c { 1 } { 2 } ) ^ { 2 } + \\\\ f r a c { 3 } { 4 } \\\\ i n [ - 1 , 1 ]',\n",
       "  '\"x ^ { 2 } + x + 1 = ( x + \\\\frac { 1 } { 2 } ) ^ { 2 } + \\\\frac { 3 } { 4 } \\\\in [ - 1 , 1 ]\"\\n'),\n",
       " ('\\\\ s i n ( \\\\ f r a c { \\\\ p i } { 6 } + x ) = 2',\n",
       "  '\\\\sin ( \\\\frac { \\\\pi } { 6 } + x ) = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { 6 } { x + 1 } = \\\\ f r a c { x - 8 } { 3 } + 2',\n",
       "  '\\\\frac { 6 } { x + 1 } = \\\\frac { x - 8 } { 3 } + 2\\n'),\n",
       " ('\\\\ a n g l e A C D + \\\\ a n g l e B A C = 1 8 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A C D + \\\\angle B A C = 1 8 0 ^ { \\\\circ }\\n'),\n",
       " ('x + 8 2 \\\\ t i m e s 6 . 2 + 6 8 \\\\ t i m e s 6 . 2 = 1 2 6 2 2',\n",
       "  'x + 8 2 \\\\times 6 . 2 + 6 8 \\\\times 6 . 2 = 1 2 6 2\\n'),\n",
       " ('2 . 4 \\\\ t i m e s ( 1 0 - 3 ) + 3 6 ( x - 1 0 ) + 1 2 = 3 2 . 4',\n",
       "  '2 . 4 \\\\times ( 1 0 - 3 ) + 3 . 6 ( x - 1 0 ) + 1 2 = 3 2 . 4\\n'),\n",
       " ('x - 4 y + 5 = 0 y S 1 5 } . 5 x } 5 x - 1 5 } 5 x } 5 x - 1 5 \\\\ n 1 5 5 \\\\ n 1 5 5 \\\\ n 1 5 5 \\\\ n 1 5 5 \\\\ n 1 5 5 \\\\ n 1 5 5',\n",
       "  'x - 4 y + 5 = 0\\n'),\n",
       " ('y = - \\\\ f r a c { 2 } { 5 }', 'y = - \\\\frac { 2 } { 5 }\\n'),\n",
       " ('\\\\ f r a c { 4 } { 5 } S _ { A B C D } = 2 ( S _ { D B _ { 1 } } + S _ { B _ { 1 } C D _ { 2 } } + S _ { D _ { 2 } B H } + S _ { C _ { 1 } A A _ { 4 } }',\n",
       "  '\\\\frac { 4 } { 5 } S _ { A B C D } = 2 ( S _ { D B _ { 1 } C _ { 1 } } + S _ { B _ { 1 } C D _ { 2 } } + S _ { D _ { 2 } B A _ { 4 } } + S _ { C _ { 1 } A A _ { 4 } }\\n'),\n",
       " ('N a O H + K N O _ { 3 } = N a N O _ { 3 } + K O H',\n",
       "  'N a O H + K N O _ { 3 } = N a N O _ { 3 } + K O H\\n'),\n",
       " ('B D = O D = \\\\ f r a c { 1 } { 2 } B C',\n",
       "  'B D = O D = \\\\frac { 1 } { 2 } B C\\n'),\n",
       " ('C a ( O H ) _ { 2 } + k C O _ { 2 } = C a C O _ { 3 } \\\\ d o w n a r r o w + 2 k O H',\n",
       "  'C a ( O H ) _ { 2 } + k _ { 2 } C O _ { 3 } = C a C O _ { 3 } \\\\downarrow + 2 k O H\\n'),\n",
       " ('\\\\ v e r t x + 2 \\\\ v e r t = 2 6 - 5 = ( 3 7 - c ) \\\\ t e x t c i r c l e d { 1 }',\n",
       "  '1 7 6 + 1 2 0 - 2 6 . 5 = 2 6 9 . 2 ( m ^ { 2 } )\\n'),\n",
       " ('\\\\ f r a c { D E } { O D } = \\\\ f r a c { B D } { D E }',\n",
       "  '\\\\frac { D E } { O D } = \\\\frac { B D } { D E }\\n'),\n",
       " ('\\\\ p m \\\\ s q r t { ( - 6 ) ^ { 2 } } = \\\\ p m 6',\n",
       "  '\\\\pm \\\\sqrt { ( - 6 ) ^ { 2 } } = \\\\pm 6\\n'),\n",
       " ('D E ^ { 2 } = 1 2 0 \\\\ c d o t D B D 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 比 D 比 D 比 D',\n",
       "  'D E ^ { 2 } = D O \\\\cdot D B\\n'),\n",
       " ('2 x + y = 3 x + 6 y = 0', '2 x + y = 3 x + 6 y = 0\\n'),\n",
       " ('0 . 1 \\\\ d o t { 0 } 0 \\\\ d o t { 8 } < 0 . 1 0 \\\\ d o t { 0 } \\\\ d o t { 8 } < 0 . \\\\ d o t { 1 } 0 0 \\\\ d o t { 2 } < 0 . 1 0 0 \\\\ d o t { 5 }',\n",
       "  '( 1 4 0 0 ) \\\\times ( 1 0 0 ) \\\\times ( 3 5 0 ) \\\\times ( 2 5 ) = ( 1 8 7 5 )\\n'),\n",
       " ('\\\\ a n g l e B A G = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e B A C = 9 0 ^ { \\\\ c i r c } - 2 \\\\ a l p h a',\n",
       "  '\\\\angle B A G = \\\\frac { 1 } { 2 } \\\\angle B A C = 9 0 ^ { \\\\circ } - 2 \\\\alpha\\n'),\n",
       " ('O A = \\\\ s q r t { 1 ^ { 2 } + 4 ^ { 2 } } = \\\\ s q r t { 1 7 }',\n",
       "  '. O A = \\\\sqrt { 1 ^ { 2 } + 4 ^ { 2 } } = \\\\sqrt { 1 7 }\\n'),\n",
       " ('( 2 n + 1 ) b _ { n + 1 } - b _ { n + 1 } = n b _ { n }',\n",
       "  '( 2 n + 1 ) b _ { n + 1 } - b _ { n + 1 } = n b _ { n }\\n'),\n",
       " ('2 1 \\\\ t i m e s 2 1 . 7 + 1 8 . 6 1 ( 1 ) = 1 1 1',\n",
       "  '2 1 \\\\times 2 1 = 4 6 1\\n'),\n",
       " ('k = \\\\ f r a c { 1 9 } { 2 7 }', 'k = \\\\frac { 1 9 } { 2 7 }\\n'),\n",
       " ('\\\\ f r a c { D C } { A O } = \\\\ f r a c { A D } { B O } = \\\\ f r a c { A C } { A B } = \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\cdot \\\\frac { D C } { A O } = \\\\frac { A D } { B O } = \\\\frac { A C } { A B } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('v = \\\\ f r a c { F + ( n + \\\\ f r a c { 3 } { 4 } ) } { M } = w R',\n",
       "  'V = \\\\frac { F + ( n + \\\\frac { 3 } { 4 } ) } { M } = W R\\n'),\n",
       " ('\" x = ( 5 0 + k _ { 0 } \\\\ t i m e s 5 0 ) c m ( k _ { 1 } = 1 , 2 , 3 \\\\ c d o t s ) \"',\n",
       "  '\"x = ( 5 0 + k _ { 1 } \\\\times 5 0 ) c m ( k _ { 1 } = 1 , 2 , 3 \\\\cdots )\"\\n'),\n",
       " ('1 4 、 - \\\\ f r a c { 4 } { 3 }', '1 4 、 - \\\\frac { 4 } { 3 }\\n'),\n",
       " ('\\\\ b e t a + \\\\ a l p h a = 2 k \\\\ p i + \\\\ f r a c { \\\\ p i } { 2 } ( k \\\\ i n z )',\n",
       "  '\\\\beta + \\\\alpha = 2 k \\\\pi + \\\\frac { \\\\pi } { 2 } ( k \\\\in z )\\n'),\n",
       " ('x _ { 1 } = - 5 x _ { 2 } = 2', 'x _ { 1 } = - 5 x _ { 2 } = 2\\n'),\n",
       " ('x _ { 1 } = 0 x _ { 2 } = 2 { 2 }', 'x _ { 1 } = 0 x _ { 2 } = 2\\n'),\n",
       " ('( a + 1 ) ^ { 2 } + ( a - 1 ) ^ { 2 } = 3 4',\n",
       "  '( a + 1 ) ^ { 2 } + ( a - 1 ) ^ { 2 } = 3 4\\n'),\n",
       " ('\\\\ s q r t { 5 x + 4 } = - x + 2', '\\\\sqrt { 5 x - 4 } = - x + 2\\n'),\n",
       " ('x = - 1', 'x _ { 2 } = - 1\\n'),\n",
       " ('1 \\\\ d i v 2 = \\\\ f r a c { 1 } { 2 } ( k g )',\n",
       "  '1 \\\\div 2 = \\\\frac { 1 } { 2 } ( k g )\\n'),\n",
       " ('\" \\\\ t h e t a , B V _ { 1 } = m \\\\ f r a c { V _ { c } } { R } \" \" \" , \" , \" , \" , \" , \" , \" , \" , \" , m , P _ { 1 } , \" , m , m = 1 , m , m = 1 , 0 , 1 , 0 , 1 , 0 , 3 , \" , \" , \"',\n",
       "  '- q _ { 1 } B V _ { 1 } = m \\\\frac { V _ { 1 } ^ { 2 } } { R }\\n'),\n",
       " ('x ^ { 2 } + \\\\ f r a c { \\\\ s q r t { 6 } } { 4 } = - \\\\ f r a c { 3 } { 4 } \\\\ s q r t { 6 }',\n",
       "  'x ^ { . } + \\\\frac { \\\\sqrt { 6 } } { 4 } = - \\\\frac { 3 } { 4 } \\\\sqrt { 6 }\\n'),\n",
       " ('S x = 5', 's x = 5\\n'),\n",
       " ('\\\\ a n g l e A H C = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A H C = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('x _ { 2 } = 2 \\\\ f r a c { 2 } { 3 }', 'x = 2 \\\\frac { 2 } { 3 }\\n'),\n",
       " ('\\\\ b e g i n { m a t r i x } 3 0 + 2 7 + 4 3 = 1 0 0 \\\\ \\\\ 5 7 \\\\ e n d { m a t r i x }',\n",
       "  '\\\\frac { 3 0 + 2 7 + 4 3 } { 5 7 } = 1 0 0\\n'),\n",
       " ('2 x - 2 5 = \\\\ f r a c { x + 1 2 } { 4 } 2 x - x = 2 8 \\\\ t i m e s 4',\n",
       "  '2 x - 2 5 = \\\\frac { x + 1 2 } { 4 } 2 x - x = 2 8 \\\\times 4\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } S _ { \\\\ D e l t a A B O } = 2 4 \\\\ t i m e s \\\\ f r a c { 1 } { 3 } = 8 = S _ { \\\\ D e l t a } A B P',\n",
       "  '\\\\frac { 1 } { 3 } S _ { \\\\Delta } A B O = 2 4 \\\\times \\\\frac { 1 } { 3 } = 8 = S _ { \\\\Delta } A B P\\n'),\n",
       " ('a = \\\\ f r a c { 2 } { 5 }', 'a = \\\\frac { 2 } { 5 }\\n'),\n",
       " ('u x + 5 u = 3 x + 5', '9 x + G = 3 x + G\\n'),\n",
       " ('\\\\ t h e r e f o r e \\\\ v e r t a \\\\ v e r t = \\\\ v e r t a \\\\ v e r t = \\\\ v e r t 3 \\\\ v e r t = 2 0 0 - 2 3 \\\\ v e r t + 4 0 b',\n",
       "  '( x - 2 b ) ( y - 2 a ) = x y - 2 a x - 2 b y + 4 a b\\n'),\n",
       " ('( x _ { 0 } - m ) y = y _ { 0 } ( x - m )',\n",
       "  '( x _ { 0 } - m ) y = y _ { 0 } ( x - m )\\n'),\n",
       " ('x > 2 . x < - 2', 'x > 2 x < - 2\\n'),\n",
       " ('x = \\\\ f r a c { 1 } { 6 } \\\\ d i v \\\\ f r a c { 3 } { 1 0 }',\n",
       "  'x = \\\\frac { 1 } { 6 } \\\\div \\\\frac { 3 } { 1 0 }\\n'),\n",
       " ('\\\\ s i n C = \\\\ f r a c { A B } { A C } = \\\\ f r a c { C } { 2 R }',\n",
       "  '\\\\sin C = \\\\frac { A B } { A C } = \\\\frac { C } { 2 R }\\n'),\n",
       " ('\" \\\\ a n g l e A = 6 0 ^ { \\\\ c i r c } , \\\\ f r a c { \\\\ c o s B } { \\\\ s i n C } \\\\ o v e r r i g h t a r r o w { A B } \" \" \" \" \" \" \" \" \" \" \" \" , \" \" \" , \" , \" , \" , 3 , 5 , 1 ,',\n",
       "  '\"\\\\angle A = 6 0 ^ { \\\\circ } , \\\\frac { \\\\cos B } { \\\\sin C } \\\\overrightarrow { A B }\"\\n'),\n",
       " ('\" ( 2 ) \\\\ b e c a u s e a = \\\\ f r a c { 1 } { 3 } , c = \\\\ f r a c { \\\\ s q r t { 2 } } { 3 } \" \" \" = \\\\ f r a c { \\\\ s q r t { 2 } } { 3 } \" \" \" \" \" \" \" \" \" \" , \" , \" , \"',\n",
       "  '\"( 2 ) \\\\because a = \\\\frac { 1 } { 3 } , c = \\\\frac { \\\\sqrt { 2 } } { 3 }\"\\n'),\n",
       " ('x _ { 2 } = \\\\ f r a c { 1 } { 4 }', 'x _ { 2 } = \\\\frac { 1 } { - 4 }\\n'),\n",
       " ('S _ { \\\\ D e l t a } = S = 1 0 m', 'S _ { a } = S = 1 0 m\\n'),\n",
       " ('\\\\ f r a c { 4 } { B D } = \\\\ f r a c { 8 } { 1 2 }',\n",
       "  '\\\\frac { 4 } { B D } = \\\\frac { 8 } { 1 2 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 4 } - = - \\\\ f r a c { 2 \\\\ s q r t { 7 } } { 7 }',\n",
       "  'y _ { 2 } = - \\\\frac { 2 \\\\sqrt { 7 } } { 7 }\\n'),\n",
       " ('\\\\ D e l t a = b ^ { 2 } - 4 a c \\\\ d i v ( - 1 ) ^ { 2 } - 4 \\\\ t i m e s 1 \\\\ t i m e s 2 = - 7 < 0',\n",
       "  '\\\\Delta = b ^ { 2 } - 4 a c \\\\div ( - 1 ) ^ { 2 } - 4 \\\\times 1 \\\\times 2 = - 7 < 0\\n'),\n",
       " ('x _ { 1 } = 7 + \\\\ s q r t { 3 }', 'x _ { 1 } = 1 + \\\\sqrt { 3 }\\n'),\n",
       " ('\\\\ a n g l e 2 = 6 0 ^ { \\\\ c i r c } 3 0 ^ { \\\\ c i r c } \\\\ t h e r e f o r e \\\\ a n g l e A B C = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle 2 = 6 0 ^ { \\\\circ } 3 0 ^ { \\\\circ } \\\\because \\\\angle A B C = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('1 \\\\ d i v 9 = \\\\ f r a c { 1 } { 9 }', '1 \\\\div 9 = \\\\frac { 1 } { 9 }\\n'),\n",
       " ('\\\\ f r a c { 2 x - 1 } { 3 } = \\\\ f r a c { 3 } { 4 } x - 5',\n",
       "  '\\\\frac { 2 x - 1 } { 3 } = \\\\frac { 3 } { 4 } x - 5\\n'),\n",
       " ('- \\\\ f r a c { 1 } { 2 } ( x - 6 ) ( x - t + 4 ) = 0',\n",
       "  '\\\\frac { 1 } { 2 } ( x - t ) ( x - t + 4 ) = 0\\n'),\n",
       " ('4 x ^ { 4 } - 5 x ^ { 2 } - 4 = 0', '4 x ^ { 4 } - 5 x ^ { 2 } - 4 = 0\\n'),\n",
       " ('A B ^ { 2 } - A D ^ { 2 } = D B', 'A B ^ { 2 } - A D ^ { 2 } = D B\\n'),\n",
       " ('1 5 - 5 = 1 0 ^ { \\\\ c i r c } C', '1 5 - 5 = 1 0 ^ { \\\\circ } C\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } ( x - ( k \\\\ t i m e s 2 2 - 1 ) ) = \\\\ f r a c { 2 } { 3 } ( 2 2 - x )',\n",
       "  '\\\\frac { 1 } { 2 } [ x - ( k \\\\times 2 2 - 1 ) ] = \\\\frac { 2 } { 3 } ( 2 2 - x )\\n'),\n",
       " ('P = \\\\ a n g l e I _ { 2 } I _ { 2 } = \\\\ a n g l e . 5 V \\\\ t i m e s \\\\ f r a c { 1 . 6 5 ^ { \\\\ c i r c } } { w R } = \\\\ f r a c { 2 . 4 v \\\\ t i m e s 1 . 6 5 ^ { \\\\ c i r c } } { w R }',\n",
       "  'P 2 = U _ { 2 } I = 2 . 5 v \\\\times \\\\frac { 1 . 6 5 w } { U R } = \\\\frac { 2 . 5 V \\\\times 1 . 6 5 W } { u _ { 2 } - 2 . 5 V }\\n'),\n",
       " ('\\\\ f r a c { 4 } { 5 } x = \\\\ f r a c { 4 } { 5 } + \\\\ f r a c { 2 5 } { 5 . } . . . . . . . . . . . . . . . .',\n",
       "  '\\\\frac { 4 } { 5 } x = \\\\frac { 4 } { 5 } + \\\\frac { 2 5 } { 5 . }\\n'),\n",
       " ('\\\\ a n g l e D O H = 9 0 ^ { \\\\ c i r c } t i m n = 9 0 ^ { \\\\ c i r c } \\\\ a n g l e 1 = 9 0 ^ { \\\\ c i r c } \\\\ b o x = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle D O H = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('W _ { 1 } = V J _ { 1 } t = U I t = 6 v \\\\ t i m e s 0 . 6 A \\\\ t i m e s 1 0 s = 3 6 J',\n",
       "  'W _ { 1 } = V _ { 1 } I _ { 1 } t = V I _ { 1 } t = 6 v \\\\times 0 . 6 A \\\\times 1 0 s = 3 6 J\\n'),\n",
       " ('4 x + 2 x - 2 = 4', '4 x + 2 x - 2 = 4\\n'),\n",
       " ('m g = G \\\\ f r a c { M m } { R ^ { 2 } }',\n",
       "  'm g = G \\\\frac { M m } { R ^ { 2 } }\\n'),\n",
       " ('4 . 5 0 6 < 4 . 5 6 < 4 . 5 6 5 < 5 . 6 5',\n",
       "  '4 . 5 0 6 < 4 . 5 6 < 4 . 5 6 5 < 5 . 6 5\\n'),\n",
       " ('\\\\ t a n \\\\ a n g l e O A B = \\\\ f r a c { O B } { O A } = \\\\ f r a c { 1 } { 3 }',\n",
       "  '\\\\tan \\\\angle O A B = \\\\frac { O B } { O A } = \\\\frac { 1 } { 3 }\\n'),\n",
       " ('6 0 m = 6 0 0 0 c m', '6 0 m = 6 0 0 0 c m\\n'),\n",
       " ('\\\\ o v e r r i g h t a r r o w { E D } = ( 0 . 4 \\\\ d i v 3 ) 比 0 , 0 , 4 ; 1 )',\n",
       "  '\"\\\\overrightarrow { E D } = ( 0 . 4 , - 3 )\"\\n'),\n",
       " ('S = 2 4 \\\\ t i m e s 4', 'S = 2 4 . 5 \\\\sqrt { 5 }\\n'),\n",
       " ('t _ { 1 } = \\\\ f r a c { V _ { 1 } } { u _ { 1 } } _ { 1 } _ { 1 }',\n",
       "  't _ { 1 } = \\\\frac { V _ { 1 } } { a _ { 1 } }\\n'),\n",
       " ('2 0 0 0 c m ^ { 2 } = ( 1 2 0 ) d m ^ { 2 }',\n",
       "  '2 0 0 0 c m ^ { 2 } = ( 1 2 0 ) d m ^ { 2 }\\n'),\n",
       " ('\\\\ a n g l e B D E = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B D E = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('= \\\\ f r a c { ( 1 8 ) } { 4 0 } \\\\ d i v 8 = 5',\n",
       "  '( \\\\frac { 2 2 + 1 8 } { 4 0 } ) \\\\div 8 = 5\\n'),\n",
       " ('\\\\ a n g l e C = 1 2 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle C = 1 2 0 ^ { \\\\circ }\\n'),\n",
       " ('2 1 0 \\\\ t i m e s \\\\ f r a c { 5 } { 9 } = 1 5 0 c m',\n",
       "  '2 1 0 \\\\times \\\\frac { 5 } { 7 } = 1 5 0 c m\\n'),\n",
       " ('C F = E A \\\\ c d o t E B = 2 k', 'C E ^ { 2 } = E A \\\\cdot E B = 2 k\\n'),\n",
       " ('b \\\\ n e q - 1 \\\\ t h e r e f o r e b + 1 \\\\ n e q 0',\n",
       "  'b \\\\neq - 1 \\\\therefore b + 1 \\\\neq 0\\n'),\n",
       " ('1 2 0 \\\\ d i v 2 5 = 4 . 8 ( d m )', '1 2 0 \\\\div 2 5 = 4 . 8 ( d m )\\n'),\n",
       " ('8 x + ( 1 0 4 + 9 6 ) = 2 8 7 k m', '8 7 + ( 1 0 4 + 9 6 ) = 2 8 7 k m\\n'),\n",
       " ('\\\\ a n g l e E O B = \\\\ a n g l e 3 + \\\\ a n g l e 4 + 4 5 ^ { \\\\ c i r c } = 1 1 0 ^ { \\\\ c i r c } (',\n",
       "  '\\\\angle E O B = \\\\angle 3 + \\\\angle 4 + \\\\angle 5 = 1 1 0 ^ { \\\\circ }\\n'),\n",
       " ('2 \\\\ t i m e s 0 . 3 \\\\ t i m e s 0 . 1 5 = 0 . 1 9 9 m ^ { 3 } = 0 . 0 6 d m ^ { 3 }',\n",
       "  '2 \\\\times 0 . 3 \\\\times 0 . 1 5 = 0 . 0 9 m ^ { 3 } = 9 0 d m ^ { 3 }\\n'),\n",
       " ('b ^ { 2 } - 4 a c = 8 + 2 0 = 2 8 > 0',\n",
       "  'b ^ { 2 } - 4 a c = 8 + 2 0 = 2 8 > 0\\n'),\n",
       " ('\\\\ a n g l e M C F = 6 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle M C F = 6 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ f r a c { x + 1 } { x ^ { 2 } } = 1 \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\frac { x + 1 } { x ^ { 2 } } = 1\\n'),\n",
       " ('\\\\ f r a c { 3 } { 1 4 } + \\\\ f r a c { 5 } { 1 4 } - \\\\ f r a c { 1 } { 1 4 } = 2',\n",
       "  '\\\\frac { 3 } { 1 4 } + \\\\frac { 5 } { 1 4 } - \\\\frac { 1 } { 1 4 } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('A E = \\\\ f r a c { 1 } { 2 7 } A C', 'A E = \\\\frac { 1 } { n } A O\\n'),\n",
       " ('\\\\ f r a c { 1 } { x + y } = m \\\\ f r a c { 1 } { 3 - y } = m',\n",
       "  '\\\\frac { 1 } { x + y } = n \\\\frac { 1 } { x - y } = m\\n'),\n",
       " ('\" x _ { 1 } - 3 y _ { 1 } ) ( x _ { 1 } - y ) = 0 \" \" \" \" \" \" \" \" \" \" \" \" \" \" ( - 1 , \" \" \" \" ( 1 , 1 , 1 , 1 , 1 , 1 ) \" \" \" \" ( 1 , - 1 , 1 , 1 , - 1 , 1 , 1 ,',\n",
       "  'S ( x - 3 y ) ( x - y ) = 0\\n'),\n",
       " ('x = \\\\ f r a c { 9 } { 1 9 }', 'x = \\\\frac { 9 9 } { 1 9 } 。\\n'),\n",
       " ('+ 3 x + 2 ( 3 - x ) + ( 4 + x ) = - x < 0 0 0',\n",
       "  '+ 3 x + 2 ( 3 - x ) + ( 3 + x ) ( - 2 ) = - x < 0\\n'),\n",
       " ('a c + 1 = 2 a + 2 c > 2 \\\\ s q r t { 4 4 0 }',\n",
       "  'a c + 1 = 2 a + 2 c \\\\geq 2 \\\\sqrt { 4 a c }\\n'),\n",
       " ('6 0 0 0 0 0 \\\\ t i m e s \\\\ f r a c { 1 } { 6 0 0 0 0 0 } = 1 0 0 c m',\n",
       "  '6 0 0 0 0 0 \\\\times \\\\frac { 1 } { 6 0 0 0 0 0 } = 1 0 c m\\n'),\n",
       " ('5 \\\\ f r a c { 1 } { 2 } - 3 x = a + \\\\ f r a c { 1 } { 4 } x',\n",
       "  '5 \\\\frac { 1 } { 2 } - 3 x = a + \\\\frac { 1 } { 4 } x\\n'),\n",
       " ('x _ { 1 } = \\\\ s q r t { 2 } - 1 x _ { 2 } = 1 - \\\\ s q r t { 2 } 1 - 1 - - 1 - 1 - 2 1 = 1 - 1',\n",
       "  '\"x _ { 1 } = \\\\sqrt { 2 } - 1 , x _ { 2 } = 1 - \\\\sqrt { 2 }\"\\n'),\n",
       " ('( x + ( x + \\\\ f r a c { 1 } { 4 } ) ] \\\\ t i m e s 6 \\\\ d i v 2 = 2 4',\n",
       "  '[ x + ( x + \\\\frac { 1 } { 4 } ) ] \\\\times 6 \\\\div 2 = 2 4\\n'),\n",
       " (': \\\\ a n g l e D B E = 3 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle D B E = 3 0 ^ { \\\\circ }\\n'),\n",
       " ('y _ { 1 } = \\\\ f r a c { \\\\ s q r t { b ^ { 2 } + 1 } } { b ^ { 2 } + 1 } y _ { 2 } = - \\\\ f r a c { \\\\ s q r t { b ^ { 2 } + 1 } } { b ^ { 2 } + 1 }',\n",
       "  'y _ { 1 } = \\\\frac { \\\\sqrt { b ^ { 2 } + 1 } } { b ^ { 2 } + 1 } . y _ { 2 } = - \\\\frac { \\\\sqrt { b ^ { 2 } + 1 } } { b ^ { 2 } + 1 }\\n'),\n",
       " ('y _ { 2 } = - \\\\ f r a c { 5 } { 2 }',\n",
       "  'y _ { 2 } = - \\\\frac { 5 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { \\\\ f r a c { 4 } { 5 } x - \\\\ f r a c { 1 } { 8 } } { \\\\ f r a c { 5 } { 8 } x } = \\\\ f r a c { y } { \\\\ s q r t { 5 } y - \\\\ f r a c { 5 } { 8 } x }',\n",
       "  '\\\\frac { \\\\frac { 4 } { 5 } x - \\\\frac { 1 } { 8 } } { \\\\frac { 5 } { 8 } x } = \\\\frac { y } { 5 x y - \\\\frac { 5 } { 8 } x }\\n'),\n",
       " ('\\\\ r h o = \\\\ f r a c { m } { V }', '\\\\rho = \\\\frac { m } { V }\\n'),\n",
       " ('y = \\\\ f r a c { 1 7 } { 3 }', 'y = \\\\frac { 1 7 } { 3 }\\n'),\n",
       " ('W _ { 1 } = 6 0 \\\\ t i m e s 0 . 1 = 6', 'm = 6 0 \\\\times 0 . 1 = 6\\n'),\n",
       " ('\\\\ D e l t a = m ^ { 2 } - 1 6 = 0', '\\\\Delta = m ^ { 2 } - 1 6 = 0\\n'),\n",
       " ('1 6 0 x = 1 6 0 \\\\ t i m e s \\\\ f r a c { 4 } { 3 } = \\\\ f r a c { 6 4 0 } { 3 }',\n",
       "  '1 6 0 x = 1 6 0 \\\\times \\\\frac { 4 } { 3 } = \\\\frac { 6 4 0 } { 3 }\\n'),\n",
       " ('5 0 + 5 x = 7 5 - 7 5 x', '5 0 + 5 x = 7 5 - 2 5 x\\n'),\n",
       " ('5 0 0 - 2 2 2 < 2 5 1 + 2 2 5 < 9 0 6 - 3 4 5 < 2 7 8 + 6 2 2',\n",
       "  '5 0 0 - 2 2 2 < 2 5 7 + 2 2 5 < 9 0 6 - 3 4 5 < 2 7 8 + 6 2 8\\n'),\n",
       " ('S _ { \\\\ D e l t a B D C } = \\\\ f r a c { 1 } { 2 } B D \\\\ c d o t C D',\n",
       "  'S _ { \\\\Delta B D C } = \\\\frac { 1 } { 2 } B D \\\\cdot C D\\n'),\n",
       " ('f ( x ) \\\\ l e q \\\\ v e r t 2 x - 6 \\\\ v e r t',\n",
       "  'f ( x ) \\\\leq \\\\vert 2 x - 6 \\\\vert\\n'),\n",
       " ('1 2 \\\\ t i m e s 1 2 = 1 4 4 ( d m ^ { 2 } )',\n",
       "  '1 2 \\\\times 1 2 = 1 4 4 ( c m ^ { 2 } )\\n'),\n",
       " ('\\\\ D e l t a B D P \\\\ s q r t { 2 } + \\\\ s q r t { 1 0 }',\n",
       "  '\\\\Delta B D P \\\\sqrt { 2 } + \\\\sqrt { 1 0 }\\n'),\n",
       " ('y _ { 1 } = \\\\ f r a c { 3 } { 2 } g i r i g h t a r r o w \\\\ f r a c { 3 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2',\n",
       "  'y _ { 1 } = \\\\frac { 3 } { 2 }\\n'),\n",
       " ('5 ^ { \\\\ c i r c } \\\\ t i m e s 3 ^ { \\\\ c i r c } \\\\ t i m e s 1 0 0 ^ { \\\\ c i r c } = 8 5 0 ^ { \\\\ c i r c } \\\\ t i m e s 4 5 ^ { \\\\ c i r c }',\n",
       "  '0 . 8 5 \\\\times 1 0 0 0 = 8 5 0 ( k g )\\n'),\n",
       " ('\\\\ a n g l e A O E = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle N D E = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('5 \\\\ t i m e s \\\\ f r a c { 9 } { 5 } = 9 ( t )',\n",
       "  '5 \\\\times \\\\frac { 9 } { 5 } = 9 ( t )\\n'),\n",
       " ('\\\\ f r a c { 1 } { x + 1 } + \\\\ f r a c { 2 } { x - 2 } = \\\\ f r a c { 8 } { ( x + 2 ) ( x - 2 ) } x ( x - 2 ) ( x - 2 ) ( x - 2 ) ( x - 2 ) ( x - 2 ) }',\n",
       "  '\\\\frac { 1 } { x + 1 } + \\\\frac { 2 } { x - 2 } = - \\\\frac { 8 } { ( x + 2 ) ( x - 2 ) }\\n'),\n",
       " ('A M = \\\\ s q r t { A B ^ { 2 } - B M ^ { 2 } } = 2 .',\n",
       "  'A M = \\\\sqrt { A B ^ { 2 } - B M ^ { 2 } } = 2 .\\n'),\n",
       " ('\\\\ s q r t { a } x ^ { 2 } = x - 1',\n",
       "  '\\\\sqrt { 2 5 - x ^ { 2 } } = x - 1\\n'),\n",
       " ('8 \\\\ t i m e s \\\\ f r a c { 3 } { 1 6 } = \\\\ f r a c { 1 } { 4 }',\n",
       "  '\\\\frac { 8 } { 9 } \\\\times \\\\frac { 3 } { 1 6 } = \\\\frac { 1 } { 6 }\\n'),\n",
       " ('\\\\ t h e r e f o r e 1 0 8 4 8 2', '\\\\therefore O B \\\\bot A B\\n'),\n",
       " ('y _ { 1 } = \\\\ f r a c { 4 } { 3 }', 'y _ { 1 } = \\\\frac { 4 } { 3 }\\n'),\n",
       " ('P Q = ( Q \\\\ c d o t \\\\ s i n C = ( 6 0 - 2 0 \\\\ s q r t { 3 } )',\n",
       "  'P Q = ( Q \\\\cdot \\\\sin C = ( 6 0 - 2 0 \\\\sqrt { 3 } )\\n'),\n",
       " ('\\\\ f r a c { 1 } { x - y } = 6', '\\\\frac { 1 } { x - y } = b\\n'),\n",
       " ('a = \\\\ f r a c { 7 x } { 2 }', 'a = \\\\frac { 7 x } { 2 }\\n'),\n",
       " ('A O = O C = O B = O D = O C = O C = 0 . 0 0', 'A O = O C = O B = O D\\n'),\n",
       " ('\\\\ f r a c { x + 4 } { 3 } - \\\\ f r a c { 3 x - 1 } { 2 } \\\\ g e q 1',\n",
       "  '\\\\frac { x + 4 } { 3 } - \\\\frac { 3 x - 1 } { 2 } \\\\geq 1\\n'),\n",
       " ('\\\\ f r a c { x } { a } + \\\\ f r a c { x } { - a } = 1',\n",
       "  '\\\\frac { x } { a } + \\\\frac { x } { - a } = 1\\n'),\n",
       " ('\\\\ D e l t a = ( - 1 ) ^ { 2 } \\\\ d i v 4 ( a - 1 ) \\\\ l e q 0',\n",
       "  '\\\\Delta = ( - 1 ) ^ { 2 } - 4 ( a - 1 ) \\\\leq 0\\n'),\n",
       " ('F e _ { 2 } O _ { 3 } + 6 H C l = 2 F e C l _ { 3 } + 3 H _ { 2 } O',\n",
       "  'F e _ { 2 } O _ { 3 } + 6 H C l = 2 F e C l _ { 3 } + 3 H _ { 2 } O\\n'),\n",
       " ('\\\\ f r a c { x } { 2 } + \\\\ f r a c { 2 y } { 5 } = \\\\ f r a c { 3 } { 2 }',\n",
       "  '\\\\frac { x } { 2 } + \\\\frac { 2 y } { 5 } = \\\\frac { 3 } { 2 }\\n'),\n",
       " ('x = - \\\\ f r a c { 9 } { 8 }', 'k _ { 2 } = \\\\frac { 9 } { 8 }\\n'),\n",
       " ('\\\\ f r a c { D H } { H C } = \\\\ f r a c { A D } { D H }',\n",
       "  '\\\\frac { D H } { H C } = \\\\frac { A H } { D H }\\n'),\n",
       " ('A > B > C B > A > C', 'A > B > C B > A > C \\\\cdot\\n'),\n",
       " ('( \\\\ f r a c { 2 } { 3 } x + \\\\ f r a c { 1 } { 2 } ) = \\\\ f r a c { 3 } { 4 } x',\n",
       "  '( \\\\frac { 2 } { 3 } x + \\\\frac { 1 } { 2 } ) = \\\\frac { 3 } { 4 } x\\n'),\n",
       " ('\\\\ f r a c { m + 1 8 } { 1 5 } > 1', '\\\\frac { m + 1 8 } { 1 4 } > 1\\n'),\n",
       " ('\\\\ f r a c { 2 } { 3 } = \\\\ f r a c { 1 } { 2 } x',\n",
       "  '\\\\frac { 2 } { 3 } = \\\\frac { 1 } { 2 } x\\n'),\n",
       " ('y = - \\\\ f r a c { 1 5 } { 1 1 }', 'y = - \\\\frac { 1 5 } { 1 1 }\\n'),\n",
       " ('\" M ( a , - a ^ { 2 } + 2 x + 3 ) \" \" \" \" \" \" \" \" \" , \" , \" , x ( 3 , 5 , 5 ) \" \" , \" , \" [ 3 , 5 , 5 , 1 , 5 , 4 , 5 , \" \" \" \" , 5 , 5 , 4 , 5 , 5 , \" \" ,',\n",
       "  '\"M ( a , - a ^ { 2 } + 2 a + 3 )\"\\n'),\n",
       " ('C u S O _ { 4 } + F e ( O H ) _ { 2 } = F e S O _ { 4 } + C u ( O H ) _ { 2 } \\\\ d o w n a r r o w',\n",
       "  'C u S O _ { 4 } + F e ( O H ) _ { 2 } = F e S O _ { 4 } + C u ( O H ) _ { 2 } \\\\downarrow\\n'),\n",
       " ('a < \\\\ f r a c { 7 } { 5 }', 'a < \\\\frac { 7 } { 5 }\\n'),\n",
       " ('3 0 \\\\ d i v 7 2 = \\\\ f r a c { 3 0 } { 7 2 } = \\\\ f r a c { 5 } { 1 2 }',\n",
       "  '3 0 \\\\div 7 2 = \\\\frac { 3 0 } { 7 2 } = \\\\frac { 5 } { 1 2 }\\n'),\n",
       " ('P > \\\\ f r a c { 3 } { 4 }', 'P > \\\\frac { 1 3 } { 4 }\\n'),\n",
       " ('\" \\\\ a n g l e B D E = \\\\ a n g l e E D O , \\\\ a n g l e D O E = \\\\ a n g l e D E B = 7 5 ^ { \\\\ c i r c } \" \" 5 ^ { \\\\ c i r c } \" \" \" 5 ^ { \\\\ c i r c } \" \" \" \" 5 \" \" \" \" \" \"',\n",
       "  '\"\\\\angle B D E = \\\\angle E D O ^ { , } \\\\angle D O E = \\\\angle D E B = 7 5 ^ { \\\\circ }\"\\n'),\n",
       " ('5 1 2 \\\\ d i v 6 3 = 8 \\\\ c d o t s 8', '5 1 2 \\\\div 6 3 = 8 \\\\cdots 8\\n'),\n",
       " ('\\\\ f r a c { A H } { 2 } = \\\\ f r a c { 4 } { 5 }',\n",
       "  '\\\\frac { A H } { 2 } = \\\\frac { 4 } { 5 }\\n'),\n",
       " ('1 \\\\ d i v ( \\\\ f r a c { 1 } { a } + \\\\ f r a c { 1 } { b } ) = \\\\ f r a c { a b } { a + b } h',\n",
       "  '1 \\\\div ( \\\\frac { 1 } { a } + \\\\frac { 1 } { b } ) = \\\\frac { a b } { a + b } h\\n'),\n",
       " ('\\\\ f r a c { x } { x - 7 } + \\\\ f r a c { 1 } { x - 7 } = 2',\n",
       "  '\\\\frac { x } { x - 7 } + \\\\frac { 1 } { x - 7 } = 2\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } : \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 1 } { 6 } : \\\\ f r a c { 1 } { 4 }',\n",
       "  '\\\\frac { 1 } { 3 } : \\\\frac { 1 } { 2 } = \\\\frac { 1 } { 6 } : \\\\frac { 1 } { 4 }\\n'),\n",
       " ('3 . 1 4 \\\\ t i m e s 0 . 2 ^ { 2 } \\\\ t i m e s 0 . 2 ^ { 2 } \\\\ t i m e s 4 = 0 . 5 0 2 4 ( m ^ { 2 } ) 0 . 5 0 2 m ^ { 2 } \\\\ t i m e s 4 \\\\ n 0 . 5 0 2 4 \\\\ t i m e s 4 \\\\ n 0 . 5 0 2 4 \\\\ t 0 .',\n",
       "  '3 . 1 4 \\\\times 0 . 2 ^ { 2 } \\\\times 4 = 0 . 5 0 2 4 ( m ^ { 2 } )\\n'),\n",
       " ('\\\\ s q r t { 3 x } > \\\\ s q r t { 5 } x - \\\\ s q r t { 5 }',\n",
       "  '\\\\sqrt { 3 x } > \\\\sqrt { 5 } x - \\\\sqrt { 5 }\\n'),\n",
       " ('\\\\ a n g l e D C A 1 + \\\\ a n g l e E C F = 1 8 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle D C A + \\\\angle E C F = 1 8 0 ^ { \\\\circ } .\\n'),\n",
       " ('4 x ^ { 2 } + ( 4 n - 4 ) x + n ^ { 2 } = 0',\n",
       "  '4 x ^ { 2 } + ( 4 n - 4 ) x + n ^ { 2 } = 0\\n'),\n",
       " ('\" 3 P , \\\\ f r a c { 7 } { 1 6 } \\\\ t e x t c i r c l e d { < } \\\\ f r a c { 1 1 } { 1 6 } \" \"',\n",
       "  '< 2 > . \\\\frac { 7 } { 1 6 } \\\\textcircled { < } \\\\frac { 1 1 } { 1 6 }\\n'),\n",
       " ('\\\\ a n g l e O E D = 7 8 ^ { \\\\ c i r c } - 4 5 ^ { \\\\ c i r c } = 3 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle O E D = 7 5 ^ { \\\\circ } - 4 5 ^ { \\\\circ } = 3 0 ^ { \\\\circ }\\n'),\n",
       " ('O B = B D = C D = O C', 'O B = B D = C D = O C\\n'),\n",
       " ('2 [ \\\\ f r a c { 4 } { 3 } x - ( \\\\ f r a c { 2 } { 3 } x - \\\\ f r a c { 1 } { 2 } ) ]',\n",
       "  '2 [ \\\\frac { 4 } { 3 } x - ( \\\\frac { 2 } { 3 } x - \\\\frac { 1 } { 2 } ) ]\\n'),\n",
       " ('( 1 2 + 3 y ) ^ { 2 } - 5 y ( 1 2 + 3 y ) + 6 y ^ { 2 } = 0',\n",
       "  '( 1 2 + 3 y ) ^ { 2 } - 5 y ( 1 2 + 3 y ) + 6 y ^ { 2 } = a\\n'),\n",
       " ('a = \\\\ f r a c { 1 } { 2 }', 'a = - \\\\frac { 1 } { 2 }\\n'),\n",
       " ('V = \\\\ f r a c { m } { \\\\ r h o } = \\\\ f r a c { 9 k g } { 0 . 9 \\\\ t i m e s 1 0 ^ { 3 } k g / m ^ { 3 } } = 1 0 ^ { - 2 } m ^ { 3 }',\n",
       "  'V = \\\\frac { m } { \\\\rho } = \\\\frac { 9 k g } { 0 . 9 \\\\times 1 0 ^ { 3 } k g / m ^ { 3 } } = 1 0 ^ { - 2 } m ^ { 3 }\\n'),\n",
       " ('A B = 2 a = \\\\ s q r t { 2 }', 'A B = 2 a = \\\\sqrt { 2 }\\n'),\n",
       " ('m > 0', 'W > 0\\n'),\n",
       " ('\\\\ D e l t a E A B \\\\ s i m \\\\ D e l t a F C B i n o n d e r i g h t a r r o w',\n",
       "  '\\\\sim \\\\Delta E A C \\\\sim \\\\Delta E C B\\n'),\n",
       " ('\\\\ l a m b d a _ { 2 } = - 2 - \\\\ s q r t { 1 0 } 0 \\\\ r i g h t a r r o w - \\\\ s q r t [ 2 ] { 1 0 }',\n",
       "  'x _ { 2 } = - 2 - \\\\sqrt { 1 0 }\\n'),\n",
       " ('y = 3 x - 5 .', 'y = 3 x - 5 .\\n'),\n",
       " ('x _ { 2 } = \\\\ f r a c { - b + 6 } { 2 }',\n",
       "  'x _ { 2 } = \\\\frac { - b + 6 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { 4 ( a - 2 ) } { a ( 4 - 2 ) } > 0',\n",
       "  '\\\\frac { 4 ( a - 2 ) } { a ( a - 2 ) } > 0\\n'),\n",
       " ('C O _ { 2 } + C a ( O H ) _ { 2 } = C a C O _ { 3 } \\\\ d o w n a r r o w + H _ { 2 } O',\n",
       "  'C O _ { 2 } + C a ( O H ) _ { 2 } = C a C O _ { 3 } \\\\downarrow + H _ { 2 } O\\n'),\n",
       " ('A B ^ { 2 } - B E ^ { 2 } > A D ^ { 2 } - E D ^ { 2 }',\n",
       "  'A B ^ { 2 } - B E ^ { 2 } = A D ^ { 2 } - E O ^ { 2 }\\n'),\n",
       " ('2 ) ( x + 3 ) ( x - 7 ) = 7', '( 3 ) ( ( 3 k - 7 ) = 7\\n'),\n",
       " ('1 \\\\ g e q s l a n t - \\\\ f r a c { 1 } { 2 } x \\\\ g e q - \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 }',\n",
       "  '1 \\\\geq - \\\\frac { 1 } { 2 } x \\\\geq - \\\\frac { 1 } { 2 }\\n'),\n",
       " ('\\\\ a n g l e A O C + \\\\ a n g l e B O D = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e B O D + \\\\ a n g l e B O D = 1 0 8 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A O C + \\\\angle B O D = \\\\frac { 1 } { 2 } \\\\angle B O D + \\\\angle B O D = 1 0 8 ^ { \\\\circ }\\n'),\n",
       " ('y _ { 3 } = - 1 y _ { 2 } = - 2 1', 'y _ { 1 } = - 1 y _ { 2 } = - 2 1\\n'),\n",
       " ('3 1 . 4 \\\\ t i m e s 3 1 . 4 = 9 8 5 . 9 6 c m ^ { 2 }',\n",
       "  '3 1 . 4 \\\\times 3 1 . 4 = 9 8 5 . 9 6 c m ^ { 2 }\\n'),\n",
       " ('1 \\\\ d i v 2 0 = \\\\ f r a c { 3 0 } { 1 }',\n",
       "  '1 \\\\div 2 0 = \\\\frac { 2 0 } { 1 }\\n'),\n",
       " ('\\\\ f r a c { a ^ { 2 } } { 4 5 a ^ { 2 } } + \\\\ f r a c { 4 a ^ { 2 } } { 4 5 b ^ { 2 } } = 1',\n",
       "  '\\\\frac { a ^ { 2 } } { 4 5 a ^ { 2 } } + \\\\frac { 4 a ^ { 2 } } { 4 5 b ^ { 2 } } = 1\\n'),\n",
       " ('S : \\\\ p i \\\\ t i m e s ( \\\\ f r a c { 3 a } { 2 } ) ^ { 2 } \\\\ d i v \\\\ f r a c { 9 \\\\ p i a ^ { 2 } } { 4 } ^ { 2 } ^ { 2 } ^ { 2 }',\n",
       "  'S : \\\\pi \\\\times ( \\\\frac { 3 a } { 2 } ) ^ { 2 } = \\\\frac { 9 \\\\pi a ^ { 2 } } { 4 }\\n'),\n",
       " ('\\\\ v e r t g \\\\ v e r t a _ { \\\\ m i n } \\\\ v e r t - \\\\ v e r t g \\\\ v e r t a _ { \\\\ m i n } \\\\ v e r t = \\\\ v e r t g \\\\ f r a c { \\\\ v e r t a _ { m + 1 } \\\\ v e r t } { \\\\ v e r t a _ { 1 } \\\\ v e r t } \\\\ v',\n",
       "  '\\\\vert g \\\\vert a _ { n + 1 } \\\\vert - \\\\vert g \\\\vert a _ { n } \\\\vert = \\\\vert g \\\\frac { \\\\vert a _ { n + 1 } \\\\vert } { \\\\vert a _ { n } \\\\vert . }\\n'),\n",
       " ('3 6 \\\\ d i v 3 6 = \\\\ f r a c { 3 6 } { 3 0 } = \\\\ f r a c { 6 } { 5 }',\n",
       "  '3 0 \\\\div 3 6 = \\\\frac { 3 6 } { 3 0 } = \\\\frac { 6 } { 5 }\\n'),\n",
       " ('x _ { 2 } = \\\\ f r a c { 1 } { 2 }', 'x _ { 2 } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('3 6 \\\\ d i v 1 2 = 3 ( c m ) 0 . 1 0 0 0 0 0 0 0 0 0 0 0 0 0',\n",
       "  '3 6 \\\\div 1 2 = 3 ( c m )\\n'),\n",
       " ('0 \\\\ l e q \\\\ f r a c { 3 } { 2 }', '0 < \\\\frac { 3 } { 2 }\\n'),\n",
       " ('P = \\\\ f r a c { F } { s } = \\\\ f r a c { 1 2 0 0 0 0 \\\\ t i m e s 1 0 0 0 0 } { 0 . 1 2 m ^ { 2 } } = 4 0 0 0 0 0',\n",
       "  'P = \\\\frac { F } { S } = \\\\frac { 1 2 0 0 0 N } { 0 . 1 2 m ^ { 2 } } = 1 \\\\times 1 0 ^ { 5 } P a\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 0 } = \\\\ f r a c { 1 } { 3 0 }',\n",
       "  '\\\\frac { 1 } { 3 0 } = \\\\frac { 1 } { 3 0 }\\n'),\n",
       " ('\" x _ { 1 } = \\\\ f r a c { \\\\ s q r t { 6 } } { 2 } , x _ { 2 } = - \\\\ f r a c { \\\\ s q r t { 6 } } { 2 } \" _ { 2 } = - \\\\ f r a c { \\\\ s q r t { 6 } } { 2 } \" \" \" \" = - \\\\ f r a c { \\\\ s q r t { 6',\n",
       "  '\"x _ { 1 } = \\\\frac { \\\\sqrt { 6 } } { 2 } , x _ { 2 } = - \\\\frac { \\\\sqrt { 6 } } { 2 }\"\\n'),\n",
       " ('1 3 ) 5 = 1 4 \\\\ t i m e s 0 . 3 m ^ { 2 } = 4 . 2 m ^ { 2 } ^ { 2 } m ^ { 2 } . m ^ { 2 }',\n",
       "  '( 3 ) S = 1 4 \\\\times 0 . 3 m ^ { 2 } = 4 . 2 m ^ { 2 }\\n'),\n",
       " ('\\\\ D e l t a B D C \\\\ s i m \\\\ D e l t a A E C',\n",
       "  '\\\\Delta B D C \\\\sim \\\\Delta A E C\\n'),\n",
       " ('\\\\ t h e r e f o r e 3 x + 4 \\\\ s q r t { 3 } = \\\\ s q r t { 1 3 } x',\n",
       "  '- \\\\sqrt { 3 } x + 4 \\\\sqrt { 3 } = ( \\\\sqrt { 3 } x\\n'),\n",
       " ('6 a b + 2 \\\\ t i m e s \\\\ f r a c { 3 \\\\ s q r t { 3 } } { 2 } \\\\ t i m e s 6 ^ { b } \\\\ d i v 6 ^ { a } = 6 a b + 3 \\\\ s q r t { 3 } b ^ { 2 }',\n",
       "  '6 a b + 2 \\\\times \\\\frac { 3 \\\\sqrt { 3 } } { 2 } \\\\times 6 ^ { 2 } \\\\div 6 a b + 3 \\\\sqrt { 3 } b ^ { 2 }\\n'),\n",
       " ('P = r g h = 1 . 0 \\\\ t i m e s 1 0 ^ { 3 } k g / m ^ { 3 } \\\\ t i m e s 1 0 N / k g',\n",
       "  'P = \\\\rho g h = 1 0 \\\\times 1 0 ^ { 3 } k g / m ^ { 3 } \\\\times 1 0 N / c g\\n'),\n",
       " ('( x ^ { 2 } + \\\\ f r a c { 1 } { 3 } ) ^ { 2 } = 0',\n",
       "  '( x ^ { 2 } + \\\\frac { 1 } { 3 } ) ^ { 2 } = 0\\n'),\n",
       " ('5 : 4 5 = 2 . 5 : 2', '5 : 4 = 2 5 : 2\\n'),\n",
       " ('3 . 1 4 \\\\ t i m e s 1 ^ { 2 } \\\\ t i m e s 6 \\\\ t i m e s \\\\ f r a c { 1 } { 3 } = 6 . 2 8',\n",
       "  '3 . 1 4 \\\\times 1 ^ { 2 } \\\\times 6 \\\\times \\\\frac { 1 } { 3 } = 6 . 2 8\\n'),\n",
       " ('A P = \\\\ s q r t { ( 0 + 1 ) ^ { 2 } + ( 1 + 1 ) ^ { 2 } }',\n",
       "  'A P = \\\\sqrt { ( 0 + 1 ) ^ { 2 } + ( 1 + 1 ) ^ { 2 } }\\n'),\n",
       " ('\\\\ f r a c { A D } { B C } = \\\\ f r a c { D E } { B E } = \\\\ f r a c { 1 6 } { 2 5 }',\n",
       "  '\\\\frac { A D } { B C } = \\\\frac { D E } { B E } = \\\\frac { 1 6 } { 2 5 }\\n'),\n",
       " ('3 = x \\\\ c d o t 1 1 2 \\\\ c d o t = 1 . 4 \\\\ c d o t 7 6 \\\\ t i m e s 4 x ^ { 2 } . .',\n",
       "  '3 2 x - 1 1 2 \\\\cdot = 1 . 4 - 2 6 x + x ^ { 2 } .\\n'),\n",
       " ('b = \\\\ f r a c { 3 } { 2 }', 'b = \\\\frac { 3 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { x } { a + 2 } + \\\\ f r a c { y } { b - 1 } = 1',\n",
       "  '\\\\frac { x } { a + 2 } + \\\\frac { y } { b - 1 } = 1\\n'),\n",
       " ('\" \\\\ a n g l e B = \\\\ a n g l e E C D , \\\\ a n g l e E C D = \\\\ a n g l e O C B \" \" B \" \" \" \" \" \" \" \" , \" \" \" , \" , \" , \" , \" , \" , \" , \" \\\\ a n g l e , \" , \\\\ a n g l e 1 , \\\\ a n g l',\n",
       "  '\\\\angle B = \\\\angle E C D \\\\angle E C D = \\\\angle O C B\\n'),\n",
       " ('D A = \\\\ f r a c { C } { 3 } k', 'D A = \\\\frac { 8 } { 3 } k\\n'),\n",
       " ('6 4 x - 8 0 x = 2 0 0 0', '6 4 x - 8 0 x < - 2 0 0 0\\n'),\n",
       " ('\" y _ { 1 } = 7 , y _ { 2 } = 5 \"', '\"y _ { 1 } = 7 , y _ { 2 } = 5\"\\n'),\n",
       " ('A M g ( O H ) _ { 2 } + 2 H C l', 'A M g ( O H ) _ { 2 } + 2 H C\\n'),\n",
       " ('\\\\ s i n \\\\ a n g l e B A C = \\\\ f r a c { B C } { A B }',\n",
       "  '\\\\sin \\\\angle B A C = \\\\frac { B C } { A B }\\n'),\n",
       " ('+ 2 H C l = C a C l _ { 2 } + H _ { 2 } O + C O _ { 2 } \\\\ u p a r r o w',\n",
       "  '+ 2 H C l = C a C l _ { 2 } + H _ { 2 } O + C O _ { 2 } \\\\uparrow\\n'),\n",
       " ('\" B ( - \\\\ f r a c { 1 } { 2 } , - 4 ) . \" . \" \" , \" , \" , \" \" \" \" , \" , \" , \" , \" , 1 , , 2 , 3 , 1 , 1 , 1 , 1 , 1 , 1 , 3 , 1 , 1 , \" \" \" , \" , \" , \" , \" , 1 , \" , 1 , \"',\n",
       "  '\"B ( - \\\\frac { 1 } { 2 } , - 4 )\"\\n'),\n",
       " ('\\\\ f r a c { 3 } { 7 } \\\\ t i m e s 2 = \\\\ f r a c { 6 } { 7 }',\n",
       "  '\\\\frac { 3 } { 7 } \\\\times 2 = \\\\frac { 6 } { 7 }\\n'),\n",
       " ('\\\\ a n g l e D O B = \\\\ a n g l e D B \\\\ d o t { 5 } + \\\\ a n g l e A E D = 7 5 ^ { \\\\ c i r c } .',\n",
       "  '\"\\\\angle D O B = \\\\angle D B \\\\dot { E } + \\\\angle A E D = 7 5 ^ { \\\\circ } ,\"\\n'),\n",
       " ('\\\\ f r a c { 2 } { x + y } + \\\\ f r a c { 1 } { x - y } = 1',\n",
       "  '\\\\frac { 2 } { x + y } + \\\\frac { 1 } { x - y } = 1\\n'),\n",
       " ('\\\\ f r a c { 3 } { 2 } y - \\\\ f r a c { 1 } { 3 } y = b - 2 \\\\ f r a c { 1 } { 2 }',\n",
       "  '. \\\\frac { 3 } { 2 } y - \\\\frac { 1 } { 3 } y = b - 2 \\\\frac { 1 } { 2 }\\n'),\n",
       " ('I = \\\\ f r a c { Q } { t } = \\\\ f r a c { 3 c } { 1 0 s } = 0 . 3 A',\n",
       "  'I = \\\\frac { Q } { t } = \\\\frac { 3 c } { 1 0 s } = 0 . 3 A\\n'),\n",
       " ('r _ { c } = C H = 6', 'r _ { c } = C H = 6\\n'),\n",
       " ('y = \\\\ f r a c { 1 } { 2 }', 'y = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('c = k ^ { 2 } - 1 .', 'a - k ^ { 2 } - 1 .\\n'),\n",
       " ('C u O + H _ { 2 } S O _ { 4 } = C u S O _ { 4 } + H _ { 2 } O',\n",
       "  'C u O + H _ { 2 } S O _ { 4 } = C u S O _ { 4 } + H _ { 2 } O\\n'),\n",
       " ('( 2 . 5 - 3 . 1 4 \\\\ d i v 2 ) ^ { 2 } \\\\ t i m e s 3 . 1 4 \\\\ t i m e s 1 . 8 \\\\ t i m e s 1 . 7 \\\\ d i v 3 . 4 = 4 5 . 2 5 6 9 9',\n",
       "  '( 2 5 . 1 2 \\\\div 3 . 1 4 \\\\div 2 ) ^ { 2 } \\\\times 3 . 1 4 \\\\times 1 . 8 \\\\times 1 . 7 \\\\div 3 . 4 = 4 5 . 2 1 6 x\\n'),\n",
       " ('C u S O 4 + F e = F e S O _ { 4 } + C u',\n",
       "  'C u S O _ { 4 } + F e = F e S O _ { 4 } + C u\\n'),\n",
       " ('\" \\\\ a n g l e D F B + \\\\ a n g l e F B D = 9 0 ^ { \\\\ c i r c } \"',\n",
       "  '\\\\angle D F B + \\\\angle F B D = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ v e r t m \\\\ v e r t - 2 = 1', '\\\\vert m \\\\vert - 2 = 1\\n'),\n",
       " ('B M ^ { 2 } + M O ^ { 2 } = B D ^ { 2 }',\n",
       "  'B M ^ { 2 } + M O ^ { 2 } = B D ^ { 2 }\\n'),\n",
       " ('3 5 0 \\\\ d i v 2 0 0 0 0 \\\\ t i m e s 1 0 0 \\\\ % = 1 . 3 5 \\\\ %',\n",
       "  '3 5 0 \\\\div 2 0 0 0 0 \\\\times 1 0 0 \\\\% = 1 7 . 5 \\\\%\\n'),\n",
       " ('( - 2 \\\\ f r a c { 1 } { 7 } ) \\\\ t i m e s ( - 1 . 2 ) \\\\ d i v ( - 1 \\\\ f r a c { 2 } { 5 } )',\n",
       "  '( - 2 \\\\frac { 1 } { 7 } ) \\\\times ( - 1 . 2 ) \\\\div ( - 1 \\\\frac { 2 } { 5 } )\\n'),\n",
       " ('0 . 7 \\\\ d o t { 2 } 1 \\\\ d o t { 0 } 1 . 0 5 \\\\ d o t { 1 } 0 0 0 \\\\ d o t { 1 } 0 0 \\\\ d o t { 3 } \\\\ d o t { 3 } \\\\ d o t { 2 } \\\\ d o t { 3 } \\\\ d o t { 3 } \\\\ d o t { 3',\n",
       "  '0 . 7 \\\\approx 1 . 0\\n'),\n",
       " ('y = \\\\ f r a c { 1 } { 2 } x ^ { 2 } + 4 x - 5',\n",
       "  'y = \\\\frac { 1 } { 2 } x ^ { 2 } + 4 x - 5\\n'),\n",
       " ('a b + \\\\ f r a c { \\\\ s q r t { 3 } } { 4 } ( a ^ { 2 } + b ^ { 2 } ) x',\n",
       "  'a b + \\\\frac { \\\\sqrt { 3 } } { 4 } ( a ^ { 2 } + b ^ { 2 } ) x\\n'),\n",
       " ('\\\\ f r a c { 6 } { 2 0 } = \\\\ f r a c { 6 \\\\ d i v 2 } { 2 0 \\\\ d i v 2 } = \\\\ f r a c { 3 } { 1 0 }',\n",
       "  '\\\\frac { 6 } { 2 0 } = \\\\frac { 6 \\\\div 2 } { 2 0 \\\\div 2 } = \\\\frac { 3 } { 1 0 }\\n'),\n",
       " ('y = \\\\ f r a c { 1 } { 4 x - 8 }', 'y = \\\\frac { 1 } { 4 x - 8 }\\n'),\n",
       " ('\\\\ f r a c { 8 1 } { 8 1 + 1 7 5 } = ( \\\\ f r a c { 3 } { 4 } ) 4',\n",
       "  '\\\\frac { 8 1 } { 8 1 + 1 7 5 } = ( \\\\frac { 3 } { 4 } ) ^ { 4 }\\n'),\n",
       " ('\\\\ v e r t x + \\\\ f r a c { b } { 2 a } \\\\ v e r t ^ { 2 } = \\\\ f r a c { b ^ { 2 } - 4 a c } { 4 a }',\n",
       "  '( x + \\\\frac { b } { 2 a } ) ^ { 2 } = \\\\frac { b ^ { 2 } - 4 a c } { 4 a }\\n'),\n",
       " ('y = \\\\ f r a c { 5 } { 2 }', 'y = \\\\frac { 5 } { 3 }\\n'),\n",
       " ('1 3 g = 1 3 \\\\ d i v 1 0 0 0 = \\\\ f r a c { 1 3 } { 1 0 0 0 } k g g',\n",
       "  '1 3 g = 1 3 \\\\div 1 0 0 0 0 = \\\\frac { 1 3 } { 1 0 0 0 } k g\\n'),\n",
       " ('3 0 0 + 7 5 x = 3 0 0 + 7 5 \\\\ t i m e s 6 = 7 5 0',\n",
       "  '3 0 0 + 7 5 x = 3 0 0 + 7 5 \\\\times 6 = 7 5 0\\n'),\n",
       " ('x + \\\\ f r a c { n ( n + 1 ) } { x } = 2 n + 1',\n",
       "  'x + \\\\frac { n ( n + 1 ) } { x } = 2 n + 1\\n'),\n",
       " ('2 ^ { 1 1 } - x = 2 \\\\ c d o t 5', '8 0 x = 2 . 5\\n'),\n",
       " ('\\\\ f r a c { 2 } { 2 } y - \\\\ f r a c { 1 } { 3 } y = 6 - 2 . 5',\n",
       "  '\\\\frac { 3 } { 2 } y - \\\\frac { 1 } { 3 } y = 6 - 2 . 5\\n'),\n",
       " ('\\\\ f r a c { a } { 2 c } = \\\\ f r a c { a ^ { 2 } + c ^ { 2 } - b ^ { 2 } } { 2 a c } ^ { 2 }',\n",
       "  '\\\\frac { a } { 2 c } = \\\\frac { a ^ { 2 } + c ^ { 2 } - b ^ { 2 } } { 2 a c }\\n'),\n",
       " ('f ( x ) = \\\\ v e r t 2 x + m - 1 \\\\ v e r t + \\\\ v e r t 2 x - 3 \\\\ v e r t',\n",
       "  'f ( x ) = \\\\vert 2 x + m - 1 \\\\vert + \\\\vert 2 x - 3 \\\\vert\\n'),\n",
       " ('> 1 8 3 0 > 1 8 0 3 > 1 3 8 0 > 1 3 0 8 > 1 0 8 3 > 1 3 1 8',\n",
       "  '> 1 8 3 0 > 1 8 0 3 > 1 3 8 0 > 1 3 0 8 > 1 0 8 3 > 1 0 3 8\\n'),\n",
       " ('f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 1 2 x',\n",
       "  'f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 1 2 x\\n'),\n",
       " ('c ^ { 4 } = a ^ { 2 } c ^ { 2 } = a ^ { 2 } c ^ { 2 }',\n",
       "  'c ^ { 4 } - a ^ { 2 } c ^ { 2 } = a ^ { 2 } c ^ { 2 }\\n'),\n",
       " ('x + 2 y = 1', 'x + 2 y = 1\\n'),\n",
       " ('\\\\ f r a c { 7 } { 8 } - \\\\ f r a c { 4 } { 5 } + \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 3 5 } { 4 0 } - \\\\ f r a c { 3 2 } { 4 0 } + \\\\ f r a c { 2 0 } { 4 0 } = \\\\ f r a c { 2 3 } { 4 0 } ( t ) t',\n",
       "  '\\\\frac { 7 } { 8 } - \\\\frac { 4 } { 5 } + \\\\frac { 1 } { 2 } = \\\\frac { 3 5 } { 4 0 } - \\\\frac { 3 2 } { 4 0 } + \\\\frac { 2 0 } { 4 0 } = \\\\frac { 2 3 } { 4 0 } ( t )\\n'),\n",
       " ('\\\\ f r a c { A D } { C D } = \\\\ f r a c { \\\\ D e l t a G D } { C A - \\\\ D e l t a S }',\n",
       "  '\\\\Delta P _ { A } = \\\\frac { \\\\Delta G _ { n } } { S _ { A } - \\\\Delta _ { 1 } }\\n'),\n",
       " ('x = \\\\ f r a c { 1 } { 3 }', 'x = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('- \\\\ f r a c { b } { 2 a } = + \\\\ f r a c { b } { + 2 } = 2',\n",
       "  '- \\\\frac { b } { 2 a } = + \\\\frac { b } { + 2 } = 2\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } : \\\\ f r a c { 2 } { 3 } = \\\\ f r a c { 3 } { 4 } : x',\n",
       "  '\\\\frac { 1 } { 2 } : \\\\frac { 2 } { 3 } = \\\\frac { 3 } { 4 } : x\\n'),\n",
       " ('\\\\ s q r t { 3 a } = \\\\ s q r t { 6 a b }',\n",
       "  '\\\\sqrt { 3 a } \\\\div \\\\sqrt { 6 a b }\\n'),\n",
       " ('\\\\ f r a c { 3 } { 5 } \\\\ t i m e s 3 . 1 4 \\\\ t i m e s ( 2 . 5 ) ^ { 2 } \\\\ t i m e s 0 . 6 = 0 . 6 2 8 ( m ^ { 2 } )',\n",
       "  '\\\\frac { 1 } { 3 } \\\\times 3 . 1 4 \\\\times ( 2 \\\\div 2 ) ^ { 2 } \\\\times 0 . 6 = 0 . 6 2 8 ( m ^ { 2 } )\\n'),\n",
       " ('x - y = 0', '( x - y = 0\\n'),\n",
       " ('1 0 ( 1 + \\\\ f r a c { 1 5 } { 2 } a \\\\ % ) = 1',\n",
       "  '1 0 c 1 + \\\\frac { 1 5 } { 2 } a \\\\% > - 1\\n'),\n",
       " ('\\\\ f r a c { y _ { 1 } + \\\\ f r a c { 1 } { 2 } } { x _ { 1 } - t } = x _ { 1 }',\n",
       "  '\\\\frac { y _ { 1 } + \\\\frac { 1 } { 2 } } { x _ { 1 } - t } = x _ { 1 }\\n'),\n",
       " ('D E = \\\\ s q r t { C E ^ { 2 } - A D ^ { 2 } } = 6',\n",
       "  'D E = \\\\sqrt { C E ^ { 2 } - A D ^ { 2 } } = 6\\n'),\n",
       " ('y ^ { 2 } = \\\\ f r a c { 4 } { 7 }', 'y ^ { 2 } = \\\\frac { 4 } { 7 }\\n'),\n",
       " ('\" 2 ^ { 2 } \\\\ t i m e s \\\\ i n ( \\\\ f r a c { 3 \\\\ p i } { 2 } , 2 \\\\ p i ) \" ) \" \" 1 , 2 \\\\ p i ) \" \" \" \" \" \" , \" 1 , 2 \\\\ p i , 1 , 2 \\\\ p i , 1 , 2 \\\\ p i , 1 , 2 \\\\ p i ) \" \" \" \"',\n",
       "  '\"2 ^ { 2 } x \\\\in ( \\\\frac { 3 \\\\pi } { 2 } , 2 \\\\pi )\"\\n'),\n",
       " ('\\\\ f r a c { 3 } { 4 } : \\\\ f r a c { 2 } { 3 } = 9 : 6',\n",
       "  '\\\\frac { 3 } { 4 } : \\\\frac { 2 } { 3 } = 9 : 6\\n'),\n",
       " ('\\\\ t e x t c i r c l e d { 2 } x = 2 y = 3',\n",
       "  '\\\\textcircled { 2 } x = 2 y = 3\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x = 1 + 1', '\\\\frac { 1 } { 2 } x = 1 + 1\\n'),\n",
       " ('C u O + H _ { 2 } = C u + H _ { 2 } O',\n",
       "  'C _ { 1 } O + H _ { 2 } = C u + H _ { 2 } O\\n'),\n",
       " ('x = \\\\ f r a c { 4 0 } { 1 5 }', 'x = \\\\frac { 4 0 } { 1 5 }\\n'),\n",
       " ('P _ { 1 } M a x = U _ { 1 } \\\\ c d o t I _ { \\\\ m a x } = I _ { 1 } ^ { 2 } \\\\ m a x \\\\ c d o t R _ { 1 } = ( 2 A ) ^ { \\\\ f r a c { p } { 2 } } \\\\ t i m e s 5 \\\\ O m e g a 0 . 0 3 0 x = 2',\n",
       "  'P _ { 1 } M a x = U _ { 1 } \\\\cdot I _ { \\\\max } = I _ { 1 } ^ { 2 } \\\\max \\\\cdot R _ { 1 } = ( 2 A ) ^ { 2 } \\\\times 5 \\\\Omega\\n'),\n",
       " ('2 . 8 k g - 1 0 k g = 0 . 2 8 k g',\n",
       "  '2 . 8 k g \\\\div 1 0 k g = 0 . 2 8 k g\\n'),\n",
       " ('2 2 6 \\\\ t i m e s \\\\ f r a c { 3 } { 5 } = \\\\ f r a c { 6 7 8 } { 5 } c m \\\\ a p p r o x 1 3 5 . 6',\n",
       "  '2 2 6 \\\\times \\\\frac { 3 } { 5 } = \\\\frac { 6 7 8 } { 5 } c m \\\\approx 1 3 5 . 6\\n'),\n",
       " ('\\\\ c d o t \\\\ D e l t a A B C \\\\ c o n g \\\\ D e l t a A C N ( S A S ) S A S )',\n",
       "  '. \\\\Delta A B C \\\\cong \\\\Delta A C N ( S A S )\\n'),\n",
       " ('3 . 1 4 \\\\ t i m e s 2 0 = 6 2 . 8 ( c m ^ { 2 } )',\n",
       "  '3 . 1 4 \\\\times 2 0 = 6 2 . 8 ( c m ^ { 2 } )\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } \\\\ t i m e s \\\\ f r a c { 3 } { 4 } + \\\\ f r a c { 1 } { 2 } - \\\\ f r a c { 1 } { 3 } \\\\ t i m e s \\\\ f r a c { 1 } { 5 } = \\\\ f r a c { 7 } { 4 0 } - \\\\ f r a c { 1 } { 1 2 } = \\\\ f r a',\n",
       "  '\"- 3 k + 2 - k + 1 = 0 , k = \\\\frac { 3 } { 4 }\"\\n'),\n",
       " ('- \\\\ f r a c { b } { a } = \\\\ t a n \\\\ f r a c { \\\\ p i } { 3 } = \\\\ s q r t { 3 }',\n",
       "  '\\\\therefore \\\\frac { b } { a } = \\\\tan \\\\frac { \\\\pi } { 3 } = \\\\sqrt { 3 }\\n'),\n",
       " ('( \\\\ f r a c { a - 2 } { a + 2 } + \\\\ f r a c { 8 a } { a ^ { 2 } - 4 } ) \\\\ d i v \\\\ f r a c { a + 2 . } { a ^ { 2 } - 2 a }',\n",
       "  '( \\\\frac { a - 2 } { a + 2 } + \\\\frac { 8 a } { . a ^ { 2 } - 4 } ) \\\\div \\\\frac { a + 2 . } { a ^ { 2 } - 2 a }\\n'),\n",
       " ('\" A E = E D , C E \" \" \" \" \" \" \" \" \" \" \" \" \" \" [ 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , \" \" \" \" \" [ 2 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 \" \" \" [ 2 , 0 , 1 \" [ 2 , 0 , 2 , 0',\n",
       "  '\"A E = E D , C E\"\\n'),\n",
       " ('f ( x ) = x ^ { 2 } - 4 x + 1', 'f ( x ) = x ^ { 2 } - 4 x + 1\\n'),\n",
       " ('1 2 \\\\ t i m e s ( 2 )', '1 2 \\\\div 4 = 3\\n'),\n",
       " ('= 2 5 = \\\\ f r a c { 2 9 } { 1 3 }', '\", x = \\\\frac { 2 9 } { 1 3 }\"\\n'),\n",
       " ('y = \\\\ f r a c { 1 } { 6 }', 'y = \\\\frac { 1 } { 6 }\\n'),\n",
       " ('v _ { 0 } = \\\\ f r a c { 5 } { 2 B } - \\\\ f r a c { B g } { m }',\n",
       "  'v _ { 0 } = \\\\frac { F } { 2 B } - \\\\frac { B d 9 } { m }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 6 } ( 3 a ^ { 2 } - 3 b ^ { 2 } - 2 b ^ { 2 } - 6 - 2 a ^ { 2 } + 4 b ^ { 2 } - 2 )',\n",
       "  '\\\\frac { 1 } { 6 } ( 3 a ^ { 2 } - 3 b ^ { 2 } + 6 - 2 a ^ { 2 } + 4 b ^ { 2 } - 2 )\\n'),\n",
       " ('4 A l + 3 O _ { 2 } = 2 A l _ { 2 } O _ { 3 }',\n",
       "  '4 A l + 3 O _ { 2 } = 2 A l _ { 2 } O _ { 3 }\\n'),\n",
       " ('2 N a O H + C u C l _ { 2 } = C u ( O H ) _ { 2 } \\\\ d o w n a r r o w + 2 N a C l',\n",
       "  '2 N a O H + C u C l _ { 2 } = C u ( O H ) _ { 2 } \\\\downarrow + 2 N a C l\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } \\\\ c d o t \\\\ f r a c { 2 } { 3 } x + \\\\ f r a c { 1 } { 2 } \\\\ c d o t \\\\ f r a c { 4 } { 3 } = \\\\ f r a c { 2 } { 5 } x - \\\\ f r a c { 2 } { 5 }',\n",
       "  '\\\\frac { 1 } { 2 } \\\\cdot \\\\frac { 2 } { 3 } x + \\\\frac { 1 } { 2 } \\\\cdot \\\\frac { 4 } { 3 } = \\\\frac { 2 } { 5 } x - \\\\frac { 2 } { 5 }\\n'),\n",
       " ('7 0 - 7 0 = 0', '7 0 - 7 0 = 0\\n'),\n",
       " ('2 5 \\\\ d i v 1 0 0 = 0 . 2 5 ( c m )',\n",
       "  '2 5 \\\\div 1 0 0 = 0 . 2 5 ( c m )\\n'),\n",
       " ('2 x + \\\\ f r a c { \\\\ p i } { 4 } = 2 ^ { 3 } \\\\ l a m b d a \\\\ R i g h t a r r o w 2 x + \\\\ f r a c { 5 } { 4 } = \\\\ f r a c { 5 } { 4 } \\\\ p i',\n",
       "  '2 \\\\pi t + \\\\frac { \\\\pi } { 4 } = \\\\frac { 3 } { 2 } \\\\pi 2 \\\\pi t = \\\\frac { 5 } { 4 } \\\\pi\\n'),\n",
       " ('y = x ^ { 2 } - 6 x + 9 = ( x - 3 ) ^ { 2 }',\n",
       "  'y = x ^ { 2 } - 6 x + 9 = ( x - 3 ) ^ { 2 . }\\n'),\n",
       " ('x _ { 5 } = \\\\ f r a c { - 5 + \\\\ s q r t { 3 5 - 1 } } { 4 }',\n",
       "  'x _ { 3 } = \\\\frac { - 5 + \\\\sqrt { x - 1 } } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 3 } { 2 x + y } = 1', '\\\\frac { 3 } { 2 x + y } = 1\\n'),\n",
       " ('5 \\\\ t i m e s \\\\ f r a c { 1 } { 3 0 } = \\\\ f r a c { 1 } { 6 }',\n",
       "  '5 \\\\times \\\\frac { 1 } { 3 0 } = \\\\frac { 1 } { 6 }\\n'),\n",
       " ('\\\\ a n g l e 2 = 1 8 0 ^ { \\\\ c i r c } - \\\\ a n g l e 1 = 1 8 0 ^ { \\\\ c i r c } - 4 0 ^ { \\\\ c i r c } = 1 4 0 ^ { \\\\ c i r c } 0 ^ { \\\\ c i r c } 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle 2 = 1 8 0 ^ { \\\\circ } - \\\\angle 1 = 1 8 0 ^ { \\\\circ } - 4 0 ^ { \\\\circ } = 1 4 0 ^ { \\\\circ }\\n'),\n",
       " ('C D \\\\ b o t M D', 'C D \\\\bot M B\\n'),\n",
       " ('A E ^ { 2 } - B E ^ { 2 } = 1 \\\\ a n g l e A E B - \\\\ a n g l e B E D',\n",
       "  '\"A E ^ { 2 } - E B - E C , \\\\angle A E C = \\\\angle A E ^ { \\\\prime } C\"\\n'),\n",
       " ('4 0 - 5 y + 3 y', '4 0 - 5 y + 3 y\\n'),\n",
       " ('x _ { 2 } = \\\\ f r a c { 2 m } { m + m }',\n",
       "  '\\\\varphi x _ { 2 } = \\\\frac { 2 _ { n } } { m + m }\\n'),\n",
       " ('x + y = 1', 'x + y = 1\\n'),\n",
       " ('\\\\ f r a c { 7 } { 1 0 0 } \\\\ c d o t s \\\\ f r a c { 3 0 } { 3 0 0 }',\n",
       "  '\\\\frac { 7 } { 1 0 0 } = \\\\frac { 3 0 } { 3 0 0 }\\n'),\n",
       " ('W _ { 1 } = F \\\\ c d o t S = 4 0 N \\\\ c d o t 1 0 0 m = 4 0 0 0 J',\n",
       "  'W = F \\\\cdot S = 4 0 N \\\\cdot 1 0 m = 4 0 0 J\\n'),\n",
       " ('( 2 a - 3 ) x = 2', '( 2 a - 3 ) x = 2\\n'),\n",
       " ('x _ { 2 } = - \\\\ f r a c { 3 } { 2 }',\n",
       "  'x _ { 2 } = - \\\\frac { 3 } { 2 }\\n'),\n",
       " ('6 1 4 \\\\ a p p r o x 6 0 0', '6 1 4 \\\\approx 6 0 0\\n'),\n",
       " ('\" E B \\\\ a p p r o x + \\\\ a n g l e B F B = 1 8 0 ^ { \\\\ c i r c } , \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" , \" , \" B , \" , \" , \" B = 1 8 0 ^ { \\\\ c i r c } , \" B = 1 8 0 ^ { \\\\ c i',\n",
       "  'E B C + \\\\angle D F B = 1 8 0 ^ { \\\\circ }\\n'),\n",
       " ('S _ { \\\\ D e l t a B O C } = O B . C E \\\\ c d o t \\\\ f r a c { 1 } { 2 } = 4 \\\\ t i m e s 3 \\\\ t i m e s \\\\ f r a c { 1 } { 2 } = 6',\n",
       "  'S _ { \\\\Delta B O C } = O B \\\\cdot C E \\\\cdot \\\\frac { 1 } { 2 } = 4 \\\\times 3 \\\\times \\\\frac { 1 } { 2 } = 6\\n'),\n",
       " ('\" ( 1 + \\\\ f r a c { 1 } { 2 } , 1 + \\\\ f r a c { 1 } { 2 } , 1 + \\\\ f r a c { 1 } { 2 } ) \\\\ t i m e s 3 = a 1 , \" \" \" \"',\n",
       "  '\"2 + 3 = - \\\\frac { 5 } { a } , 2 \\\\times 3 = \\\\frac { 1 } { a } ,\"\\n'),\n",
       " ('\\\\ f r a c { b + c } { a } = \\\\ f r a c { a + c } { b } = \\\\ f r a c { a + b } { c } = 2',\n",
       "  '\\\\frac { b + c } { a } = \\\\frac { a + c } { b } = \\\\frac { a + b } { c } = 2\\n'),\n",
       " ('2 0 1 8 ( x ^ { 2 } y - x y ^ { 2 } + x y ) - x y ^ { 2 } ( 2 x ^ { 2 } - 6 x y + 2 x )',\n",
       "  '3 x y ( x ^ { 2 } y - x y ^ { 2 } + x y ) - x y ^ { 2 } ( 2 x ^ { 2 } - 6 x y + 2 x )\\n'),\n",
       " ('\\\\ f r a c { D E } { B C } = \\\\ f r a c { 1 } { 3 } .',\n",
       "  '\\\\frac { D E } { B C } = \\\\frac { 1 } { 3 } .\\n'),\n",
       " ('2 = 8 < x \\\\ l e q 4', '\\\\therefore - 3 < x \\\\leq 4\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } + \\\\ f r a c { 1 } { 5 } = \\\\ f r a c { 8 } { 1 5 } = 1 5',\n",
       "  '\\\\frac { 1 } { 3 } + \\\\frac { 1 } { 5 } = \\\\frac { 8 } { 1 5 }\\n'),\n",
       " ('+ 2 \\\\ c d o t ( - \\\\ f r a c { 1 } { 3 } ) \\\\ c d o t 3 b + 9 b',\n",
       "  '2 \\\\cdot ( - \\\\frac { 1 } { 3 } ) \\\\cdot 3 6 + 9 6\\n'),\n",
       " ('= \\\\ f r a c { B H } { A H } = \\\\ t a n \\\\ a n g l e D A G',\n",
       "  '= \\\\frac { - B H } { A H } = \\\\tan \\\\angle D A G\\n'),\n",
       " ('2 x = 2 k \\\\ p i \\\\ d i v \\\\ f r a c { 4 } { 5 } \\\\ t i m e s \\\\ f r a c { 5 } { 4 } \\\\ t i m e s \\\\ f r a c { 4 } { 9 } \\\\ t i m e s \\\\ f r a c { 4 } { 9 }',\n",
       "  '2 x = 2 k \\\\pi ^ { 2 } \\\\pm \\\\frac { \\\\pi } { 6 } ( k \\\\in z )\\n'),\n",
       " ('4 . 5 0 6 < 4 . 5 6 < 4 . 5 6 5 < 5 . 6 5',\n",
       "  '4 . 5 0 6 < 4 . 5 6 < 4 . 5 6 5 < 5 . 6 5\\n'),\n",
       " ('x = \\\\ f r a c { 1 1 } { 3 }', 'x = \\\\frac { 1 1 } { 3 }\\n'),\n",
       " ('\\\\ a n g l e B O D = \\\\ a n g l e C O D = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e B O C = \\\\ f r a c { 1 } { 2 } \\\\ t i m e s 1 2 0 ^ { \\\\ c i r c } = 6 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B O D = \\\\angle C O D = \\\\frac { 1 } { 2 } \\\\angle B O C = \\\\frac { 1 } { 2 } \\\\times 1 2 0 ^ { \\\\circ } = 6 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ a n g l e A O C + \\\\ a n g l e C O D + \\\\ a n g l e D O E = 1 8 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A O C + \\\\angle C O D + \\\\angle D O E = 1 8 0 ^ { \\\\circ }\\n'),\n",
       " ('x - \\\\ f r a c { 1 } { 3 } x = \\\\ f r a c { 2 } { 3 } x',\n",
       "  'x - \\\\frac { 1 } { 3 } x = \\\\frac { 2 } { 3 } x\\n'),\n",
       " ('3 、 2 4 \\\\ d i v 3 = 3 \\\\ c d o t s 3', '3 、 2 4 \\\\div 7 = 3 \\\\cdots 3\\n'),\n",
       " ('\\\\ f r a c { B D } { D A } = \\\\ f r a c { B E } { E M } = 1',\n",
       "  '\\\\frac { B D } { D A } = \\\\frac { B E } { E M } = 1\\n'),\n",
       " ('\\\\ f r a c { 1 } { x + y } = m r i g h t a r r o w m g h i n g h t a r r o w 1 0 0 0 \\\\ d i v n = 1 0 0 0 0 0 0 \\\\ d i v 1 0 0 0',\n",
       "  '\\\\frac { 1 } { x + y } = m\\n'),\n",
       " ('2 N a O H + B a C O _ { 3 } \\\\ d o w n a r r o w',\n",
       "  '2 N a O H + B a C O _ { 3 } \\\\downarrow\\n'),\n",
       " ('\" \\\\ a n g l e 2 = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e A O D = 6 5 ^ { \\\\ c i r c } , \" \" \" \" \" \" \" \" \" \" , \" , \" \\\\ a n g l e 2 , \" , \" \\\\ a n g l e 3 , 1 , 5 , 6 , \" , \" , \"',\n",
       "  '\\\\angle 2 = \\\\frac { 1 } { 2 } \\\\angle A O D = 6 5 ^ { \\\\circ }\\n'),\n",
       " ('x = 3 i c i r c l e d { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 }',\n",
       "  'x = 6\\n'),\n",
       " ('\" x _ { 1 } = 1 8 , y _ { 1 } = - 9 \" \" \" \" \" \" \" \" \" 1 , y _ { 1 } = - 9 \" \" \" \" , y _ { 1 } , y _ { 1 } \" 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1',\n",
       "  '\"x = 1 8 , y = - 9\"\\n'),\n",
       " ('\\\\ f r a c { M P } { B P } = \\\\ f r a c { P M } { P A } = \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\frac { M P } { B P } = \\\\frac { P M } { P A } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('1 2 3 \\\\ d i v 3 = 4 1', '1 2 3 \\\\div 3 = 4 1\\n'),\n",
       " ('x = 0 . 6', 'x = 0 . 6\\n'),\n",
       " ('\\\\ f r a c { 7 } { 9 } = \\\\ f r a c { 1 } { 9 } = \\\\ f r a c { 6 } { 9 } = \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { 7 } { 9 } = \\\\frac { 1 } { 9 } = \\\\frac { 6 } { 9 } = \\\\frac { 2 } { 3 }\\n'),\n",
       " ('\\\\ f r a c { 5 0 } { 3 } + b = 1 0', '\\\\frac { 5 0 } { 3 } + b = 1 0\\n'),\n",
       " ('- \\\\ f r a c { 9 } { 5 } x - \\\\ f r a c { 7 } { 3 } x \\\\ l e q 2 - 3 \\\\ f r a c { 1 } { 5 }',\n",
       "  '- \\\\frac { 9 } { 5 } x - \\\\frac { 2 } { 3 } x \\\\leq 2 - 3 \\\\frac { 1 } { 5 }\\n'),\n",
       " ('3 0 \\\\ % + 2 0 \\\\ % y = 1 3 0', '3 0 \\\\% + 2 0 \\\\% y = 1 3 0\\n'),\n",
       " ('2 x + 9 9 = 1 0 x - 5 x', '2 8 + 9 3 = 3 0 x - 5 x\\n'),\n",
       " ('( b - a ) - ( c + 8 ) - b - c - a = d',\n",
       "  '( b - c ) - ( a + d ) = b - c - a - d\\n'),\n",
       " ('y = \\\\ f r a c { 6 } { k + 4 }', 'y = \\\\frac { 6 } { k + 4 }\\n'),\n",
       " ('5 8 . 0 9 < 5 8 . 1 7 < 5 8 . 5 2 < 5 9 . 0 2',\n",
       "  '5 8 . 0 9 < 5 8 . 1 7 < 5 8 . 5 2 < 5 9 . 0 2\\n'),\n",
       " ('\\\\ f r a c { x - 1 } { 6 } - \\\\ f r a c { 3 x - 1 } { 2 } > 2 - \\\\ f r a c { x } { 2 }',\n",
       "  '\\\\frac { x - 1 } { 6 } - \\\\frac { 3 x - 1 } { 2 } > 2 - \\\\frac { x } { 2 }\\n'),\n",
       " ('r = d \\\\ d i v 2 = 1 0 \\\\ d i v 2 = 5',\n",
       "  'r = d \\\\div 2 = 1 0 \\\\div 2 = 5\\n'),\n",
       " ('5 7 6 + 3 0 + 2 7 0 = 8 7 6 ( c m ^ { 2 } )',\n",
       "  '5 7 6 + 3 0 + 2 7 0 = 8 7 6 、 ( m ^ { 2 } )\\n'),\n",
       " ('x = - \\\\ f r a c { 2 } { 7 + 4 \\\\ s q r t { 3 } } + 4 \\\\ s q r t { 3 } \\\\ s q r t { 3 }',\n",
       "  'x = - \\\\frac { 2 } { 7 + 4 \\\\sqrt { 3 } }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x = \\\\ f r a c { 2 } { 3 } \\\\ t i m e s \\\\ f r a c { 3 } { 4 }',\n",
       "  '\\\\frac { 1 } { 2 } x = \\\\frac { 2 } { 3 } \\\\times \\\\frac { 3 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 2 } { x - 1 } = 1', '\\\\frac { 2 } { x - 1 } = 1\\n'),\n",
       " ('x = - \\\\ f r a c { 2 } { 3 }', 'x = - \\\\frac { 2 } { 3 }\\n'),\n",
       " ('O G / / B C s i n t h i n s 7 \\\\ t h e r e f o r e \\\\ f r a c { 1 } { 2 }',\n",
       "  'O G / / B C\\n'),\n",
       " ('1 5 \\\\ t i m e s \\\\ f r a c { 2 x - 7 } { 3 } = 1 5 \\\\ t i m e s \\\\ f r a c { 3 - 2 x } { 5 }',\n",
       "  '1 5 \\\\times \\\\frac { 2 x - 7 } { 3 } = 1 5 \\\\times \\\\frac { 3 - 2 x } { 5 }\\n'),\n",
       " ('\\\\ a n g l e B O D > = 8 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B O D = 8 0 ^ { \\\\circ }\\n'),\n",
       " ('1 2 \\\\ t i m e s 6 \\\\ t i m e s 2 = 1 4 4 ( d m ^ { 3 } ) = 1 4 4 ( L ) )',\n",
       "  '1 2 \\\\times 6 \\\\times 2 = 1 4 4 ( d m ^ { 3 } ) = 1 4 4 ( L )\\n'),\n",
       " ('= \\\\ f r a c { 1 } { 2 } x - \\\\ f r a c { 2 } { 3 } x = 5 + 1',\n",
       "  '2 \\\\frac { 1 } { 2 } x - \\\\frac { 2 } { 3 } x = 5 + 1\\n'),\n",
       " ('1 - \\\\ f r a c { 3 } { 4 } < \\\\ f r a c { 2 } { 2 } x < 1 - \\\\ f r a c { 1 } { 4 } < 1 - ( 1 - \\\\ f r a c { 3 } { 4 } )',\n",
       "  '1 - \\\\frac { 1 } { 4 } \\\\sin ^ { 2 } \\\\alpha = 1 - \\\\frac { 1 } { 4 } ( 1 - \\\\cos ^ { 2 } \\\\alpha )\\n'),\n",
       " ('3 6 \\\\ d i v 4 = \\\\ s q u a r e', '3 6 \\\\div 4 = \\\\boxed { 9 }\\n'),\n",
       " ('\\\\ o v e r r i g h t a r r o w { a } + 3 \\\\ o v e r r i g h t a r r o w { b } S i g h t a r r o w { a }',\n",
       "  'a + 3 b\\n'),\n",
       " ('0 . 0 8 = \\\\ f r a c { 8 } { 1 0 0 } = \\\\ f r a c { 2 } { 2 5 }',\n",
       "  '0 . 0 8 = \\\\frac { 8 } { 1 0 0 } = \\\\frac { 2 } { 2 5 }\\n'),\n",
       " ('x = \\\\ f r a c { 1 } { 2 }', 'x = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('P \\\\ c d o t Q : t = \\\\ f r a c { \\\\ l a m b d a } { V } = \\\\ f r a c { 1 . 2 m } { 2 0 / s } = 0 . 6 s s s s',\n",
       "  'P \\\\cdot Q : t = \\\\frac { \\\\lambda } { V } = \\\\frac { 1 . 2 m } { 2 0 / s } = 0 . 6 s\\n'),\n",
       " ('\" x = k \\\\ p i + \\\\ f r a c { \\\\ p i } { 2 } , k \\\\ i n z ] \" \" [ \" \" \" \" \" [ \" [ \" [ \\\\ f r a c { 1 } { 2 } , \\\\ f r a c { \\\\ p i } { 2 } ] \" \" [ k \\\\ i n [ 0 , \\\\ f r a c { \\\\ p i } { 2 } ]',\n",
       "  '\"x = k \\\\pi + \\\\frac { \\\\pi } { 2 } , k \\\\in z ]\"\\n'),\n",
       " ('1 . 6 0 0 \\\\ d i v 1 0 0 0 = 1 . 6 ( d m )',\n",
       "  '1 6 0 0 \\\\div 1 0 0 0 = 1 . 6 ( d m )\\n'),\n",
       " ('f ( - \\\\ f r a c { \\\\ p i } { 4 } ) = f ( \\\\ f r a c { 3 } { 4 } \\\\ p i ) = \\\\ f r a c { \\\\ s q r t { 2 } } { 2 }',\n",
       "  'f ( - \\\\frac { \\\\pi } { 4 } ) = f ( \\\\frac { 3 } { 4 } \\\\pi ) = \\\\frac { \\\\sqrt { 2 } } { 2 }\\n'),\n",
       " ('x = - \\\\ f r a c { \\\\ s q r t { 1 0 } } { 2 }',\n",
       "  'x = - \\\\frac { \\\\sqrt { 1 0 } } { 2 }\\n'),\n",
       " ('x + 5 ( m - 8 ) = m + 2 0', 'x + 5 ( m - 8 ) = m + 2 0\\n'),\n",
       " ('f ( \\\\ f r a c { 1 } { 5 } ) = \\\\ c o s \\\\ f r a c { 5 \\\\ p i } { 3 } = \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 1 } { 3 }',\n",
       "  'f ( 5 ) = \\\\cos \\\\frac { 5 \\\\pi } { 3 } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('x = \\\\ f r a c { 1 0 } { 1 3 }', 'x = \\\\frac { 2 0 } { 1 3 }\\n'),\n",
       " ('\\\\ v e r t x \\\\ v e r t - b \\\\ l e q x \\\\ l e q 3 \\\\ t e x t c i r c l e d { 2 }',\n",
       "  '\\\\{ x 1 - b \\\\leq x \\\\leq 3 \\\\}\\n'),\n",
       " ('x - \\\\ f r a c { 2 \\\\ s q r t { 2 } + 2 \\\\ s q r t { 2 8 } } { 4 } = \\\\ f r a c { \\\\ s q r t { 2 } \\\\ p m \\\\ s q r t { 2 } } { 2 }',\n",
       "  'x = \\\\frac { 2 \\\\sqrt { 2 } + \\\\sqrt { 2 8 } } { 4 } = \\\\frac { \\\\sqrt { 2 } \\\\pm \\\\cdot \\\\sqrt { 7 } } { 2 }\\n'),\n",
       " ('\\\\ a n g l e 1 = \\\\ a n g l e 2 = \\\\ a n g l e 3 = \\\\ a n g l e D E N',\n",
       "  '\\\\angle 1 = \\\\angle 2 = \\\\angle 3 = \\\\angle D E C\\n'),\n",
       " ('a = \\\\ f r a c { 2 l n x } { x ^ { 2 } }',\n",
       "  'a = \\\\frac { 2 \\\\ln x } { x ^ { 2 } }\\n'),\n",
       " ('B = \\\\ f r a c { M o ( N ) } { M } ^ { \\\\ p r i m e } ^ { \\\\ p r i m e } ^ { \\\\ p r i m e } ^ { \\\\ p r i m e } ^ { \\\\ p r i m e } ^ { \\\\ p r i m e }',\n",
       "  'B = \\\\frac { M o ( V _ { 0 } + V _ { 1 } ) } { 6 e V }\\n'),\n",
       " ('A C = \\\\ f r a c { 1 } { 2 } B E', 'A O = \\\\frac { 1 } { 2 } B C\\n'),\n",
       " ('1 0 0 m = 1 0 0 0 0 c m', '1 0 0 m = 1 0 0 0 0 c m\\n'),\n",
       " ('x = \\\\ f r a c { 1 } { 5 }', 'x = \\\\frac { 1 } { 5 }\\n'),\n",
       " ('\\\\ c o s C = \\\\ f r a c { c ^ { 2 } + b ^ { 2 } - c ^ { 2 } } { 2 a b }',\n",
       "  '\\\\cos c = \\\\frac { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } } { 2 a b }\\n'),\n",
       " ('6 ( \\\\ f r a c { x } { 6 } - \\\\ f r a c { 6 a - 1 } { 3 } ) = 6 ( x - \\\\ f r a c { 5 a - 1 } { 2 } )',\n",
       "  '6 ( \\\\frac { x } { 6 } - \\\\frac { 6 a - 1 } { 3 } ) = 6 ( x - \\\\frac { 5 a - 1 } { 2 } )\\n'),\n",
       " ('Z n + 2 H C l = H _ { 2 } \\\\ u p a r r o w +',\n",
       "  'Z n + 2 H C l = H _ { 2 } \\\\uparrow +\\n'),\n",
       " ('\\\\ f r a c { 1 } { ( n + 1 ) ^ { 2 } } \\\\ l e q \\\\ f r a c { 1 } { n ( n + 1 ) } = \\\\ f r a c { 1 } { n } - \\\\ f r a c { 1 } { n + 1 }',\n",
       "  '\\\\frac { 1 } { ( n + 1 ) ^ { 2 } } < \\\\frac { 1 } { n ( n + 1 ) } = \\\\frac { 1 } { n } - \\\\frac { 1 } { n + 1 }\\n'),\n",
       " ('A D \\\\ c d o t B C = A B \\\\ c d o t P C', '、 A D 、 B C = A B 、 D C\\n'),\n",
       " ('x _ { 1 } - 2 y = - 0', 'x - 2 y = 0\\n'),\n",
       " ('Z n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\ u p a r r o w',\n",
       "  'Z n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\uparrow\\n'),\n",
       " ('( b + 1 5 ) x = 5 4', '( 6 + 1 5 ) x = 5 4\\n'),\n",
       " ('\\\\ a n g l e D O E = \\\\ a n g l e D B E + \\\\ a n g l e A C B = 7 5 ^ { \\\\ c i r c }',\n",
       "  'D O E = \\\\angle D B E + \\\\angle A E B = 7 5 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ c d o t \\\\ a n g l e B C D = \\\\ f r a c { 1 } { 2 } \\\\ t i m e s 5 0 ^ { \\\\ c i r c } = 2 5 ^ { \\\\ c i r c }',\n",
       "  '\\\\cdot \\\\angle B C D = \\\\frac { 1 } { 2 } \\\\times 5 0 ^ { \\\\circ } = 2 5 ^ { \\\\circ }\\n'),\n",
       " ('A N = \\\\ f r a c { 1 } { 2 } \\\\ t i m e s 2 = 1',\n",
       "  'A N = \\\\frac { 1 } { 2 } A _ { 2 } = 1\\n'),\n",
       " ('\" D E = \\\\ f r a c { 1 } { 2 } A C > E F = \\\\ f r a c { 1 } { 2 } A B , \" \" \" \" \" \" \" , \" \" \" , \" \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" , \" ,',\n",
       "  '\"D E = \\\\frac { 1 } { 2 } A C > E F = \\\\frac { 1 } { 2 } A B ,\"\\n'),\n",
       " ('1 2 \\\\ t i m e s 4 = 4 8 ( c m )', '1 2 \\\\times 4 = 4 8 ( c m )\\n'),\n",
       " ('u _ { e } = \\\\ a l p h a - \\\\ a l p h a V',\n",
       "  'u _ { e } = \\\\alpha - \\\\alpha V\\n'),\n",
       " ('2 n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\ u p a r r o w',\n",
       "  '2 n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\uparrow\\n'),\n",
       " ('b _ { n + 1 } - b _ { n } = - 1', 'b _ { n + 1 } - b _ { n } = - 1\\n'),\n",
       " ('1 2 . 5 6 + 1 2 5 . 6 = 1 3 8 . 1 6 ( m ^ { 2 } )',\n",
       "  '1 2 . 5 6 + 1 2 5 . 6 = 1 3 8 . 1 6 ( m ^ { 2 } )\\n'),\n",
       " ('\" \\\\ a n g l e P A B = 1 8 0 ^ { \\\\ c i r c } , \\\\ a n g l e B A E = 1 8 0 ^ { \\\\ c i r c } + 6 0 ^ { \\\\ c i r c } = 5 5 ^ { \\\\ c i r c } + 9 0 ^ { \\\\ c i r c } = 2 \" \" \" \" \" \" \" \" \" \" \"',\n",
       "  '2 7 + 3 0 = 5 7 6 3 - 5 = 5 8 3 7 + 6 = 4 3 1 9 + 5 = 2 4\\n'),\n",
       " ('\\\\ a n g l e A O C = 3 6 0 ^ { \\\\ c i r c } - \\\\ a n g l e A O B - \\\\ a n g l e B O C = 3 6 0 ^ { \\\\ c i r c } - 1 5 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A O C = 3 6 0 ^ { \\\\circ } - \\\\angle A O B - \\\\angle B O C = 3 6 0 ^ { \\\\circ } - 1 5 0 ^ { \\\\circ }\\n'),\n",
       " ('S = 6 0 - 1 5 = 4 5 ( d m ^ { 3 } )',\n",
       "  'S = 6 0 - 1 5 = 4 5 ( d m ^ { 2 } )\\n'),\n",
       " ('y = 2 x - 3', 'y = 2 x - 3\\n'),\n",
       " ('\\\\ f r a c { 1 } { 1 2 } x = 7 + \\\\ f r a c { 1 } { 6 } - 1',\n",
       "  '\\\\frac { 1 } { 1 2 } x = 7 + \\\\frac { 1 } { 6 } - 1\\n'),\n",
       " ('x \\\\ d i v 1 . 2 = 4', 'x \\\\div 1 . 2 = 4\\n'),\n",
       " ('7 0 3 \\\\ a p p r o x 7 0 0', '7 0 3 \\\\approx 7 0 0\\n'),\n",
       " ('A B = 2 \\\\ s q r t { 5 }', 'A B = 2 \\\\sqrt { 5 }\\n'),\n",
       " ('\\\\ a n g l e B A E + \\\\ a n g l e D A F = 4 5 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B A E + \\\\angle D A F = 4 5 ^ { \\\\circ }\\n'),\n",
       " ('x = - \\\\ f r a c { 2 } { 5 }', 'x = - \\\\frac { 1 } { 5 }\\n'),\n",
       " ('x _ { 2 } = 1 - \\\\ s q r t { 3 } i n g Z \\\\ t i m e s 6 0 \\\\ t i m e s 6 0 \\\\ n o t i m e s 6 0 \\\\ t i m e s 6 0 \\\\ t i m e s 6 0 \\\\ t i m e s 6 0 \\\\ t i',\n",
       "  'x _ { 2 } = 1 - \\\\sqrt { 3 }\\n'),\n",
       " ('\\\\ t a n \\\\ f r a c { \\\\ f r a c { 1 4 } { 3 } \\\\ p i } { - 2 } = - \\\\ f r a c { 7 } { 3 } \\\\ p i',\n",
       "  '\\\\tan \\\\frac { \\\\frac { 1 4 } { 3 } \\\\pi } { - 2 \\\\cdot } = - \\\\frac { 7 } { 3 } \\\\pi\\n'),\n",
       " ('x = \\\\ f r a c { 2 } { 7 } \\\\ s q r t { 4 }', 'x = \\\\sqrt [ 3 ] { 4 }\\n'),\n",
       " ('( x - a ) ^ { 2 } + ( y - 1 ) ^ { 2 } = a ^ { 2 } - 1',\n",
       "  '( x - a ) ^ { 2 } + ( y - 1 ) ^ { 2 } = a ^ { 2 } - 1\\n'),\n",
       " ('a ^ { 2 } + a = - 1 .', 'a ^ { 2 } + a = - 1\\n'),\n",
       " ('6 . m = \\\\ p m 2 . 5', '6 . m = \\\\pm 2 . 5\\n'),\n",
       " ('f ( x ) - 9 x = 3 x ^ { 2 } + 1 2 b - 1 1 x + c = 0',\n",
       "  'f ^ { \\\\prime } ( x ) - 9 x = 3 x ^ { 2 } + ( 2 b - 9 ) x + c = 0\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } B O A O C = \\\\ f r a c { 1 } { 2 } B D \\\\ c d o t P A + \\\\ f r a c { 1 } { 2 } B Q \\\\ c d o t \\\\ o v e r r i g h t a r r o w { D C } + \\\\ f r a c { 1 } { 2 } P D \\\\ o v e r r i g h t a',\n",
       "  ': \\\\frac { 1 } { 2 } B a \\\\times Q C = \\\\frac { 1 } { 2 } E P \\\\cdot P C + \\\\frac { 1 } { 2 } ( B a + E P ) Q P\\n'),\n",
       " ('R _ { 2 } = \\\\ f r a c { U } { I _ { 2 } } = \\\\ f r a c { 1 2 V } { 2 A } = 6 \\\\ O m e g a',\n",
       "  'R _ { 2 } = \\\\frac { U } { I _ { 2 } } = \\\\frac { 1 2 V } { 2 A } = 6 \\\\Omega\\n'),\n",
       " ('\\\\ f r a c { 2 0 1 2 } { 2 0 1 3 } x = 2 0 1 2',\n",
       "  '\\\\frac { 2 0 1 2 } { 2 0 1 3 } x = 2 0 1 2\\n'),\n",
       " ('\\\\ f r a c { 1 0 0 . \\\\ p i } { 3 } = \\\\ f r a c { 2 \\\\ p i } { 3 } = \\\\ f r a c { 2 \\\\ p i } { 3 }',\n",
       "  '\\\\frac { 6 0 . \\\\pi . 2 ^ { 2 } } { 3 6 0 } = \\\\frac { 2 \\\\pi } { 3 }\\n'),\n",
       " ('y _ { 2 } = \\\\ f r a c { 5 } { 2 }', 'y _ { 2 } = \\\\frac { 5 } { 2 }\\n'),\n",
       " ('+ \\\\ f r a c { 1 } { 3 } = \\\\ f r a c { 4 } { 1 0 } - \\\\ f r a c { 2 } { 5 } = 1 0 \\\\ f r a c { 2 } { 5 } 天',\n",
       "  '+ \\\\frac { 1 } { 3 } = \\\\frac { 4 } { 1 0 } = \\\\frac { 2 } { 5 }\\n'),\n",
       " ('3 x = \\\\ f r a c { 3 } { 5 } \\\\ t i m e s \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c {',\n",
       "  '3 x = \\\\frac { 3 } { 5 } \\\\times \\\\frac { 1 } { 2 }\\n'),\n",
       " ('\\\\ a n g l e C A B = 1 3 4 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle C A B = 1 3 4 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ t h e t a ) ^ { 3 } = a + 0 \\\\ t i m e s 3', 'O B = O A + O B\\n'),\n",
       " ('0 . 5 < 0 . 5 1 < 0 . 5 0 1 < 0 . 5 1 1 1 1 1 1',\n",
       "  '0 . 5 < 0 . 5 1 < 0 . 5 0 1 < 0 . 5 1 1\\n'),\n",
       " ('( 1 - \\\\ f r a c { 1 } { 1 2 } \\\\ t i m e s 8 ) \\\\ d i v ( 1 4 - 9 ) = \\\\ f r a c { 1 } { 1 8 }',\n",
       "  '( 1 - \\\\frac { 1 } { 1 2 } \\\\times 8 ) \\\\div ( 1 4 - 8 ) = \\\\frac { 1 } { 1 8 }\\n'),\n",
       " ('\\\\ f r a c { 9 0 ^ { \\\\ c i r c } \\\\ t i m e s \\\\ p i \\\\ t i m e s \\\\ p i \\\\ t i m e s 1 2 ^ { 2 } } { 3 6 0 ^ { \\\\ c i r c } } =',\n",
       "  '\\\\frac { 9 0 ^ { \\\\circ } \\\\times \\\\pi \\\\times x \\\\times 1 2 ^ { 2 } } { 3 6 0 ^ { \\\\circ } }\\n'),\n",
       " ('D H = \\\\ f r a c { 1 } { 2 } A D = \\\\ f r a c { 1 } { 2 } x',\n",
       "  'D H = \\\\frac { 1 } { 2 } A D = \\\\frac { 1 } { 2 } x\\n'),\n",
       " ('\\\\ s q r t { 3 } + - 2 \\\\ s q r t { 3 } + 2 \\\\ s q r t { 3 } - 4 = - 3',\n",
       "  '\\\\sqrt { 3 } - 1 - 2 \\\\sqrt { 3 } + 2 \\\\sqrt { 3 } - 4 = - 3\\n'),\n",
       " ('x = 2 0 0 0 0 0', 'x = 2 0 0 0 0\\n'),\n",
       " ('x = - \\\\ f r a c { 1 } { 5 }', 'x = - \\\\frac { 1 } { 5 }\\n'),\n",
       " ('\\\\ v e r t - \\\\ f r a c { 5 } { 2 4 } \\\\ v e r t = \\\\ f r a c { 9 } { 4 }',\n",
       "  '\\\\vert - \\\\frac { 1 } { 2 } - \\\\frac { 1 } { 4 } \\\\vert = \\\\frac { 9 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { x _ { 2 } y _ { 1 } - x _ { 1 } y _ { 2 } } { y _ { 1 } - y _ { 2 } } = 4',\n",
       "  '\\\\frac { x _ { 2 } y _ { 1 } - x _ { 1 } y _ { 2 } } { y _ { 1 } - y _ { 2 } } = 4\\n'),\n",
       " ('3 x + 1 > 5 x - 5 1 7 x + 7 x + 7 x ) 7 x + 1 7 x ) 1 7 x 1 7 1 万 1 7 1 万 7 x > 1 7 x > 1 7 x > 7 x > 7 x > 7 x > 7 x > 7 x',\n",
       "  '3 x + 1 > 5 x - 5\\n'),\n",
       " ('\" P _ { 4 } ( - 1 + \\\\ s q r t { 3 } , 2 \" \" \" \" \" \" \" \" \" \" \" \" \" , \" , \" , 1 , 1 , 3 , 1 , 1 , 1 , 1 , 3 , 1 , 1 , 1 , 3 , 1 , 1 , 3 , 1 , 1 , 1 , 3 , 1 , 1 , 1 \" \" \" \"',\n",
       "  '\"P _ { 4 } ( - 1 + \\\\sqrt { 3 } , 2\"\\n'),\n",
       " ('\\\\ a n g l e A D B = \\\\ a n g l e A ^ { \\\\ p r i m e } D ^ { \\\\ p r i m e } B ^ { \\\\ p r i m e } = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A D B = \\\\angle A D ^ { \\\\prime } B ^ { \\\\prime } = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ f r a c { A E } { D E } = \\\\ f r a c { A D } { D F } = \\\\ f r a c { D E } { E F }',\n",
       "  '\"\\\\frac { A E } { D E } = \\\\frac { A D } { D F } = \\\\frac { D E } { E F } ,\"\\n'),\n",
       " ('\\\\ a n g l e B O F = \\\\ f r a c { 1 } { 2 } a',\n",
       "  '\\\\angle B O F = \\\\frac { 1 } { 2 } a\\n'),\n",
       " ('k = - \\\\ f r a c { 1 1 } { 2 }', 'k = - \\\\frac { 1 1 } { 2 }\\n'),\n",
       " ('y =', 'y =\\n'),\n",
       " ('A _ { 2 } O _ { 3 } + 2 H C l = B _ { 2 } C O _ { 3 } + H _ { 2 } O + C O _ { 2 } \\\\ u p a r r o w O _ { 3 } + H _ { 2 } O r i g h t a r r o w + O _ { 2 } \\\\ u p a r r o w',\n",
       "  '8 \\\\cdots 8 5 1 2 \\\\div 6 7 = 7 \\\\cdots 4 3 5 1 2 \\\\div 8 5 = 6 \\\\cdots 2\\n'),\n",
       " ('y = \\\\ f r a c { 1 } { 2 } x ^ { 2 } - x - \\\\ f r a c { 3 } { 2 } - \\\\ f r a c { 3 } { 2 }',\n",
       "  'y = \\\\frac { 1 } { 2 } x ^ { 2 } - x - \\\\frac { 3 } { 2 }\\n'),\n",
       " ('x ^ { 2 } - x - 6 + 4 = 2 x ^ { 2 } - 8',\n",
       "  'x ^ { 2 } - x - 6 + 4 = 2 x ^ { 2 } - 8\\n'),\n",
       " ('1 5 0 - 8 = 8 0 \\\\ t i m e s 1 5', '1 5 0 x = 8 0 \\\\times 1 5\\n'),\n",
       " ('y = \\\\ s q r t { \\\\ f r a c { 1 1 } { 9 } x ^ { 2 } - 4 x + 3 6 } ( 0 < x < 9 )',\n",
       "  'y = \\\\sqrt { \\\\frac { 1 1 } { 9 } x ^ { 2 } - 4 x + 3 6 } ( 0 < x < 9 )\\n'),\n",
       " ('V _ { c } = \\\\ f r a c { 1 } { 2 } m / s',\n",
       "  'V _ { c } = \\\\frac { 1 } { 2 } m / s\\n'),\n",
       " ('\\\\ t h e r e f o r e m _ { 1 } m _ { n } = 2 m _ { 2 } ( m x ) = 4',\n",
       "  '\"y _ { \\\\min } = 2 , y _ { m } G x = 4\"\\n'),\n",
       " (') = \\\\ f r a c { 3 } { 6 } = \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 1 } { 2 }',\n",
       "  ') = \\\\frac { 3 } { 6 } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('x = - \\\\ s q r t { 4 3 }', 'x = - \\\\sqrt { 4 3 }\\n'),\n",
       " ('x _ { 1 } = 1 + \\\\ s q r t { 3 }', 'x 1 = 1 + \\\\sqrt { 3 }\\n'),\n",
       " ('\\\\ f r a c { A F } { C D } = \\\\ f r a c { E F } { E D } = \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { A F } { C D } = \\\\frac { E F } { E D } = \\\\frac { 2 } { 3 }\\n'),\n",
       " ('3 . 1 4 \\\\ t i m e s ( 5 ^ { 2 } - 4 ^ { 2 } ) = 2 8 . 2 6 ( c m ^ { 2 } ) 2 8 . 2 6 \\\\ t i m e s 8 0 0 = 2 2 6 0 . 8',\n",
       "  '3 . 1 4 \\\\times ( 5 ^ { 2 } - 4 ^ { 2 } ) = 2 8 . 2 6 ( c m ^ { 2 } ) 2 8 . 2 6 \\\\times 8 0 = 2 2 6 0 . 8\\n'),\n",
       " ('O A : M ^ { 2 } = 1 2', 'O A \\\\cdot M P = 1 2\\n'),\n",
       " ('F = P S = 1 0 0 0 P a \\\\ t i m e s 0 . 0 5 m ^ { 2 } = 5 0 N',\n",
       "  'F = P S = 1 0 0 0 P a \\\\times 0 . 0 5 m ^ { 2 } = 5 0 N\\n'),\n",
       " ('\\\\ f r a c { 9 x } { 1 0 0 } = \\\\ f r a c { 4 0 } { 5 0 } - \\\\ f r a c { 2 3 } { 2 5 }',\n",
       "  '\\\\frac { 9 2 } { 1 0 0 } = \\\\frac { 4 6 } { 3 0 } = \\\\frac { 2 3 } { 2 5 }\\n'),\n",
       " ('B D \\\\ b o t B C', '. B D \\\\bot B C\\n'),\n",
       " ('m = \\\\ f r a c { 2 0 1 8 } { \\\\ s q r t { 2 0 1 9 } - 1 } = \\\\ s q r t { 2 0 1 9 } + 1',\n",
       "  'M = \\\\frac { 2 0 1 8 } { \\\\sqrt { 2 0 1 9 } - 1 } = \\\\sqrt { 2 0 1 9 } + 1\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } \\\\ l e q k < 1 1', '\\\\frac { 1 } { 3 } < k < 1\\n'),\n",
       " ('x ^ { 2 } + \\\\ f r a c { 1 } { x ^ { 2 } } = 2',\n",
       "  'x ^ { 2 } + \\\\frac { 1 } { x ^ { 2 } } = 2\\n'),\n",
       " ('x = - 2', 'x = - 2\\n'),\n",
       " ('2 \\\\ t i m e s \\\\ f r a c { x - 1 } { 2 } = 2 \\\\ t i m e s 1 - 2 \\\\ t i m e s \\\\ f r a c { 3 x - 1 } { 2 }',\n",
       "  '2 \\\\times \\\\frac { x + 1 } { 2 } = 2 \\\\times 1 - 2 \\\\times \\\\frac { 3 x - 1 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { 9 } { 1 6 } x = \\\\ f r a c { 1 2 } { 1 6 } x + 4',\n",
       "  '\\\\frac { 9 } { 1 6 } x = \\\\frac { 1 2 } { 1 6 } x + 4\\n'),\n",
       " ('\" x _ { 1 } = \\\\ f r a c { 3 } { 2 } , x _ { 2 } = 2 \" \" \" \" \" \" \" \" \" 2 , \" \" \"',\n",
       "  '\"x _ { 1 } = \\\\frac { 3 } { 2 } , x _ { 2 } = 2\"\\n'),\n",
       " ('O D = \\\\ f r a c { 3 } { 4 }', 'O D = \\\\frac { 3 } { 4 }\\n'),\n",
       " ('C F = 3 \\\\ s q r t { 2 } + \\\\ f r a c { 3 } { 2 } \\\\ s q r t { 2 } = \\\\ f r a c { 9 } { 2 } \\\\ s q r t { 2 }',\n",
       "  'C F = 3 \\\\sqrt { 2 } + \\\\frac { 3 } { 2 } \\\\sqrt { 2 } = \\\\frac { 9 } { 2 } \\\\sqrt { 2 }\\n'),\n",
       " ('3 x - x + 6 = \\\\ f r a c { 1 2 x } { 5 } - \\\\ f r a c { 2 } { 5 }',\n",
       "  '3 x - x + 6 = \\\\frac { 1 2 x } { 5 } - \\\\frac { 2 } { 5 }\\n'),\n",
       " ('1 \\\\ d i v 3 = \\\\ f r a c { 1 } { 3 } ( k g )',\n",
       "  '1 \\\\div 3 = \\\\frac { 1 } { 3 } ( k g )\\n'),\n",
       " ('\\\\ a n g l e A B C = 3 0 ^ { \\\\ c i r c } . O B =',\n",
       "  '\\\\angle A B C = 3 0 ^ { \\\\circ } . O B =\\n'),\n",
       " ('( _ { 1 0 } \\\\ t i m e s 1 ) ^ { 6 } ( 5 2 y ) ^ { 4 }',\n",
       "  'C _ { 1 0 } ^ { 4 } x ^ { 6 } ( \\\\sqrt { 2 } y ) ^ { 4 }\\n'),\n",
       " ('\" \\\\ t h e r e f o r e A = 0 . 5 = O B , O \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" , 0 , 0 , 0 , 0 , 0 , \" \" \" \" \" \" \" \" \" \" \" \" , 0 , 0 , 0 , 0 , 0 , 0',\n",
       "  '\"O A = O C = O B , O\"\\n'),\n",
       " ('y _ { 4 } = \\\\ f r a c { 1 } { 2 }', 'y _ { 4 } = - \\\\frac { V } { 2 }\\n'),\n",
       " ('P = \\\\ f r a c { F } { S }', 'P = \\\\frac { F } { S }\\n'),\n",
       " ('c > 2 x i n g h t e r i g h t a r r o w \\\\ f r a c { 1 } { 2 } \\\\ s q r t [ 3 ] { 2 } \\\\ f r a c { 1 } { 2 } \\\\ f r a c { 1 } { 2 } \\\\ s q r t [ 3 ] { 2 } \\\\ f r a c { 1 }',\n",
       "  'e ^ { 2 } > 2\\n'),\n",
       " ('x < \\\\ f r a c { 2 } { 3 }', 'x < \\\\frac { 2 } { 3 }\\n'),\n",
       " ('x = - \\\\ f r a c { 3 } { 2 }', 'x = - \\\\frac { - 3 } { 2 }\\n'),\n",
       " ('\\\\ a n g l e D F A = 1 5 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle D E A = 1 5 ^ { \\\\circ }\\n'),\n",
       " ('k = - \\\\ f r a c { 1 0 } { 2 0 }', 'k = - \\\\frac { 1 } { 2 0 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { x + y } : \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\frac { 1 } { x - y } : \\\\frac { 1 } { 2 }\\n'),\n",
       " ('\\\\ s q r t { 2 5 - x ^ { 2 } } = x - 1',\n",
       "  '\\\\sqrt { 2 5 } x ^ { 2 } = x - 1\\n'),\n",
       " ('2 2 . 4 5 - 1 9 . 1 5 6 = 3 . 9 4 \\\\ d i v 3 . 0 1 2',\n",
       "  '2 2 . 4 5 - 1 9 . 1 5 6 = 3 . 3 9 4 = 3 . 0 2\\n'),\n",
       " ('\\\\ f r a c { 8 } { 1 2 } x + \\\\ f r a c { 9 } { 1 2 } x - \\\\ f r a c { 1 0 } { 1 2 } x = \\\\ f r a c { 4 } { 4 } + \\\\ f r a c { 9 } { 4 } + \\\\ f r a c { 9 } { 4 } = \\\\ f r a c { 1 0 } { 4 }',\n",
       "  '\\\\frac { 8 } { 1 2 } x + \\\\frac { 9 } { 1 2 } x - \\\\frac { 1 0 } { 1 2 } x = \\\\frac { 4 } { 4 } + \\\\frac { 8 } { 4 } + \\\\frac { 9 } { 4 } - \\\\frac { 1 0 } { 4 }\\n'),\n",
       " ('C _ { 6 } ^ { 0 } \\\\ t i m e s ( \\\\ f r a c { 1 } { 8 } ) ^ { 0 } \\\\ t i m e s ( \\\\ f r a c { 1 } { 8 } ) ^ { 3 } = \\\\ f r a c { 3 4 3 } { 5 6 2 } ^ { 3 }',\n",
       "  'C _ { 3 } ^ { 0 } \\\\times ( \\\\frac { 1 } { 8 } ) ^ { 0 } \\\\times ( \\\\frac { 1 } { 8 } ) ^ { 3 } = \\\\frac { 3 4 3 } { 5 . 2 }\\n'),\n",
       " ('\\\\ t a n \\\\ a n g l e B C D = \\\\ f r a c { B F } { C F } = \\\\ f r a c { \\\\ f r a c { 3 } { 2 } \\\\ s q r t { 2 } } { \\\\ f r a c { 9 } { 2 } \\\\ s q r t { 2 } } = \\\\ f r a c { 1 } { 3 }',\n",
       "  '\\\\tan \\\\angle B C D = \\\\frac { B F } { C F } = \\\\frac { \\\\frac { 3 } { 2 } \\\\sqrt { 2 } } { \\\\frac { 9 } { 2 } \\\\sqrt { 2 } } = \\\\frac { 1 } { 3 }\\n'),\n",
       " ('\\\\ r h o = \\\\ f r a c { m } { v }', '\\\\rho = \\\\frac { M } { V }\\n'),\n",
       " ('3 = - \\\\ f r a c { 7 } { 5 }', '3 = - \\\\frac { 7 } { 5 }\\n'),\n",
       " ('a ^ { - 1 } = \\\\ f r a c { 1 } { a }',\n",
       "  'a ^ { - 1 } = \\\\frac { 1 } { a }\\n'),\n",
       " ('\\\\ f r a c { a } { \\\\ s i n B } = \\\\ f r a c { 1 8 } { \\\\ s i n C }',\n",
       "  '\\\\frac { b } { \\\\sin B } = \\\\frac { C } { \\\\sin C }\\n'),\n",
       " ('0 . 2 \\\\ n 2 \\\\ D I V 1 . 4 \\\\ n 1 4 \\\\ n 2 n 2', '- 0 . 5 = 1 8\\n'),\n",
       " ('= 6 \\\\ t i m e s \\\\ s q r t { x }', '= 6 \\\\times \\\\sqrt { x }\\n'),\n",
       " ('G = m g = 5 0 k g \\\\ t i m e s 9 . 8 N / k g = 4 9 0 N . 0 . 0 1 N',\n",
       "  'G = m g = 5 0 k g \\\\times 9 . 8 N / k g = 4 9 0 N\\n'),\n",
       " ('\\\\ s q u a r e A B C D s t 1 2 \\\\ t i m e s 1 2 \\\\ t i m e s 1 2 \\\\ t i m e s 1 2 \\\\ t i m e s 1 2',\n",
       "  '\\\\square A B C D\\n'),\n",
       " ('r = 8 \\\\ d i v 2 = 4 ( c m )', 'r = 8 \\\\div 2 = 4 ( c m )\\n'),\n",
       " ('3 \\\\ t i m e s ( - 8 ) + 8 = \\\\ f r a c { 3 \\\\ t i m e s ( - 8 ) } { 4 } + m',\n",
       "  '3 \\\\times ( - 8 ) + 8 = \\\\frac { 3 \\\\times ( - 8 ) } { 4 } + m\\n'),\n",
       " ('a _ { n } = 2 ^ { n } ( n \\\\ i n N ^ { \\\\ a s t } ) S',\n",
       "  '. a _ { n } = 2 ^ { n } ( n \\\\in N ^ { \\\\ast } )\\n'),\n",
       " ('S _ { n } = \\\\ v e r t S _ { n } ^ { n } + q ^ { n } \\\\ v e r t n',\n",
       "  'S _ { n } = \\\\beta n ^ { 2 } + q n\\n'),\n",
       " ('+ \\\\ s q r t { 2 x + y } = \\\\ p m \\\\ s q r t { 2 x + y } = \\\\ p m \\\\ s q r t { 6 }',\n",
       "  '\\\\pm \\\\sqrt { 2 x + y } = \\\\pm \\\\sqrt { 2 \\\\times 5 - 4 } = \\\\pm \\\\sqrt { 6 }\\n'),\n",
       " ('2 N a _ { 2 } O _ { 2 } + 2 C O _ { 2 } = 2 N a _ { 2 } C O _ { 3 }',\n",
       "  '2 N a _ { 2 } O _ { 2 } + 2 C O _ { 2 } = 2 N a _ { 2 } C O _ { 3 }\\n'),\n",
       " ('\\\\ f r a c { x } { 1 5 } = \\\\ f r a c { 8 \\\\ t i m e s 3 } { 1 5 \\\\ t i m e s 3 } = \\\\ f r a c { 2 4 } { 4 1 }',\n",
       "  '\\\\frac { 8 } { 1 5 } = \\\\frac { 8 \\\\times 3 } { 1 5 \\\\times 3 } = \\\\frac { 2 4 } { 4 5 }\\n'),\n",
       " ('1 5 \\\\ d i v 5 \\\\ t i m e s 1 = 3 = 3', '1 5 \\\\div 5 \\\\times 1 = 3\\n'),\n",
       " ('y = ( x - 4 0 ) [ 5 0 0 - 1 0 ( x - 5 0 ) ]',\n",
       "  'y = ( x - 4 0 ) [ 5 0 0 - 1 0 ( x - 5 0 ) ]\\n'),\n",
       " ('\\\\ f r a c { 1 } { 4 } + \\\\ f r a c { 1 } { 5 } = \\\\ f r a c { 9 } { 2 0 }',\n",
       "  '\\\\frac { 1 } { 4 } + \\\\frac { 1 } { 5 } = \\\\frac { 9 } { 2 0 }\\n'),\n",
       " ('5 \\\\ f r a c { 1 } { 2 } + \\\\ f r a c { 1 } { 3 } x = \\\\ f r a c { 3 } { 2 } x - 1 \\\\ f r a c { 2 } { 2 }',\n",
       "  '5 \\\\frac { 1 } { 2 } + \\\\frac { 1 } { 3 } x = \\\\frac { 3 } { 2 } x - 1 \\\\frac { 1 } { 2 }\\n'),\n",
       " ('y : \\\\ f r a c { 1 } { 6 } x = + 1', 'y - \\\\frac { 1 } { 6 } z = + 1\\n'),\n",
       " ('x = \\\\ f r a c { 3 2 0 } { 4 1 }', 'x = \\\\frac { 3 2 0 } { 4 1 }\\n'),\n",
       " ('t = \\\\ f r a c { 1 } { 2 } \\\\ t i m e s ( 0 . 3 s + 0 . 5 s ) = 0 . 4 s',\n",
       "  't = \\\\frac { 1 } { 2 } \\\\times ( 0 . 3 s + 0 . 5 s ) = 0 . 4 s\\n'),\n",
       " ('1 0 0 + 2 0 0 + 5 0 0 + 5 0 0 + 5 0 0',\n",
       "  '1 0 0 + 2 0 0 + 3 0 0 + 5 0 0 = 1 1 0 0\\n'),\n",
       " ('( a + 2 b ) ( 2 a - 4 b )', '( a + 2 b ) ( 2 a - 4 b )\\n'),\n",
       " ('x = 2 \\\\ s q r t { 3 }', 'x = 2 \\\\sqrt { 3 }\\n'),\n",
       " ('\" x _ { 1 } = \\\\ f r a c { 5 } { 3 } , x _ { 2 } = - 1 \" \" \" \" \" \" \" , y \" \" \" \" , x _ { 3 } , y \" \" \" \" , \" , \" , x _ { 2 } , \" , \" , \" , \" , \" , \" , 1 , \" , \" , 1 , \"',\n",
       "  'x _ { 1 } = \\\\frac { 5 } { 3 } \\\\times 2 = - 1\\n'),\n",
       " ('\\\\ s q r t [ 3 ] { 4 + \\\\ f r a c { 1 7 } { 2 7 } } = \\\\ f r a c { 5 } { 3 } = \\\\ f r a c { 5 } { 3 }',\n",
       "  '\\\\sqrt [ 3 ] { 4 + \\\\frac { 1 7 } { 2 7 } } = \\\\frac { 5 } { 3 }\\n'),\n",
       " ('\\\\ f r a c { 2 x + 3 } { 4 9 } = \\\\ f r a c { 4 x - 1 } { 7 } - \\\\ f r a c { 5 } { 2 1 }',\n",
       "  '\\\\frac { 2 x + 3 } { 4 9 } = \\\\frac { 4 x - 1 } { 7 } - \\\\frac { 5 } { 2 1 }\\n'),\n",
       " ('\\\\ a n g l e C O D = \\\\ a n g l e C A D = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle C O D = \\\\angle C A P = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ f r a c { 4 } { 5 } : \\\\ f r a c { 3 } { 4 } = 1 6 : 1 5',\n",
       "  '\\\\frac { 4 } { 5 } : \\\\frac { 3 } { 4 } = 1 6 : 1 5\\n'),\n",
       " ('y ^ { 2 } = \\\\ f r a c { 1 } { b ^ { 2 } + 1 }',\n",
       "  'y ^ { 2 } = \\\\frac { 1 } { b ^ { 2 } + 1 }\\n'),\n",
       " ('\\\\ f r a c { x ^ { 2 } } { 4 } + \\\\ f r a c { y ^ { 2 } } { 3 } = 1',\n",
       "  '\\\\frac { x ^ { 2 } } { 4 } + \\\\frac { y ^ { 2 } } { 3 } = 1\\n'),\n",
       " ('k ^ { 2 } - 1 2 < 0', 'k ^ { 2 } - 1 2 < 0\\n'),\n",
       " ('x _ { 1 } = \\\\ f r a c { \\\\ s q r t { m + 3 } } { 4 + 1 }',\n",
       "  'x _ { 1 } = \\\\frac { \\\\sqrt { 3 a + 3 } } { a + 1 }\\n'),\n",
       " ('\\\\ f r a c { A E } { A C } = \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { A E } { A C } = \\\\frac { 2 } { 3 }\\n'),\n",
       " (': t h e t i m e = ( x \\\\ d i v y \\\\ d i v 2 ) k m m',\n",
       "  ': t h e t i m e = ( x \\\\div y \\\\div 2 ) k m\\n'),\n",
       " ('4 y ^ { 2 } + 1 2 y + 9 + 9 y ^ { 2 } = 9',\n",
       "  '4 y ^ { 2 } + 1 2 y + 9 t y ^ { 2 } = 9\\n'),\n",
       " ('f ( x ) = 3 x ^ { 2 } + 1 2 x + 1 2',\n",
       "  'f ^ { \\\\prime } ( x ) = 3 x ^ { 2 } + 1 2 x + 1 2\\n'),\n",
       " ('1 + x - 3 = 0', 'x + y - 3 = 0\\n'),\n",
       " ('W = 4 0 0 ( x + 3 y )', 'W = 4 0 0 ( x + 3 y )\\n'),\n",
       " (': \\\\ f r a c { 1 } { a ^ { 2 } } = \\\\ s q r t { \\\\ f r a c { a ^ { 2 } + b ^ { 2 } } { a ^ { 2 } } } = \\\\ s q r t { 1 + \\\\ f r a c { b ^ { 2 } } { a ^ { 2 } } } \\\\ R i g h t a r r o w \\\\ s q r t { 2 }',\n",
       "  '\\\\frac { c } { a } = \\\\sqrt { \\\\frac { a ^ { 2 } + b ^ { 2 } } { a } } = \\\\sqrt { 1 + \\\\frac { b ^ { 2 } } { a ^ { 2 } } } > \\\\sqrt { 2 }\\n'),\n",
       " ('C _ { 3 } ^ { 2 } ( \\\\ f r a c { 1 } { 2 } ) ^ { 3 } \\\\ d i v \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 3 } { 1 6 }',\n",
       "  'C _ { 3 } ^ { 2 } ( \\\\frac { 1 } { 2 } ) ^ { 3 } \\\\cdot \\\\frac { 1 } { 2 } = \\\\frac { 3 } { 1 6 }\\n'),\n",
       " ('a = \\\\ f r a c { 1 } { 2 }', 'a = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('3 0 m = 3 0 0 0 c m', '3 0 m = 3 0 0 0 c m\\n'),\n",
       " ('1 3 + x = \\\\ f r a c { 1 } { 2 } ( 4 7 + x )',\n",
       "  '1 3 + x = \\\\frac { 1 } { 2 } ( 4 7 + x )\\n'),\n",
       " ('m ^ { 2 } - 5 m - 6 > 0', 'm ^ { 2 } - 5 m - 6 = 0\\n'),\n",
       " ('\\\\ f r a c { 9 } { 4 } + \\\\ f r a c { 3 } { 4 }',\n",
       "  '\\\\frac { 9 } { 4 } + \\\\frac { 3 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 1 5 } + \\\\ f r a c { 4 } { 1 5 } + \\\\ f r a c { 7 } { 1 5 } = \\\\ f r a c { 4 } { 5 }',\n",
       "  '\\\\frac { 1 } { 1 5 } + \\\\frac { 4 } { 1 5 } + \\\\frac { 7 } { 1 5 } = \\\\frac { 4 } { 5 }\\n'),\n",
       " ('x = \\\\ s q r t { 2 } + 1', 'x = \\\\sqrt { 2 } + 1\\n'),\n",
       " ('q = \\\\ f r a c { 4 \\\\ p i + 2 g } { 2 F }',\n",
       "  'q = \\\\frac { 4 \\\\pi \\\\rho r ^ { 3 } g } { 3 E }\\n'),\n",
       " ('\\\\ f r a c { x } { 3 } = \\\\ f r a c { y } { 4 } = x : y = ( 3 ) : ( 4 )',\n",
       "  '\\\\frac { x } { 3 } = \\\\frac { y } { 4 } = x : y = ( 3 ) : ( 4 )\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x + 3 = 1', '\\\\frac { 1 } { 2 } x + 3 = 1\\n'),\n",
       " ('2 n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\ u p a r r o w',\n",
       "  'Z n + H _ { 2 } S O _ { 4 } = Z n S O _ { 4 } + H _ { 2 } \\\\uparrow\\n'),\n",
       " ('\\\\ f r a c { 3 } { 4 } \\\\ t i m e s \\\\ f r a c { 3 \\\\ t i m e s 5 } { 5 4 \\\\ t i m e s 5 } = \\\\ f r a c { 1 5 } { 2 0 }',\n",
       "  '\\\\frac { 3 } { 4 } \\\\times \\\\frac { 3 \\\\times 5 } { 5 4 \\\\times 5 } = \\\\frac { 1 5 } { 2 0 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } ( \\\\ f r a c { 3 } { 2 } x - 1 ) = \\\\ f r a c { 3 } { 2 }',\n",
       "  '\\\\frac { 1 } { 3 } ( \\\\frac { 3 } { 2 } x - 1 ) = \\\\frac { 3 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { 2 4 } { 6 } = 4 = 4 u p = 1 u p u p = 1 4 u p \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\frac { 2 4 } { 6 } = 4\\n'),\n",
       " ('\\\\ f r a c { 1 } { 7 } [ \\\\ f r a c { 1 } { 5 } ( \\\\ f r a c { x + 2 } { 3 } + 4 ) + 6 ] + 8 = 9',\n",
       "  '\\\\frac { 1 } { 7 } [ \\\\frac { 1 } { 5 } ( \\\\frac { x + 2 } { 3 } + 4 ) + 6 ] + 8 = 9\\n'),\n",
       " ('a d = b c = 8', 'a d = b c = 8\\n'),\n",
       " ('x = \\\\ f r a c { 3 } { 4 }', 'x = \\\\frac { 3 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } + \\\\ f r a c { 3 - a } { 3 } = \\\\ f r a c { a } { b } - 3 \\\\ t i m e s \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\frac { 1 } { 2 } + \\\\frac { 3 - a } { 3 } = \\\\frac { a } { b } - 3 \\\\times \\\\frac { 1 } { 2 }\\n'),\n",
       " ('c = 4', 'c = 4\\n'),\n",
       " ('x = - 2 \\\\ p m \\\\ s q r t { 1 0 }', 'x = - 2 \\\\pm \\\\sqrt { 1 0 }\\n'),\n",
       " ('\\\\ c d o t a ^ { 2 } - 4 a + 5 + a ^ { 2 } - 2 a + 1 0 = 1 7',\n",
       "  '\\\\cdot a ^ { 2 } - 4 a + 5 + a ^ { 2 } - 2 a + 1 0 = 1 7\\n'),\n",
       " ('a _ { n + 1 } = \\\\ f r a c { n + 1 } { 2 } a _ { n + 1 } + a _ { n } - 2 a _ { n } - a _ { n } - 1',\n",
       "  'a _ { n + 1 } = \\\\frac { n + 1 } { 2 } a _ { n + 1 } + a _ { n } - \\\\frac { n } { 2 } a _ { n } - a _ { n } - \\\\dot { 1 }\\n'),\n",
       " ('\\\\ a n g l e A + \\\\ a n g l e A C D = 4 5 ^ { \\\\ c i r c } + 9 5 ^ { \\\\ c i r c } = 1 8 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A + \\\\angle A C D = 4 5 ^ { \\\\circ } + 1 3 5 ^ { \\\\circ } = 1 8 0 ^ { \\\\circ }\\n'),\n",
       " ('M G B _ { 5 } + \\\\ a n g l e B G H + \\\\ a n g l e E G M = 1 8 0 ^ { \\\\ c i r c }',\n",
       "  'M G B + \\\\angle B G H + \\\\angle E G M = 1 8 0 ^ { \\\\circ }\\n'),\n",
       " ('S _ { \\\\ D e l t a A B C } = ( 1 + ( n - 1 ) \\\\ t i m e s 3 = 2 n - 2',\n",
       "  '\\\\frac { 1 } { a _ { n } } = \\\\frac { 1 } { a _ { 1 . } } + ( n - 1 ) \\\\times 3 = 3 n - 2\\n'),\n",
       " ('x = \\\\ f r a c { 1 5 } { 4 }', 'x = \\\\frac { 1 5 } { 4 }\\n'),\n",
       " ('\\\\ s q r t { 3 } + \\\\ s q r t { 3 } = \\\\ s q r t { 6 }',\n",
       "  '\\\\sqrt { 3 } + \\\\sqrt { 3 } = \\\\sqrt { 6 }\\n'),\n",
       " ('r = 1 8 . 8 4 \\\\ d i v 3 . 1 4 \\\\ d i v 2 = 3 ( m )',\n",
       "  'r = 1 8 . 8 4 \\\\div 3 . 1 4 \\\\div 2 = 3 ( m )\\n'),\n",
       " ('\\\\ f r a c { a b } { 3 } = S _ { \\\\ D e l t a } a b c = 7',\n",
       "  '\\\\frac { a b } { 2 } = S _ { \\\\Delta } a b c = 7\\n'),\n",
       " ('x _ { 1 } \\\\ c d o t y _ { 1 } + x _ { 2 } \\\\ c d o t ( x _ { 2 } - x _ { 1 } - 1 ) + ( y _ { 2 } - y _ { 1 } ) = 3',\n",
       "  '\"x _ { 2 } , y _ { 2 } \\\\in z ( x _ { i } - x _ { i } - 1 ) + ( y _ { i } - y _ { 2 - 1 } ) = 3\"\\n'),\n",
       " ('\\\\ f r a c { B G } { A G } = \\\\ f r a c { B H } { A F }',\n",
       "  '\\\\frac { B G } { A G } = \\\\frac { B H } { A F }\\n'),\n",
       " ('b - 3 > 0 \\\\ R i g h t a r r o w b > 3', 'k - 3 > 0 \\\\Rightarrow k > 3\\n'),\n",
       " ('- x - 8 = \\\\ f r a c { 1 } { x }', '- x . - 8 = \\\\frac { 1 } { x }\\n'),\n",
       " ('x = \\\\ f r a c { \\\\ p i } { 6 } o r \\\\ f r a c { 2 \\\\ p i } { 3 }',\n",
       "  'x = \\\\frac { \\\\pi } { 6 } o r \\\\frac { 2 \\\\pi } { 3 }\\n'),\n",
       " ('f ^ { \\\\ p r i m e } ( x ) = 3 x ^ { 2 } + 2 b x + 6',\n",
       "  'f ^ { \\\\prime } ( x ) = 3 x ^ { 2 } + 2 b x + 6\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } : \\\\ f r a c { 1 } { 6 } = \\\\ f r a c { 1 } { 2 } : \\\\ f r a c { 1 } { 4 }',\n",
       "  '\\\\frac { 1 } { 3 } : \\\\frac { 1 } { 6 } = \\\\frac { 1 } { 2 } : \\\\frac { 1 } { 4 }\\n'),\n",
       " ('\\\\ f r a c { a } { a - 1 } = a - 1 u p a r r o w u p a r r o w u p a r r o w u p \\\\ u p a r r o w \\\\ u p a r r o w \\\\ u p a r r o w \\\\ u p a r r o w \\\\ u',\n",
       "  '\\\\frac { a } { a - 1 } = a - 1\\n'),\n",
       " ('9 x - 4 x = 6 + 3 + 2', '9 x - 4 x = 6 + 3 + 2\\n'),\n",
       " ('Z n + 2 H C l = Z n C l _ { 2 } + H _ { 2 } \\\\ u p a r r o w',\n",
       "  'Z n + 2 H C l = Z n C l _ { 2 } + H _ { 2 } \\\\uparrow\\n'),\n",
       " ('\\\\ t h e r e f o r e 1 k _ { 2 } = \\\\ f r a c { 6 } { 9 } = 2 9',\n",
       "  'z _ { 1 } + z _ { 2 } = - \\\\frac { b } { - a } = 2 p\\n'),\n",
       " ('1 . 2 x + m - 1 1 + 3 - 2 x \\\\ l e q 6 - 2 x',\n",
       "  '\\\\vert 2 x + m - 1 \\\\vert + 3 - 2 x \\\\leq 6 - 2 x\\n'),\n",
       " ('( 1 ) 4 ( 2 x - 1 ) < 3 、 4 x + 2 ) 1 x + 2 x + 2 1 ) 1 x + 2 1 1 x + 2 1 1 x + 2 1 1 x + 2 1 1 x + 2 1 1 1 x + 2 1 x + 2 1 1 x + 1 1 x + 2',\n",
       "  '( 1 ) 4 ( 2 x - 1 ) < 3 ( 4 x + 2 )\\n'),\n",
       " ('\\\\ f r a c { m ( m + 1 ) } { ( m - 1 ) ^ { 2 } } \\\\ c d o t \\\\ f r a c { m - 1 } { m } = \\\\ f r a c { m + 1 } { m - 1 }',\n",
       "  ': \\\\frac { m ( m + 1 ) } { ( m - 1 ) ^ { 2 } } \\\\cdot \\\\frac { m - 1 } { m } = \\\\frac { m + 1 } { m - 1 }\\n'),\n",
       " ('\\\\ f r a c { D G } { B A } = \\\\ f r a c { A D } { A B } = \\\\ f r a c { 1 } { 3 }',\n",
       "  '\\\\frac { D G } { B A } = \\\\frac { A D } { A B } = \\\\frac { 1 } { 3 }\\n'),\n",
       " ('S _ { 1 } = \\\\ f r a c { v } { h } = 2 . 5 \\\\ t i m e s 1 0 ^ { - 2 } m ^ { 2 }',\n",
       "  'S _ { 1 } = \\\\frac { V } { h } = 2 . 5 \\\\times 1 0 ^ { - 2 } m ^ { 2 }\\n'),\n",
       " ('x _ { 2 } = - \\\\ f r a c { 5 } { 2 }',\n",
       "  'x _ { 2 } = - \\\\frac { 5 } { 2 }\\n'),\n",
       " ('x ( \\\\ f r a c { 1 } { 2 } - \\\\ f r a c { 1 } { 2 } )',\n",
       "  'x _ { 1 } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('\\\\ t e x t c i r c l e d { 2 } C a ( O H ) _ { 2 } + N a _ { 2 } C O _ { 3 } = C a C O _ { 3 }',\n",
       "  'C a ( O H ) _ { 2 } + N a _ { 2 } C O _ { 3 } = C a C O _ { 3 }\\n'),\n",
       " ('\" ( 7 \\\\ e p s i l o n ) P ( \\\\ e p s i l o n ) = \\\\ f r a c { 1 } { 5 2 } \" \" \" \" \" \" \" \" \" ( 1 , - , - \\\\ f r a c { 1 } { 5 2 } , - \\\\ f r a c { 1 } { 5 2 } ) \" \" \" \" \" \" \" (',\n",
       "  'P ( A ) P ( B ) = \\\\frac { 1 } { 5 2 }\\n'),\n",
       " ('b = 3 - \\\\ s q r t { 7 }', 'b = 3 - \\\\sqrt { 7 }\\n'),\n",
       " ('5 \\\\ d i v 8 = \\\\ f r a c { 5 } { 8 }', '5 \\\\div 8 = \\\\frac { 5 } { 8 }\\n'),\n",
       " ('S _ { \\\\ D e l t a } \\\\ a n g l e O C C = \\\\ f r a c { 1 } { 2 } x',\n",
       "  'S _ { \\\\Delta } a b c = \\\\frac { 1 } { 2 } x\\n'),\n",
       " ('A E = \\\\ a n g l e P O', 'A E \\\\bot P N\\n'),\n",
       " ('1 8 0 ^ { \\\\ c i r c } - 4 0 ^ { \\\\ c i r c } - 9 0 ^ { \\\\ c i r c } = 5 0 ^ { \\\\ c i r c }',\n",
       "  '1 8 0 ^ { \\\\circ } - 4 0 ^ { \\\\circ } - 9 0 ^ { \\\\circ } = 5 0 ^ { \\\\circ }\\n'),\n",
       " ('x = 6 \\\\ d i v \\\\ f r a c { 2 } { 7 }', 'x = 6 \\\\div \\\\frac { 2 } { 7 }\\n'),\n",
       " ('\\\\ s q r t [ 3 ] { ( \\\\ s q r t { 2 a - 1 } ) } = - \\\\ s q r t [ 3 ] { 5 }',\n",
       "  '\\\\sqrt [ 3 ] { ( \\\\sqrt { 2 a - 1 } ) } = 3 \\\\sqrt { 5 }\\n'),\n",
       " ('\\\\ t h e r e f o r e \\\\ f r a c { 1 } { 2 } = \\\\ f r a c { 3 ^ { 2 } - 2 b c - ( \\\\ s q r t { 3 } ) ^ { 2 } } { 2 b c } ^ { 2 } ^ { 2 }',\n",
       "  '\\\\therefore \\\\frac { 1 } { 2 } = \\\\frac { 3 ^ { 2 } - 2 b c - ( \\\\sqrt { 3 } ) ^ { 2 } } { 2 b c }\\n'),\n",
       " ('\\\\ f r a c { f ( x ) } { f ( x ) } = \\\\ f r a c { e ^ { 2 x } } { x } < 0',\n",
       "  'f ( x ) = \\\\frac { e ^ { 2 } x } { x } < 0\\n'),\n",
       " ('S _ { \\\\ D e l t a A B C } = \\\\ f r a c { 1 } { 2 } b c \\\\ s i n A = \\\\ f r a c { 1 } { 2 } b c \\\\ c d o t \\\\ f r a c { 5 } { 1 3 } = 1 0',\n",
       "  'S _ { \\\\Delta A B C } = \\\\frac { 1 } { 2 } b c \\\\sin A = \\\\frac { 1 } { 2 } b c \\\\cdot \\\\frac { 5 } { 1 3 } = 1 0\\n'),\n",
       " ('= 1 6 a + 5 - 8 1 b', '= 1 6 a ^ { 4 } - 8 1 b\\n'),\n",
       " ('x = k \\\\ p i - \\\\ f r a c { \\\\ p i } { 1 2 }',\n",
       "  'x = k \\\\pi - \\\\frac { \\\\pi } { 1 2 }\\n'),\n",
       " ('2 L i O H + C O _ { 2 } = L i _ { 2 } C O _ { 3 } + H _ { 2 } O .',\n",
       "  '2 L i O H + C O _ { 2 } = L i _ { 2 } C O _ { 3 } + H _ { 2 } O .\\n'),\n",
       " ('6 2 8 + ( 3 . 1 4 \\\\ d i v 3 . 1 4 \\\\ d i v 2 ) ^ { 2 } \\\\ t i m e s 2 \\\\ t i m e s 3 . 1 4 = 9 8 5 ( c m ^ { 2 } ) 0 8 5 ( c m ^ { 2 } )',\n",
       "  '6 2 8 + ( 3 1 . 4 \\\\div 3 . 1 4 \\\\div 2 ) ^ { 2 } \\\\times 2 \\\\times 3 . 1 4 = 7 8 5 ( c m ^ { 2 } )\\n'),\n",
       " ('6 + 1 = 7 1', '6 + 1 = 7\\n'),\n",
       " ('\\\\ f r a c { 1 } { 3 } + \\\\ f r a c { 1 } { 3 } = \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { 1 } { 3 } + \\\\frac { 1 } { 3 } = \\\\frac { 2 } { 3 }\\n'),\n",
       " ('\\\\ s q r t { a ^ { 2 } + 1 } = \\\\ s q r t { a + ( a ) ^ { 2 } + 1 }',\n",
       "  '\\\\sqrt { a ^ { 2 } + 1 } = \\\\sqrt { ( 1 + a ) ^ { 2 } + 1 }\\n'),\n",
       " ('C E = B E = 2 \\\\ s q r t { 2 }', 'C E = B E = 2 5 2\\n'),\n",
       " ('\\\\ R i g h t a r r o w d _ { A } \\\\ R i g h t a r r o w B C',\n",
       "  '\\\\rightarrow d _ { A } \\\\rightarrow B C\\n'),\n",
       " ('9 6 \\\\ d i v 6 0 = 1 \\\\ c d o t s 2 0', '9 6 \\\\div 6 0 = 1 \\\\cdots 3 6\\n'),\n",
       " ('\\\\ f r a c { B E } { A B } = \\\\ f r a c { A E } { A C }',\n",
       "  '\\\\frac { B E } { A B } = \\\\frac { A E } { A C }\\n'),\n",
       " ('m - 2 0 + 2 0 = 6 + 2 0', 'm - 2 0 + 2 0 = 6 + 2 0\\n'),\n",
       " ('f ( - x ) = \\\\ f r a c { 1 } { 2 ^ { x } } + 2 ^ { x } - 1',\n",
       "  'f ( - x ) = \\\\frac { 1 } { 2 ^ { x } } + 2 ^ { x } - 1\\n'),\n",
       " ('7 . 5 2 \\\\ d i v 7 . 0 1 \\\\ d i v 0 . 5 1 . 6 8 8 0 = 7 6 . 6 8 5 5 . 1 2 2',\n",
       "  '7 . 5 2 - 7 . 0 1 = 0 . 5 1 7 6 . 8 0 - 7 1 . 6 8 = 5 . 1 2 .\\n'),\n",
       " ('\\\\ f r a c { 1 } { R } < \\\\ f r a c { 1 } { 2 5 \\\\ c d o t 2 ^ { x } }',\n",
       "  '\\\\frac { 1 } { k } < \\\\frac { 1 } { 2 ^ { x 1 } 2 ^ { x 2 } }\\n'),\n",
       " ('N a _ { 2 } S O _ { 4 } + B a C l _ { 2 } = B a C l _ { 2 } \\\\ d o w n a r r o w + 2 N a C l',\n",
       "  'N a _ { 2 } S O _ { 4 } + B a C l _ { 2 } = B a C l _ { 2 } \\\\downarrow + 2 N a C l\\n'),\n",
       " ('A D = \\\\ f r a c { 1 } { 2 } A C = 1 2 c m',\n",
       "  'A D = \\\\frac { 1 } { 2 } A C = 1 2 c m .\\n'),\n",
       " ('\\\\ f r a c { B D } { A B } = \\\\ f r a c { A B } { B C }',\n",
       "  '\\\\frac { B D } { A B } = \\\\frac { A B } { B C }\\n'),\n",
       " ('y = \\\\ s q r t { 2 0 0 }', 'y = \\\\sqrt { 2 0 0 }\\n'),\n",
       " ('2 8 . 2 6 \\\\ d i v 3 = 9 . 4 2 ( c m ^ { 3 } )',\n",
       "  '2 8 . 2 6 \\\\div 3 = 9 . 4 2 ( c m ^ { 3 } )\\n'),\n",
       " ('5 6 + 4 0 0 = 9 6 9 5 - 4 4 8 =', '5 6 + 4 0 = 9 6 9 5 - 4 0 =\\n'),\n",
       " ('2 5 \\\\ t i m e s 1 0 \\\\ t i m e s 1 4 = 3 5 0 0 ( c m ^ { 3 } )',\n",
       "  '2 5 \\\\times 1 0 \\\\times 1 4 = 3 5 0 0 ( c m ^ { 3 } )\\n'),\n",
       " ('\\\\ f r a c { 2 } { 3 } \\\\ l e q \\\\ t h e t a \\\\ l e q \\\\ f r a c { 2 } { 3 } \\\\ p i',\n",
       "  '\\\\cdot \\\\frac { x } { 3 } \\\\leq \\\\theta \\\\leq \\\\frac { 2 } { 3 } \\\\pi\\n'),\n",
       " ('\\\\ c o s C = \\\\ f r a c { \\\\ s q r t { 3 } } { 2 }',\n",
       "  '\\\\cos C = \\\\frac { \\\\sqrt { 3 } } { 2 }\\n'),\n",
       " ('R = \\\\ f r a c { U } { I }', 'R = \\\\frac { U } { I }\\n'),\n",
       " ('\\\\ a r c s i n ( \\\\ f r a c { 1 } { 2 } + \\\\ f r a c { 1 } { \\\\ s q r t { 2 } } ) = \\\\ f r a c { 1 } { 2 } + \\\\ f r a c { 1 } { \\\\ s q r t { 2 } } + \\\\ f r a c { 1 } { \\\\ s q r t { 2 } } + \\\\ f r a c { 1 } { \\\\',\n",
       "  '4 a + c = ( 4 a + c ) ( \\\\frac { 1 } { a } + \\\\frac { 1 } { c } ) = 5 x + \\\\frac { c } { a } + \\\\frac { 4 a } { 4 c } \\\\geq 5 + \\\\sqrt [ 2 ] { \\\\frac { c } { a } \\\\cdot \\\\frac { 4 a } { c } } = 9 .\\n'),\n",
       " ('\\\\ f r a c { 2 x ( - 1 ) - a } { 3 } - \\\\ f r a c { - 1 - a } { 2 } = - 1 : - 1',\n",
       "  '\\\\frac { 2 x ( - 1 ) - a } { 3 } - \\\\frac { - 1 - a } { 2 } = - 1 = - 1\\n'),\n",
       " ('k = \\\\ f r a c { 1 } { 2 }', 'k = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('2 a + \\\\ f r a c { 2 } { 3 } \\\\ p i = 2 k \\\\ p i + \\\\ f r a c { 2 } { 3 } \\\\ p i',\n",
       "  '2 \\\\alpha + \\\\frac { 2 } { 3 } \\\\pi = 2 k \\\\pi + \\\\frac { 2 } { 3 } \\\\pi\\n'),\n",
       " ('\\\\ f r a c { 5 } { 7 } \\\\ t i m e s 5', '\\\\frac { 4 } { 7 } \\\\times 5\\n'),\n",
       " ('1 0 \\\\ t i m e s 3 0 \\\\ t i m e s 3 0 = 2 7 0 0 0 ( c m ^ { 3 } )',\n",
       "  '3 0 \\\\times 3 0 \\\\times 3 0 = 2 7 0 0 0 ( c m ^ { 3 } )\\n'),\n",
       " ('\\\\ f r a c { 4 0 0 0 0 } { x } - \\\\ f r a c { 4 0 0 0 + 8 0 0 0 } { ( 1 + 2 0 \\\\ % ) x } = 3',\n",
       "  '\\\\frac { 4 0 0 0 } { x } - \\\\frac { 4 0 0 0 + 8 0 } { ( 1 + 2 0 \\\\% ) x } = 3\\n'),\n",
       " ('\\\\ f r a c { 4 } { 2 4 } x = - \\\\ f r a c { 3 } { 2 }',\n",
       "  '\\\\frac { 4 } { 2 4 } x = - \\\\frac { 3 } { 2 }\\n'),\n",
       " ('( - \\\\ f r a c { 1 } { n } ) ( - 2 ) = - 1',\n",
       "  '( - \\\\frac { 1 } { n } ) ( - 2 ) = - 1\\n'),\n",
       " ('\\\\ a n g l e F D B = \\\\ a n g l e B A D + \\\\ a n g l e A D D , \\\\ a n g l e B A D = \\\\ a n g l e C D',\n",
       "  '\"\\\\angle F D E = \\\\angle B A D + \\\\angle A B D , \\\\angle B A D = \\\\angle C\"\\n'),\n",
       " ('\\\\ f r a c { A D } { A E } = k', '\\\\frac { A D } { A E } = k\\n'),\n",
       " ('B a C l _ { 2 } + N a O H = B a C O _ { 3 } \\\\ d o w n a r r o w + 2 N a C l',\n",
       "  'B a C l _ { 2 } + N a _ { 2 } C O _ { 3 } = B a C O _ { 3 } t + 2 N a C l\\n'),\n",
       " ('\\\\ s i n a = - \\\\ s q r t { 1 - \\\\ c o s ^ { 2 } a } = - \\\\ f r a c { 3 } { 5 } = - \\\\ f r a c { 2 } { 5 }',\n",
       "  '\\\\sin a = - \\\\sqrt { 1 - \\\\cos ^ { 2 } a } = - \\\\frac { 3 } { 5 }\\n'),\n",
       " ('\\\\ a n g l e B A M = 2 2 . 5 ^ { \\\\ c i r c } r i g h t a r r o w . 5 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B A M = 2 2 . 5 ^ { \\\\circ }\\n'),\n",
       " ('6 0 m = \\\\ f r a c { 3 } { 5 } d m', '6 c m = \\\\frac { 3 } { 5 } d m\\n'),\n",
       " ('x = \\\\ f r a c { 1 } { 1 2 } \\\\ d i v \\\\ f r a c { 5 } { 8 }',\n",
       "  'x = \\\\frac { 1 } { 1 2 } \\\\div \\\\frac { 5 } { 8 }\\n'),\n",
       " ('S _ { \\\\ D e l t a } = 2 s = 2 \\\\ t i m e s 2 5 0 c m = 5 0 0 c m ^ { 2 }',\n",
       "  'S _ { x } = 2 s = 2 \\\\times 2 5 0 c m = 5 0 0 c m ^ { 3 }\\n'),\n",
       " ('e ^ { 2 } = \\\\ f r a c { m } { v }', '\\\\rho = \\\\frac { m } { V }\\n'),\n",
       " ('C D = \\\\ f r a c { C A \\\\ c d o t B C } { A B \\\\ c d o t } = \\\\ f r a c { 1 2 } { 5 }',\n",
       "  'C D = \\\\frac { C A \\\\cdot B C } { A B \\\\cdot } = \\\\frac { 1 2 } { 5 }\\n'),\n",
       " (': C a ( O H ) _ { 2 } = C a ( O H ) _ { 2 } \\\\ d o w n a r r o w = a ( O H ) _ { 2 } \\\\ d o w n a r r o w + C a C l _ { 2 } = 1 a C l _ { 2 } o o',\n",
       "  '\": a + ( b ) = a + ( a ) = a + ( a + 1 ) = a ^ { 2 } + a , a ( - E 1 , 0 ) .\"\\n'),\n",
       " ('1 8 \\\\ t i m e s 2 = 3 6 m ^ { 2 }', '1 8 \\\\times 2 = 3 6 m ^ { 2 }\\n'),\n",
       " ('B _ { 4 } C O _ { 3 } + H N O _ { 3 } = B _ { a } N O _ { 3 } + C O _ { 2 } \\\\ u p a r r o w + H _ { 2 } O',\n",
       "  'B a C O _ { 3 } + H N O _ { 3 } = B a N O _ { 3 } + C O _ { 2 } \\\\uparrow + H _ { 2 } O\\n'),\n",
       " ('( 1 ) \\\\ b e c a u s e \\\\ D e l t a ( - 1 \\\\ c d o t 0 )',\n",
       "  '\"( 1 ) \\\\because A ( - 1 , 0 )\"\\n'),\n",
       " ('9 - 2 x + 6 = x - 3 - 6 x 0', '9 - 2 x + 6 = x - 3 - 6 x\\n'),\n",
       " ('3 x = \\\\ f r a c { m } { 4 }', '3 x = \\\\frac { m } { 4 }\\n'),\n",
       " ('3 b + \\\\ f r a c { 5 } { 3 } \\\\ t i m e s 2',\n",
       "  '3 b + \\\\frac { 5 } { 3 } \\\\times 2\\n'),\n",
       " ('x = - \\\\ f r a c { 3 } { 7 }', 'x = - \\\\frac { 3 } { 7 }\\n'),\n",
       " ('\\\\ f r a c { B D } { C E } = \\\\ f r a c { A B } { A C } = k',\n",
       "  '\\\\frac { B D } { C E } = \\\\frac { A B } { A C } = k\\n'),\n",
       " ('\\\\ a n g l e C O E = 1 8 0 ^ { \\\\ c i r c } - \\\\ a n g l e E O D = 1 8 0 ^ { \\\\ c i r c } - 5 5 ^ { \\\\ c i r c } . 1 =',\n",
       "  '\\\\angle C O E = 1 8 0 ^ { \\\\circ } - \\\\angle E O D = 1 8 0 ^ { \\\\circ } - 5 5 ^ { \\\\circ } = 1 2\\n'),\n",
       " ('S : 2 y = - \\\\ f r a c { 4 } { x }', '( 1 ) y = - \\\\frac { 4 } { x }\\n'),\n",
       " (': 3 \\\\ t i m e s 3 \\\\ t i m e s 1 8 = 1 6 2 c m ^ { 2 }',\n",
       "  '3 \\\\times 3 \\\\times 1 8 = 1 6 2 c m ^ { 2 }\\n'),\n",
       " ('\\\\ f r a c { 1 5 } { 4 } < - ( 6 - 2 ) ^ { 2 } + 4 \\\\ l e 4',\n",
       "  '- \\\\frac { 1 5 } { 4 } < - ( 6 - 2 ) ^ { 2 } + 4 \\\\leq 4\\n'),\n",
       " ('\\\\ f r a c { 6 5 } { 3 } = 2 1 \\\\ f r a c { 2 } { 3 }',\n",
       "  '\\\\frac { 6 5 } { 3 } = 2 1 \\\\frac { 2 } { 3 }\\n'),\n",
       " ('x = \\\\ f r a c { 3 } { 2 }', 'x = \\\\frac { 3 } { 2 }\\n'),\n",
       " ('S = - 2 5 0 \\\\ t i m e s \\\\ f r a c { 3 0 0 } { 1 7 } + 5 0 0 0',\n",
       "  'S = - 2 5 0 \\\\times \\\\frac { 3 0 0 } { 1 7 } + 5 0 0 0\\n'),\n",
       " ('\\\\ f r a c { x ( x + 1 ) } { 2 } = 1 0',\n",
       "  '\\\\frac { x ( x + 1 ) } { 2 } = 1 0\\n'),\n",
       " ('A P = 4 0 + 3 0 ( \\\\ s q r t { 3 } - 1 )',\n",
       "  'A P = 4 0 + 3 0 ( \\\\sqrt { 3 } - 1 )\\n'),\n",
       " ('\\\\ a n g l e O C F = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e A C D',\n",
       "  '\\\\angle O C F = \\\\frac { 1 } { 2 } \\\\angle A C D\\n'),\n",
       " ('- 6 . 5 5 ) ] = 4 9 \\\\ f r a c { 7 } { 1 0 } - [ - 2 3 \\\\ f r a c { 3 } { 4 } + ( - 6 \\\\ f r a c { 1 1 } { 2 0 } ) ]',\n",
       "  '- 6 . 5 5 ) ] = 4 9 \\\\frac { 7 } { 1 0 } - [ - 2 3 \\\\frac { 3 } { 4 } + ( - 6 \\\\frac { 1 1 } { 2 0 } ) ]\\n'),\n",
       " ('F e + H _ { 2 } S O 4 = H _ { 2 } \\\\ u p a r r o w + F e S O 4',\n",
       "  'F e + H _ { 2 } S O 4 = H _ { 2 } \\\\uparrow + F e S O 4\\n'),\n",
       " ('x = \\\\ f r a c { 3 } { 4 } + \\\\ f r a c { 1 } { 8 }',\n",
       "  'x = \\\\frac { 3 } { 4 } + \\\\frac { 1 } { 8 }\\n'),\n",
       " ('1 0 \\\\ t i m e s 2 = 2 0 ( c m )', '1 0 \\\\times 2 = 2 0 ( c m )\\n'),\n",
       " ('m c = 1 2 \\\\ t i m e s 0 . 1 = 1 . 2 g',\n",
       "  'm c = 1 2 \\\\times 0 . 1 = 1 . 2 g\\n'),\n",
       " ('\\\\ f r a c { x } { x + 1 } = - 2 2', '\\\\frac { x } { x + 1 } = - 2\\n'),\n",
       " ('\\\\ a n g l e E C D = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle E C D = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } \\\\ t i m e s ^ { 2 } + x y + \\\\ f r a c { 1 } { 2 } y',\n",
       "  '\\\\frac { 1 } { 2 } x ^ { 2 } + x y + \\\\frac { 1 } { 2 } y ^ { 2 }\\n'),\n",
       " ('( x ^ { 2 } + y + z ^ { 2 } ) ( 1 + 4 + 9 ) \\\\ g e q ( x + 2 y + 3 z ) ^ { 2 }',\n",
       "  '( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ) ( 1 + 4 + 9 ) \\\\geqslant ( x + 2 y + 3 z ) ^ { 2 }\\n'),\n",
       " ('\" P ( 1 ) \\\\ a n g l e A D , \\\\ a n g l e 2 A 2 , A 1 3 \" \" \" \" \" \" \" \" \" \" \" \" \" , \" , \" , \" , \" , \" , \" , \" , \" , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1',\n",
       "  '\"D C D \\\\bot A D , D A \\\\bot A B\"\\n'),\n",
       " ('f ( x ) = \\\\ v e r t x - a \\\\ v e r t + 3 x',\n",
       "  'f ( x ) = \\\\vert x - a \\\\vert + 3 x\\n'),\n",
       " ('\\\\ f r a c { 5 } { 6 } \\\\ t e x t c i r c l e d { > } \\\\ f r a c { 5 } { 8 }',\n",
       "  '\\\\frac { 5 } { 6 } \\\\textcircled { < } \\\\frac { 5 } { 8 }\\n'),\n",
       " ('Z n + H _ { 2 } S O _ { 4 } = H _ { 2 } \\\\ u p a r r o w + Z n S O _ { 4 }',\n",
       "  'Z n + H _ { 2 } S O _ { 4 } = H _ { 2 } \\\\uparrow + Z n S O _ { 4 }\\n'),\n",
       " ('n = \\\\ f r a c { 2 } { 3 } = \\\\ f r a c { 2 } { 3 } g h c a r i g h t ) { \\\\ f r a c { 2 } { 3 } } \\\\ f r a c { 2 } { 3 } } { 3 } \\\\ f r a c { 2 } { 3 } \\\\ f r a c { 2',\n",
       "  'n = \\\\frac { 2 } { 3 }\\n'),\n",
       " ('V _ { \\\\ D e l t a A = V } = \\\\ f r a c { 2 m g \\\\ t i m e s R } { 3 R ^ { 2 } }',\n",
       "  'V _ { a b } = V = \\\\frac { 2 m g \\\\cdot R } { 3 B ^ { 2 } l ^ { 2 } }\\n'),\n",
       " ('. x = 6 + 4 \\\\ s q r t [ 6 ] { 2 }', '. x = 6 + 4 \\\\sqrt { 2 }\\n'),\n",
       " ('8 0 0 0 \\\\ t i m e s \\\\ f r a c { 1 } { 4 0 0 0 } = 2 c m',\n",
       "  '8 0 0 0 \\\\times \\\\frac { 1 } { 4 0 0 0 } = 2 c m\\n'),\n",
       " ('x ^ { 2 } + \\\\ s q r t { 3 } t x + t ^ { 2 } - 1 = 0 .',\n",
       "  '. x ^ { 2 } + \\\\sqrt { 3 } t x + t ^ { 2 } - 1 = 0 .\\n'),\n",
       " ('\\\\ s i n A = \\\\ f r a c { a } { 2 R }', '\\\\sin A = \\\\frac { a } { 2 R }\\n'),\n",
       " ('k ^ { 2 } = \\\\ f r a c { 1 } { 2 }', 'k ^ { 2 } = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('x = \\\\ f r a c { 2 2 0 } { 1 7 }', 'x = \\\\frac { 2 2 0 } { 1 2 }\\n'),\n",
       " ('1 4 4 \\\\ d i v 2 = 7 2 ( m ^ { 2 } )',\n",
       "  '1 4 4 \\\\div 2 = 7 2 ( m ^ { 2 } )\\n'),\n",
       " ('\" \\\\ o v e r r i g h t a r r o w { P A } = ( - 6 - x _ { 1 } , - \\\\ s q r t { 2 0 ( 1 - \\\\ f r a c { x ^ { 2 } } { 3 6 } ) } ) \" \" \" ) \" \" \" \" \" \" , \" \" , \" , 1 , \" , 1 , \"',\n",
       "  '\\\\overrightarrow { P A } = ( - 6 - x _ { 1 } - \\\\sqrt { 2 0 ( 1 - \\\\frac { x ^ { 2 } } { 3 6 } ) }\\n'),\n",
       " ('A C = b c', 'a c = b c\\n'),\n",
       " ('b = \\\\ f r a c { 8 } { 2 } h = \\\\ f r a c { 8 } { 2 } P } > 2',\n",
       "  '6 = \\\\frac { 8 } { 2 } h\\n'),\n",
       " ('d _ { n } = \\\\ f r a c { a _ { n } } { n ( n + 1 ) }',\n",
       "  'd _ { n } = \\\\frac { a _ { n } } { n ( n + 1 ) }\\n'),\n",
       " ('\\\\ D e l t a A B E \\\\ c o n g \\\\ D e l t a D B E',\n",
       "  '\\\\Delta A B E \\\\cong \\\\Delta D B E\\n'),\n",
       " ('m : 2 0 = 2 0 0 5 0 - 1 = - \\\\ f r a c { 7 } { 2 5 } = - \\\\ f r a c { 7 } { 2 5 }',\n",
       "  '3 0 : 2 \\\\theta = 2 \\\\cos ^ { 2 } \\\\theta - 1 = - \\\\frac { 7 } { 2 5 }\\n'),\n",
       " ('1 \\\\ d i v 1 0 0 = 0 . 0 1 ( c m )', '1 \\\\div 1 0 0 = 0 . 0 1 ( c m )\\n'),\n",
       " ('2 m - 3 + 5 - m = 0', '2 m - 3 + 5 - m = 0\\n'),\n",
       " ('( 1 2 - 4 y ) ( 1 2 - 3 y ) = 0', '( 1 2 - 4 y ) ( 1 2 - 3 y ) = 0\\n'),\n",
       " ('2 \\\\ t i m e s 1 8 = 3 6 . 1 2 \\\\ t i m e s 3 = 3 1',\n",
       "  '2 \\\\times 1 8 = 3 6 . 1 2 \\\\times 3 = 3 6\\n'),\n",
       " ('M N = 3 0 \\\\ s q r t { 3 } - 3 0 \\\\ a p p r o x 3 3 0 ( \\\\ s q r t { 3 } - 1 )',\n",
       "  'M N = 3 0 \\\\sqrt { 3 } - 3 0 \\\\approx = 3 0 ( \\\\sqrt { 3 } - 1 )\\n'),\n",
       " ('a + 1 5 - 1 5 = 2 5 - 1 5', 'a + 1 5 - 1 5 = 2 5 - 1 5\\n'),\n",
       " ('A H \\\\ b o t B C', 'A H \\\\bot B C\\n'),\n",
       " ('3 1 0 \\\\ d i v 5 0 = 6 \\\\ c d o t s 1 0',\n",
       "  '3 1 0 \\\\div 5 0 = 6 \\\\cdots 1 0\\n'),\n",
       " ('\\\\ t h e r e f o r e \\\\ w i d e h a t { B D } = \\\\ w i d e h a t { C D }',\n",
       "  '\\\\therefore \\\\widehat { B D } = \\\\overline { C D }\\n'),\n",
       " ('\\\\ f r a c { 3 + m } { 4 } \\\\ l e q 0',\n",
       "  '\\\\frac { 3 - 4 m } { 4 } \\\\leq 0\\n'),\n",
       " ('m = t _ { 1 } \\\\ c d o t V = 5 S . 0 . 3 m / s = 1 . 5 m',\n",
       "  'h _ { 1 } = t _ { 1 } \\\\cdot V = 5 S . 0 . 3 m / s = 1 . 5 m\\n'),\n",
       " ('x ^ { 2 } = 5 + 5 + 6 = 0', 'x ^ { 2 } - 5 x + 6 = 0\\n'),\n",
       " ('[ ( 5 6 + 4 4 ) \\\\ d i v 2 ] \\\\ t i m e s ( 5 6 - 4 4 ) = 6 0 0',\n",
       "  '[ ( 5 6 + 4 4 ) \\\\div 2 ] \\\\times ( 5 6 - 4 4 ) = 6 0 0\\n'),\n",
       " ('\\\\ o v e r r i g h t a r r o w { C B } = \\\\ o v e r r i g h t a r r o w { C _ { 1 } } \\\\ o v e r r i g h t a r r o w { C D }',\n",
       "  '\"\\\\overrightarrow { C B } = 1 1 , 0 0\"\\n'),\n",
       " ('( x + 1 ) \\\\ t i m e s 0 \\\\ c d o t 2 x = \\\\ f r a c { 1 } { 2 }',\n",
       "  '( \\\\sqrt { 5 } + 1 ) x \\\\cdot 2 x \\\\times \\\\frac { 1 } { 2 }\\n'),\n",
       " ('( \\\\ f r a c { 3 } { 2 } + 1 ) \\\\ t i m e s \\\\ f r a c { 2 } { 3 } = 2',\n",
       "  '( \\\\frac { 3 } { 2 } a + b ) ^ { 2 } _ { ; } \\\\geq 0\\n'),\n",
       " ('4 - 4 ( 2 k - 4 ) > 0', '4 - 4 ( 2 k - 4 ) > 0\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x - 1 = 1', '\\\\frac { 1 } { 2 } x - 1 = 1\\n'),\n",
       " ('2 ^ { n } b _ { n } = 2 ^ { a _ { n } } - a ^ { a _ { n - 1 } } = 2 ^ { 2 }',\n",
       "  '2 ^ { n } b _ { n } = 2 ^ { C m } - a ^ { a n - 1 } = 2 ^ { 2 }\\n'),\n",
       " ('A E = 3 \\\\ s q r t { 3 } x = 5 \\\\ s q r t { 3 } \\\\ s q r t { 3 } \\\\ f r a c { 1 } { 3 } \\\\ s q r t { 3 } \\\\ s q r t { 3 } 3 \\\\ s q r t { 3 } 3 \\\\ s q r t { 3 } 3',\n",
       "  'A E = 3 \\\\sqrt { 3 }\\n'),\n",
       " ('- \\\\ f r a c { 1 } { 2 } x - 3 x = 6 - 8 + 3',\n",
       "  '- \\\\frac { 1 } { 2 } x - 3 x = 6 - 8 + 3\\n'),\n",
       " ('\\\\ a n g l e 1 = \\\\ f r a c { 7 } { 2 }', 'y = \\\\frac { 7 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { G D } { A D } = \\\\ f r a c { 3 x } { 3 - 2 x } = \\\\ f r a c { 4 } { 5 }',\n",
       "  '\\\\frac { G D } { A D } = \\\\frac { 3 x } { 3 - 2 x } = \\\\frac { 4 } { 5 }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 6 } < \\\\ f r a c { 5 } { 6 } < \\\\ f r a c { 1 } { 6 } \\\\ f r a c { 1 } { 6 } \\\\ f r a c { 1 } { 6 } \\\\ f r a c {',\n",
       "  '\\\\frac { 4 } { 6 } < \\\\frac { 5 } { 6 } < 1 < \\\\frac { 3 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { x - 4 } { 3 } - 8 = - \\\\ f r a c { x + 2 } { 2 }',\n",
       "  '\\\\frac { x - 4 } { 3 } - 8 = - \\\\frac { x + 2 } { 2 }\\n'),\n",
       " ('W = - x ^ { 2 } + 7 x + 4 8 = - ( x - 3 . 5 ) ^ { 2 } + 6 0 . 2 5',\n",
       "  'W = - x ^ { 2 } + 7 x + 4 8 = - ( x - 3 . 5 ) ^ { 2 } + 6 0 . 2 5\\n'),\n",
       " ('t = 0 . S _ { 3 } = ( - \\\\ f r a c { 1 } { 4 } T',\n",
       "  't = 0 . 5 s = 1 \\\\frac { 1 } { 4 } T\\n'),\n",
       " ('8 0 0 \\\\ t i m e s 3 = 2 4 0 0 ( m )', '8 0 0 \\\\times 3 = 2 4 0 0 ( m )\\n'),\n",
       " ('\\\\ t a n \\\\ f r a c { \\\\ t h e t a } { 2 } = \\\\ f r a c { \\\\ s q r t { 2 } } { 2 }',\n",
       "  '\\\\tan \\\\frac { \\\\theta } { 2 } = \\\\frac { \\\\sqrt { \\\\xi } } { 2 }\\n'),\n",
       " ('B E = \\\\ f r a c { 1 } { 2 } C E', 'B E = \\\\frac { 1 } { 2 } C E\\n'),\n",
       " ('b _ { n } = \\\\ f r a c { 1 + 2 n a _ { n } } { a _ { n } } = \\\\ f r a c { 1 } { a _ { n } } + 2 n = \\\\ f r a c { 1 } { 2 ^ { n - 1 } } + 2 n',\n",
       "  'b _ { n } = \\\\frac { 1 + 2 n a _ { n } } { a _ { n } } = \\\\frac { 1 } { a _ { n } } + 2 n = \\\\frac { 1 } { 2 ^ { n - 1 } } + 2 n\\n'),\n",
       " ('x = \\\\ f r a c { 5 } { 3 } + k', 'x = \\\\frac { 5 } { 3 } + k\\n'),\n",
       " ('4 \\\\ t i m e s 8 \\\\ t i m e s 2 = 6 4 ( c m ^ { 2 } )',\n",
       "  '4 \\\\times 8 \\\\times 2 = 6 4 ( c m ^ { 2 } )\\n'),\n",
       " ('\\\\ t a n 6 3 4 ^ { \\\\ c i r c } = \\\\ f r a c { C E } { B E }',\n",
       "  '\\\\tan 6 3 . 4 ^ { \\\\circ } = \\\\frac { C E } { B E }\\n'),\n",
       " ('\\\\ v e r t \\\\ o v e r r i g h t a r r o w { a } + \\\\ o v e r r i g h t a r r o w { b } \\\\ v e r t = \\\\ s q r t { 1 3 } .',\n",
       "  '\\\\vert \\\\overrightarrow { a } + \\\\overrightarrow { b } \\\\vert = \\\\sqrt { 1 3 } .\\n'),\n",
       " ('\\\\ f r a c { 1 } { 5 } \\\\ t i m e s \\\\ f r a c { 3 } { 4 } + \\\\ f r a c { 1 } { 5 } \\\\ t i m e s \\\\ f r a c { 3 } { 4 } \\\\ n = \\\\ f r a c { 3 } { 4 } \\\\ t i m e s \\\\ l e f t ( \\\\ f r a c { 1 } { 5 } + \\\\ f r a c',\n",
       "  'B a C l _ { 2 } + N a _ { 2 } C O _ { 3 } = B a C O\\n'),\n",
       " ('\\\\ a n g l e C P Q = \\\\ a n g l e B = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle C P Q = \\\\angle B = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('x _ { 4 } = - \\\\ s q r t { 6 } - 2', 'x _ { 4 } = - \\\\sqrt { 6 0 } - 2\\n'),\n",
       " ('( m - n ) ^ { 2 } = 1', '( m - n ) ^ { 2 } = 1\\n'),\n",
       " ('\\\\ a n g l e A B C = \\\\ a n g l e A = 4 5 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A B C = \\\\angle A = 4 5 ^ { \\\\circ }\\n'),\n",
       " ('n = \\\\ f r a c { 4 } { 3 }', 'x = \\\\frac { 4 } { 3 }\\n'),\n",
       " ('x _ { 1 } + x _ { 2 } = \\\\ f r a c { 1 2 k ^ { 2 } } { 3 k ^ { 2 } + 1 } = \\\\ f r a c { 1 2 k ^ { 2 } } { 3 k ^ { 2 } + 1 }',\n",
       "  'x _ { 1 } + x _ { 2 } = \\\\frac { 1 2 k ^ { 2 } } { 3 k ^ { 2 } + 1 }\\n'),\n",
       " ('B = B _ { 1 } + k x', 'B = B _ { 0 } + k x\\n'),\n",
       " ('b = - \\\\ f r a c { 1 } { 3 }', 'b = - \\\\frac { 1 } { 3 }\\n'),\n",
       " ('y _ { 2 } = - \\\\ f r a c { 1 } { 3 }',\n",
       "  'y _ { 2 } = - \\\\frac { 1 } { 3 }\\n'),\n",
       " ('f ^ { \\\\ p r i m e } ( x ) = - \\\\ f r a c { 1 } { x ^ { 2 } } - 1 + \\\\ f r a c { a } { x } = \\\\ f r a c { a x - x ^ { 2 } - 1 } { x ^ { 2 } } ^ { 2 }',\n",
       "  'f ^ { \\\\prime } ( x ) = - \\\\frac { 1 } { x ^ { 2 } } - 1 + \\\\frac { a } { x } = \\\\frac { a x - x ^ { 2 } - 1 } { x ^ { 2 } }\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } Q = 1 0 . 5', '\\\\frac { 1 } { 2 } x = 1 0 . 5\\n'),\n",
       " ('F _ { 1 } L _ { 1 } = F _ { 2 } L _ { 2 }',\n",
       "  'F _ { 1 } L _ { 1 } = F _ { 2 } L 2\\n'),\n",
       " ('\" c d = 1 , m ^ { 2 } = 4 \" \" \" \" \" \" \" \" = 1 \" \" \" \" = 1 , \" \" = 1 , \" , m , - 1 , - 1 , - 1 , - 1 \" = 2 , - 1 \" \" = 2 1 , - 1 , - 1 , - 1 , - \" = - 1 , - 1 , \" = - 1 , - 1',\n",
       "  '\"c d = 1 , m ^ { 2 } = 4\"\\n'),\n",
       " ('6 0 k g \\\\ t i m e s \\\\ f r a c { 1 } { 6 } = 1 0 k g',\n",
       "  '6 0 k g \\\\times \\\\frac { 1 } { 6 } = 1 0 k g\\n'),\n",
       " ('\\\\ v e r t A B \\\\ v e r t = \\\\ f r a c { \\\\ s q r t { 2 } } { 2 } \\\\ v e r t x _ { 1 } \\\\ v e r t = \\\\ f r a c { \\\\ s q r t { 2 } } { 2 } \\\\ v e r t x _ { 2 } \\\\ v e r t = \\\\ f r a c { \\\\ s q r t { 2 } } { 2 } \\\\',\n",
       "  '3 . 1 4 \\\\times 2 \\\\times 1 . 8 \\\\div 3 \\\\times 1 . 8 = 1 3 . 5 6 4 8\\n'),\n",
       " ('6 > \\\\ o v e r r i g h t a r r o w { 1 } g i n o r r i g h t a r r o w { 1 } \\\\ o v e r r i g h t a r r o w { 1 } \\\\ o v e r r i g h t a r r o w { 1 } \\\\ o v e',\n",
       "  '6 > \\\\square\\n'),\n",
       " ('x = \\\\ f r a c { 2 4 } { 7 }', 'x = \\\\frac { 2 4 } { 7 }\\n'),\n",
       " ('x ^ { 2 } - a x - 3 x + 3 = x ^ { 2 } - x',\n",
       "  'x ^ { 2 } - a x - 3 x + 3 = x ^ { 2 } - x\\n'),\n",
       " ('\\\\ t h e r e f o r e \\\\ a n g l e A O C = 3 6 0 ^ { \\\\ c i r c } - \\\\ a n g l e A O B - \\\\ a n g l e B O C',\n",
       "  '\\\\angle A O C = 3 6 0 ^ { \\\\circ } - \\\\angle A O B - \\\\angle B O C\\n'),\n",
       " ('\\\\ f r a c { 1 } { 2 } x = 1 0', '\\\\frac { 1 } { 2 } x = 1 0\\n'),\n",
       " ('y ^ { \\\\ p r i m e } = \\\\ f r a c { 2 } { y } - 1 = 0',\n",
       "  'y - \\\\frac { 2 } { y } - 1 = 0\\n'),\n",
       " ('\" x _ { 1 } = C \\\\ l a m b d a , x _ { 2 } = \\\\ f r a c { a + 1 } { a - 1 } \" \" \" \" \" \" \" \" \" , \" , \" , \" , \" \\\\ o v e r l i n e { 1 } , x = \\\\ f r a c { a + 1 } { a - 1 } \" \"',\n",
       "  '\"x _ { 1 } = a , x _ { 2 } = \\\\frac { a + 1 } { a - 1 }\"\\n'),\n",
       " ('1 5 ) k = - 8', '1 5 ) k = - 8\\n'),\n",
       " ('d = \\\\ f r a c { \\\\ v e r t - a b \\\\ v e r t } { c } = \\\\ f r a c { \\\\ s q r t { 2 } } { 4 } c',\n",
       "  'd = \\\\frac { \\\\vert - a b \\\\vert } { c } = \\\\frac { \\\\sqrt { 3 } } { 4 } c\\n'),\n",
       " ('r - \\\\ l a m b d a \\\\ f r a c { 1 } { 2 } \\\\ l a m b d a = \\\\ f r a c { 2 \\\\ s q r t { 1 7 } } { 2 }',\n",
       "  'P \\\\cdot 1 \\\\frac { 1 } { 2 } ( - 2 )\\n'),\n",
       " ('\\\\ f r a c { 4 } { 3 } x = \\\\ f r a c { 1 } { 2 }',\n",
       "  '\\\\frac { 4 } { 3 } x = \\\\frac { 1 } { 2 }\\n'),\n",
       " ('a _ { 0 } - a _ { 1 } + a _ { 2 } \\\\ c d o t s - a _ { 9 } = 5 9',\n",
       "  'a _ { 0 } - a _ { 1 } + a _ { 2 } \\\\cdots - a _ { 9 } = 5 9\\n'),\n",
       " ('x ^ { 2 } + 2 x - 6 = 0', 'x ^ { 2 } + 2 x - 6 = 0\\n'),\n",
       " ('S _ { \\\\ D e l t a A O F } = 3 S _ { \\\\ D e l t a B O F } = \\\\ f r a c { 9 } { 2 } = \\\\ f r a c { 9 } { 2 }',\n",
       "  'S _ { \\\\Delta A O F } = \\\\frac { 3 } { 2 } S _ { \\\\Delta B O F } = \\\\frac { 9 } { 2 }\\n'),\n",
       " ('\\\\ f r a c { I _ { 1 } } { I _ { 2 } } = \\\\ f r a c { R _ { 2 } } { R _ { 1 } }',\n",
       "  '\\\\frac { I _ { 1 } } { I _ { 2 } } = \\\\frac { R _ { 2 } } { R _ { 1 } }\\n'),\n",
       " ('1 1 1 + 6 < \\\\ f r a c { 9 + 9 } { 1 8 }',\n",
       "  '\\\\boxed { 1 1 } + 6 < \\\\frac { 9 + 9 } { 1 8 }\\n'),\n",
       " ('a = \\\\ f r a c { 2 } { 3 } \\\\ c d o t b = - \\\\ f r a c { 2 } { 3 }',\n",
       "  'a = \\\\frac { 2 } { 3 } \\\\cdot b = - \\\\frac { 2 } { 3 }\\n'),\n",
       " ('\\\\ a n g l e B C D = 9 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle B C D = 9 0 ^ { \\\\circ }\\n'),\n",
       " ('S = \\\\ f r a c { 1 } { 2 } a b \\\\ s i n C = \\\\ f r a c { 1 } { 2 } \\\\ t i m e s 6 \\\\ t i m e s \\\\ f r a c { \\\\ s q r t { 3 } } { 2 } = \\\\ f r a c { 3 \\\\ s q r t { 3 } } { 2 }',\n",
       "  'S = \\\\frac { 1 } { 2 } a b \\\\sin C = \\\\frac { 1 } { 2 } \\\\times 6 \\\\times \\\\frac { \\\\sqrt { 3 } } { 2 } = \\\\frac { 3 \\\\sqrt { 3 } } { 2 }\\n'),\n",
       " ('\\\\ f r a c { b } { 3 a } + \\\\ f r a c { a } { 2 b }',\n",
       "  '\\\\frac { b } { 3 a } + \\\\frac { a } { 2 b }\\n'),\n",
       " ('2 \\\\ s q r t { 4 2 } > 2 \\\\ s q r t { 4 0 }',\n",
       "  '2 \\\\sqrt { 4 2 } > 2 \\\\sqrt { 4 0 }\\n'),\n",
       " ('5 5 \\\\ t i m e s ( 1 5 \\\\ d i v 2 ) = 4 1 2 . 5 ( L ) )',\n",
       "  '5 5 \\\\times ( 1 5 \\\\div 2 ) = 4 1 2 . 5 ( L )\\n'),\n",
       " ('\" x _ { 1 } y _ { 1 } + x _ { 2 } y _ { 1 } = x _ { 2 } - x _ { 1 } x _ { 2 } y _ { 1 } + ( x _ { 2 } - y ) = 7 1 \" \" \" , x _ { 1 } y _ { 2 } = 7 1 \" \" \" { 1 } \" \" \"',\n",
       "  '\"x y ( x + y ) = 8 8 0 , x y + ( x + y ) = 7 1\"\\n'),\n",
       " ('x _ { 1 } > - 1 3', 'x _ { 1 } = 1 3\\n'),\n",
       " ('8 b : y = \\\\ p m \\\\ f r a c { \\\\ s q r t { 1 1 } } { 1 1 } ( x + \\\\ s q r t { 2 } ) )',\n",
       "  'A B : y = \\\\pm \\\\frac { \\\\sqrt { 1 1 } } { 1 1 } ( x + \\\\sqrt { 3 } )\\n'),\n",
       " ('x ^ { 2 } + 2 x y + y ^ { 2 } - 2 x y - 2 5 = 0',\n",
       "  'x ^ { 2 } + 2 x y + y ^ { 2 } - 2 x y - 2 5 = 0\\n'),\n",
       " ('B D ^ { 2 } = B C ^ { 2 } + D C ^ { 2 } ^ { 2 }',\n",
       "  'B D ^ { 2 } = B C ^ { 2 } + D C ^ { 2 }\\n'),\n",
       " ('\\\\ a n g l e F N Q = \\\\ f r a c { 1 } { 2 } \\\\ a n g l e D N F',\n",
       "  '\\\\angle F N Q = \\\\frac { 1 } { 2 } \\\\angle D N F\\n'),\n",
       " ('2 4 \\\\ t i m e s \\\\ f r a c { 5 x } { 2 4 } = 2 4 \\\\ t i m e s \\\\ f r a c { 3 x - 9 } { 8 } - 2 4 \\\\ t i m e s 2',\n",
       "  '2 4 \\\\times \\\\frac { 5 x } { 2 4 } = 2 4 \\\\times \\\\frac { 3 x - 4 } { 8 } - 2 4 \\\\times 2\\n'),\n",
       " ('O H = \\\\ f r a c { 1 } { 2 } O C _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 } _ { 1 }',\n",
       "  '0 H = \\\\frac { 1 } { 2 } O C .\\n'),\n",
       " ('a = 4 c : 6', 'a = 4 c = 6\\n'),\n",
       " ('\" \\\\ b e c a u s e a x + b y = 7 \\\\ t e x t c i r c l e d { 1 } \" \" \" \" \" \" \" \" \" \" \" \" \" \" \" , 1 , 0 , 1 , 0 , 1 , 0 , \" \" \" \" \" , 1 , 0 , 1 , 0 , 0 \" \" \" \"',\n",
       "  'a x + b y = 7 \\\\textcircled { 1 }\\n'),\n",
       " ('a c = 3 6 0', 'a c = 3 6 0\\n'),\n",
       " ('3 . 1 4 \\\\ t i m e s ( \\\\ f r a c { 8 } { 2 } ) ^ { 2 } x - 1 2',\n",
       "  '3 . 1 4 \\\\times ( \\\\frac { 8 } { 2 } ) ^ { 2 } x - 1 2\\n'),\n",
       " ('C D = B D = A D = \\\\ f r a c { 1 } { 2 } A B',\n",
       "  'C D = B D = A D = \\\\frac { 1 } { 2 } A B ^ { 2 }\\n'),\n",
       " ('1 \\\\ d i v 8 1 = \\\\ f r a c { 1 } { 8 1 }',\n",
       "  '1 \\\\div 8 1 = \\\\frac { 1 } { 8 1 }\\n'),\n",
       " ('y = \\\\ p m 6', 'y = \\\\pm 6\\n'),\n",
       " ('\" y = 1 - \\\\ s q r t { 3 } k + \\\\ f r a c { t } { 2 } , k + \\\\ f r a c { \\\\ s q r t { 3 } } { 2 } t ) \" \" [ 1 , \\\\ f r a c { \\\\ s q r t { 3 } } { 2 } t ) \" ) \" \" ) \" \" 1 , 2 \\\\ n e q 0 , \" )',\n",
       "  '\"y = 1 - \\\\sqrt { 3 } k + \\\\frac { t } { 2 } , k + \\\\frac { \\\\sqrt { 3 } } { 2 } t )\"\\n'),\n",
       " ('\\\\ c o s A = \\\\ c o s 3 0 ^ { \\\\ c i r c } = \\\\ f r a c { A D } { A C } = \\\\ f r a c { \\\\ s q r t { 3 } } { 2 } = \\\\ f r a c { \\\\ s q r t { 3 } } { 2 }',\n",
       "  '\\\\cos A = \\\\cos 3 0 ^ { . } = \\\\frac { A D } { A C } = \\\\frac { \\\\sqrt { 3 } } { 2 }\\n'),\n",
       " ('C O = 6 0 - 2 0 \\\\ s q r t { 3 }', 'C O = 6 0 - 2 0 \\\\sqrt { 3 }\\n'),\n",
       " ('\\\\ f r a c { 2 3 } { 2 0 } > 1 7', '\\\\frac { 2 3 } { 2 0 } > 1\\n'),\n",
       " ('\\\\ b e c a u s e \\\\ a n g l e 1 = \\\\ a n g l e L',\n",
       "  '\\\\because \\\\angle C = \\\\angle C\\n'),\n",
       " ('\\\\ t e x t c i r c l e d { 2 } : 2 . 2 5 \\\\ t i m e s 1 . 5 = 3 4 . 7 5 ( d m ^ { 3 } )',\n",
       "  '\\\\textcircled { 3 } : 2 . 2 5 \\\\times 1 5 = 3 4 . 7 5 ( d m ^ { 3 } )\\n'),\n",
       " ('\\\\ a n g l e A = 8 0 ^ { \\\\ c i r c } \\\\ a n g l e C = 2 0 ^ { \\\\ c i r c }',\n",
       "  '\\\\angle A = 8 0 ^ { \\\\circ } \\\\angle C = 2 0 ^ { \\\\circ }\\n'),\n",
       " ('P ^ { \\\\ p r i m e } = \\\\ f r a c { w } { s }', 'F = \\\\frac { w } { s }\\n'),\n",
       " ('\\\\ f r a c { 2 \\\\ p i } { w } = \\\\ p i',\n",
       "  '\\\\frac { 2 \\\\pi } { w } = \\\\pi\\n'),\n",
       " ('x _ { 4 } = - \\\\ f r a c { 1 } { 3 }',\n",
       "  'x _ { 8 } = - \\\\frac { 1 } { 3 }\\n'),\n",
       " ...]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pairs = combine_pred_gt('results/test_res/hme100k_predict_res.txt', test_dataset_df)\n",
    "pairs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------------------------ Calculating the Exp Rate ------------------------------\n",
      "ExpRate: 47.49%\n",
      "ExpRate (≤1 error): 55.46%\n",
      "ExpRate (≤2 errors): 62.42%\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(47.49461535335474, 55.455764619823626, 62.421262242451334)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "compute_exprate(pairs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "'\\\\ f r a c { 2 \\\\ p i } { w } = \\\\ p i' ==  '\\\\ f r a c { 2 \\\\ p i } { w } = \\\\ p i'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'2.1.2+cu118'"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
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       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>formula</th>\n",
       "      <th>filename</th>\n",
       "      <th>image</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>\\alpha _ { 1 } ^ { r } \\gamma _ { 1 } + \\dots ...</td>\n",
       "      <td>7944775fc9.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>\\Gamma ( z + 1 ) = \\int _ { 0 } ^ { \\infty } \\...</td>\n",
       "      <td>6968dfca15.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>\\frac { d } { d s } { \\bf C } _ { i } = \\frac ...</td>\n",
       "      <td>6cead0df53.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Z = \\sum _ { s p i n s } \\prod _ { c u b e s }...</td>\n",
       "      <td>5381b22df4.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>\\left\\{ Q ^ { i } , Q ^ { j } \\right\\} = c ^ {...</td>\n",
       "      <td>27f2b37ce9.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>\\breve { c } _ { n , \\nu } = \\sum _ { m = n } ...</td>\n",
       "      <td>51a257cdf5.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>R ( g ) = - f \\left[ 3 \\left[ ( \\operatorname ...</td>\n",
       "      <td>5108925e21.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>{ \\frac { d } { d s } } { \\frac { 1 } { \\Gamma...</td>\n",
       "      <td>3882dd3d43.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>c _ { \\alpha } = \\sum _ { \\beta \\in \\Lambda _ ...</td>\n",
       "      <td>7d1fe2cc05.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>{ \\cal L } = - { \\frac { 1 } { 4 } } F _ { \\mu...</td>\n",
       "      <td>450b24df87.png</td>\n",
       "      <td>{'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                             formula        filename  \\\n",
       "0  \\alpha _ { 1 } ^ { r } \\gamma _ { 1 } + \\dots ...  7944775fc9.png   \n",
       "1  \\Gamma ( z + 1 ) = \\int _ { 0 } ^ { \\infty } \\...  6968dfca15.png   \n",
       "2  \\frac { d } { d s } { \\bf C } _ { i } = \\frac ...  6cead0df53.png   \n",
       "3  Z = \\sum _ { s p i n s } \\prod _ { c u b e s }...  5381b22df4.png   \n",
       "4  \\left\\{ Q ^ { i } , Q ^ { j } \\right\\} = c ^ {...  27f2b37ce9.png   \n",
       "5  \\breve { c } _ { n , \\nu } = \\sum _ { m = n } ...  51a257cdf5.png   \n",
       "6  R ( g ) = - f \\left[ 3 \\left[ ( \\operatorname ...  5108925e21.png   \n",
       "7  { \\frac { d } { d s } } { \\frac { 1 } { \\Gamma...  3882dd3d43.png   \n",
       "8  c _ { \\alpha } = \\sum _ { \\beta \\in \\Lambda _ ...  7d1fe2cc05.png   \n",
       "9  { \\cal L } = - { \\frac { 1 } { 4 } } F _ { \\mu...  450b24df87.png   \n",
       "\n",
       "                                               image  \n",
       "0  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "1  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "2  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "3  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "4  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "5  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "6  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "7  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "8  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  \n",
       "9  {'bytes': b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHD...  "
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_parquet('dataset/test_parquets/im2latex_test.parquet')\n",
    "df.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "file = '7d1fe2cc05.png'\n",
    "img_dict = df.loc[df[\"filename\"] == file, \"image\"].values[0]\n",
    "img_dict['bytes']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'c _ { \\\\ a l p h a } = \\\\ s u m _ { \\\\ b e t a \\\\ i n \\\\ L a m b d a _ { R } } \\\\ e p s i l o n ( \\\\ a l p h a , \\\\ b e t a ) | \\\\ b e t a + \\\\ b a r { p } > < \\\\ b e t a + \\\\ b a r { p } |'"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "file = '7d1fe2cc05.png'\n",
    "form = df.loc[df[\"filename\"] == file, \"formula\"].values[0]\n",
    "' '.join(c for st in form.split() for c in list(st))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>formula</th>\n",
       "      <th>filename</th>\n",
       "      <th>image</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>\\frac { 5 } { 1 4 } = \\frac { 1 } { x }\\n</td>\n",
       "      <td>test_2.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>- 2 x ^ { 4 } - x ^ { 2 } + 5 = ( - x ^ { 2 } ...</td>\n",
       "      <td>test_3.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>S _ { \\Delta D B F } = \\frac { 3 } { 8 } S _ {...</td>\n",
       "      <td>test_5.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>: x _ { 1 } = - \\frac { 3 } { 5 }\\n</td>\n",
       "      <td>test_6.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7 5 \\div ( 7 5 + 2 5 ) = 7 5 \\div 1 0 0 = 7 5 ...</td>\n",
       "      <td>test_7.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>y = 3 . y = - 1\\n</td>\n",
       "      <td>test_8.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>x _ { 1 } = - 6 x _ { 2 } = 4\\n</td>\n",
       "      <td>test_9.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>2 3 . 5 + \\frac { 1 } { 2 } x = 1 3 + x\\n</td>\n",
       "      <td>test_10.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>\\frac { 9 } { 1 0 } \\div \\frac { 3 } { 5 } = \\...</td>\n",
       "      <td>test_13.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>= \\frac { ( 7 + 1 8 ) + ( 9 - 1 4 ) i } { 5 } ...</td>\n",
       "      <td>test_14.png</td>\n",
       "      <td>b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                             formula     filename  \\\n",
       "0          \\frac { 5 } { 1 4 } = \\frac { 1 } { x }\\n   test_2.png   \n",
       "1  - 2 x ^ { 4 } - x ^ { 2 } + 5 = ( - x ^ { 2 } ...   test_3.png   \n",
       "2  S _ { \\Delta D B F } = \\frac { 3 } { 8 } S _ {...   test_5.png   \n",
       "3                : x _ { 1 } = - \\frac { 3 } { 5 }\\n   test_6.png   \n",
       "4  7 5 \\div ( 7 5 + 2 5 ) = 7 5 \\div 1 0 0 = 7 5 ...   test_7.png   \n",
       "5                                  y = 3 . y = - 1\\n   test_8.png   \n",
       "6                    x _ { 1 } = - 6 x _ { 2 } = 4\\n   test_9.png   \n",
       "7          2 3 . 5 + \\frac { 1 } { 2 } x = 1 3 + x\\n  test_10.png   \n",
       "8  \\frac { 9 } { 1 0 } \\div \\frac { 3 } { 5 } = \\...  test_13.png   \n",
       "9  = \\frac { ( 7 + 1 8 ) + ( 9 - 1 4 ) i } { 5 } ...  test_14.png   \n",
       "\n",
       "                                               image  \n",
       "0  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "1  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "2  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "3  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "4  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "5  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "6  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "7  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "8  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  \n",
       "9  b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\...  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_parquet('dataset/test_parquets/hmer_test.parquet')\n",
    "df.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\\\ f r a c { 3 5 } { 3 5 } - \\\\ f r a c { 2 0 } { 3 5 } = \\\\ f r a c { 1 5 } { 3 5 }'"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "file = 'test_15525.png'\n",
    "form = df.loc[df[\"filename\"] == file, \"formula\"].values[0]\n",
    "' '.join(c for st in form.split() for c in list(st))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "' \\\\ f r a c { 5 } { 1 4 } = \\\\ f r a c { 1 } { x } i c i r c f r a c { 1 } { x }'"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "preds = []\n",
    "with open('results/test_res/hme100k_predict_res.txt', 'r') as f:\n",
    "    for line_num, line in enumerate(f, 1):\n",
    "        _, pred = line.strip().split(':', 1)\n",
    "        preds.append(pred) \n",
    "\n",
    "preds[0]\n",
    "\n",
    "# import re\n",
    "\n",
    "# # 给定字符串\n",
    "# string = 'S _ { \\Delta D B F } = \\frac { 3 } { 8 } S _ { \\Delta A B C } = \\frac { 3 } { 8 } \\times \\frac { 1 5 } { 2 } = \\frac { 4 5 } { 1 6 }'\n",
    "\n",
    "# # 使用正则表达式替换 \\ 和后续字符\n",
    "# # r'\\\\(.)' 匹配反斜杠后跟一个字符，并将其替换为 '\\', '字符'\n",
    "# modified_string = re.sub(r'\\\\(.)', r'\\\\ \\1', string)\n",
    "\n",
    "# # 将 '\\\\' 转换为 '\\'\n",
    "# final_string = modified_string.replace('\\\\', '\\\\\\\\')  # 先转换为 '\\\\' 以便下步替换\n",
    "# final_string = final_string.replace('\\\\\\\\', '\\\\')  # 将双反斜杠替换为单反斜杠\n",
    "\n",
    "# # 将最终字符串转换为字符列表\n",
    "# char_list = list(final_string)\n",
    "\n",
    "# # 去除空格字符\n",
    "# filtered_char_list = [char for char in char_list if char != ' ']\n",
    "\n",
    "# # 打印结果\n",
    "# ' '.join(filtered_char_list)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'S _ { \\\\ D e l t a D B F } = \\\\ f r a c { 3 } { 8 } S _ { \\\\ D e l t a A B C } = \\\\ f r a c { 3 } { 8 } \\\\ t i m e s \\\\ f r a c { 1 5 } { 2 } = \\\\ f r a c { 4 5 } { 1 6 }'"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "original_list = ['S', '_', '{', '\\\\', 'D', 'e', 'l', 't', 'a', 'D', 'B', 'F', '}', '=', '\\\\', 'f', 'r', 'a', 'c', '{', '3', '}', '{', '8', '}', 'S', '_', '{', '\\\\', 'D', 'e', 'l', 't', 'a', 'A', 'B', 'C', '}', '=', '\\\\', 'f', 'r', 'a', 'c', '{', '3', '}', '{', '8', '}', '\\\\', 't', 'i', 'm', 'e', 's', '\\\\', 'f', 'r', 'a', 'c', '{', '1', '5', '}', '{', '2', '}', '=', '\\\\', 'f', 'r', 'a', 'c', '{', '4', '5', '}', '{', '1', '6', '}']\n",
    "\n",
    "# 替换 '\\\\' 为 '\\'\n",
    "modified_list = [item.replace('\\\\\\\\', '\\\\') if item == '\\\\\\\\' else item for item in original_list]\n",
    "\n",
    "' '.join(modified_list)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'m ^ { 2 } + 1 = 2 m = 1\\n'"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "file = 'test_5525.png'\n",
    "df.loc[df[\"filename\"] == file, \"formula\"].values[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "[nltk_data] Downloading package punkt to /root/nltk_data...\n",
      "[nltk_data]   Package punkt is already up-to-date!\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import nltk\n",
    "nltk.download('punkt')  # for BLEU"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "import torch.nn as nn\n",
    "import torch.optim as optim\n",
    "from transformers import ViTModel, BertTokenizer, BertConfig\n",
    "import os\n",
    "import xml.etree.ElementTree as ET\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "from PIL import Image\n",
    "from torch.utils.data import Dataset, DataLoader\n",
    "import torchvision.transforms as transforms\n",
    "from collections import defaultdict\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "train_img_base_dir = 'dataset/crohme/train/extracted_img'\n",
    "# os.path.exists(train_img_base_dir)\n",
    "os.listdir(train_img_base_dir)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. 数据集准备"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_combined_csv(inkml_folder, latex_folder, img_folder, output_csv):\n",
    "    data = []\n",
    "    \n",
    "    # 遍历 INKML 文件夹，建立文件路径的索引\n",
    "    inkml_file_paths = {}\n",
    "    \n",
    "    for root, _, files in os.walk(inkml_folder):\n",
    "        for file in files:\n",
    "            if file.endswith('.inkml'):\n",
    "                inkml_file_paths[file] = os.path.join(root, file)\n",
    "    \n",
    "    # 遍历 LaTeX 表达式文件夹\n",
    "    for root, _, files in os.walk(latex_folder):\n",
    "        for file in files:\n",
    "            if file.endswith('.lg'):  # 假设 LaTeX 表达式文件的后缀为 .lg\n",
    "                latex_file_path = os.path.join(root, file)\n",
    "                \n",
    "                with open(latex_file_path, 'r', encoding='utf-8') as f:\n",
    "                    latex_lines = f.readlines()\n",
    "                    \n",
    "                    for inkml_file_name, latex in zip(inkml_file_paths.keys(), latex_lines):\n",
    "                        latex = latex.strip()  # 去除空白符\n",
    "                        \n",
    "                        # 找到已经转换的 PNG 图像文件\n",
    "                        image_file_name = os.path.basename(inkml_file_name).replace('.inkml', '.png')\n",
    "                        image_file_path = os.path.join(img_folder, image_file_name)\n",
    "                        \n",
    "                        if os.path.exists(image_file_path):\n",
    "                            data.append([image_file_path, latex])\n",
    "                        else:\n",
    "                            print(f\"Warning: Image file not found for {image_file_path}\")\n",
    "\n",
    "    # 将结果保存为 CSV 文件\n",
    "    if data:\n",
    "        df = pd.DataFrame(data, columns=['image_path', 'latex'])\n",
    "        df.to_csv(output_csv, index=False)\n",
    "        print(f\"CSV 文件已创建，路径为: {output_csv}\")\n",
    "    else:\n",
    "        print(\"No matching data found. Please check your file paths.\")\n",
    "\n",
    "# 使用文件夹路径来调用函数\n",
    "inkml_folder = 'TC11_CROHME23/INKML/train/CROHME2019'  # InkML 文件夹路径\n",
    "latex_folder = 'TC11_CROHME23/SymLG/train/CROHME2019_train'  # LaTeX 表达式文件夹路径\n",
    "img_folder = 'TC11_CROHME23/IMG/train/CROHME2019'  # 已转换的 PNG 图像文件存储路径\n",
    "output_csv = 'crohme_labels.csv'  # 输出 CSV 文件路径\n",
    "\n",
    "create_combined_csv(inkml_folder, latex_folder, img_folder, output_csv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.  数据加载"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "\" 定义数据集与数据加载器 \"\n",
    "class CROHMEDataset(Dataset):\n",
    "    def __init__(self, csv_file, transform=None):\n",
    "        self.data = pd.read_csv(csv_file)\n",
    "        self.transform = transform\n",
    "\n",
    "    def __len__(self):\n",
    "        return len(self.data)\n",
    "\n",
    "    def __getitem__(self, idx):\n",
    "        img_path = self.data.iloc[idx, 0]\n",
    "        latex_expr = self.data.iloc[idx, 1]\n",
    "        \n",
    "        image = Image.open(img_path).convert(\"RGB\")\n",
    "\n",
    "        if self.transform:\n",
    "            image = self.transform(image)\n",
    "        \n",
    "        return image, latex_expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "class ViTEncoder(nn.Module):\n",
    "    def __init__(self, pretrained_model_name='google/vit-base-patch16-224-in21k'):\n",
    "        super(ViTEncoder, self).__init__()\n",
    "        self.encoder = ViTModel.from_pretrained(pretrained_model_name)\n",
    "        \n",
    "    def forward(self, x):\n",
    "        outputs = self.encoder(pixel_values=x)\n",
    "        return outputs.last_hidden_state\n",
    "\n",
    "class TransformerDecoder(nn.Module):\n",
    "    def __init__(self, vocab_size, hidden_dim, num_layers, num_heads):\n",
    "        super(TransformerDecoder, self).__init__()\n",
    "        self.embedding = nn.Embedding(vocab_size, hidden_dim)\n",
    "        self.positional_encoding = nn.Parameter(torch.zeros(1, 500, hidden_dim))\n",
    "        self.decoder_layer = nn.TransformerDecoderLayer(hidden_dim, num_heads)\n",
    "        self.transformer_decoder = nn.TransformerDecoder(self.decoder_layer, num_layers)\n",
    "        self.fc_out = nn.Linear(hidden_dim, vocab_size)\n",
    "\n",
    "    def forward(self, encoder_outputs, tgt, tgt_mask):\n",
    "        tgt_embedded = self.embedding(tgt) + self.positional_encoding[:, :tgt.size(1), :]\n",
    "        outputs = self.transformer_decoder(tgt_embedded, encoder_outputs, tgt_mask=tgt_mask)\n",
    "        outputs = self.fc_out(outputs)\n",
    "        return outputs\n",
    "\n",
    "# 组合模型\n",
    "class ImageToLatexModel(nn.Module):\n",
    "    def __init__(self, vocab_size, hidden_dim=512, num_layers=6, num_heads=8):\n",
    "        super(ImageToLatexModel, self).__init__()\n",
    "        self.encoder = ViTEncoder()\n",
    "        self.decoder = TransformerDecoder(vocab_size, hidden_dim, num_layers, num_heads)\n",
    "\n",
    "    def forward(self, x, tgt, tgt_mask):\n",
    "        encoder_outputs = self.encoder(x)\n",
    "        outputs = self.decoder(encoder_outputs, tgt, tgt_mask)\n",
    "        return outputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def train_one_epoch(model, train_loader, optimizer, criterion):\n",
    "    model.train()\n",
    "    epoch_loss = 0\n",
    "    for images, latex_exprs in train_loader:\n",
    "        images = images.to(device)\n",
    "        latex_exprs = latex_exprs.to(device)  # 假设已编码为张量\n",
    "        optimizer.zero_grad()\n",
    "\n",
    "        tgt_input = latex_exprs[:, :-1]\n",
    "        tgt_output = latex_exprs[:, 1:]\n",
    "\n",
    "        tgt_mask = nn.Transformer.generate_square_subsequent_mask(tgt_input.size(1)).to(device)\n",
    "        output = model(images, tgt_input, tgt_mask)\n",
    "\n",
    "        loss = criterion(output.view(-1, output.size(-1)), tgt_output.view(-1))\n",
    "        loss.backward()\n",
    "        optimizer.step()\n",
    "        \n",
    "        epoch_loss += loss.item()\n",
    "    \n",
    "    return epoch_loss / len(train_loader)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "if __name__ == \"__main__\":\n",
    "\n",
    "    transform = transforms.Compose([\n",
    "        transforms.Resize((224, 224)),\n",
    "        transforms.ToTensor(),\n",
    "    ])\n",
    "\n",
    "    train_dataset = CROHMEDataset('crohme_labels.csv', transform=transform)\n",
    "    train_dataloader = DataLoader(train_dataset, batch_size=32, shuffle=True, num_workers=4)\n",
    "\n",
    "    # device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n",
    "    device = torch.device(\"mps\" if torch.backends.mps.is_available() else \"cpu\")\n",
    "\n",
    "    # 假设我们已经准备好词汇表并对目标文本进行了编码\n",
    "    vocab_size = 10000  # 假设词汇表大小\n",
    "    model = ImageToLatexModel(vocab_size).to(device)\n",
    "\n",
    "    optimizer = torch.optim.Adam(model.parameters(), lr=1e-4)\n",
    "    criterion = nn.CrossEntropyLoss(ignore_index=0)\n",
    "\n",
    "\n",
    "    # 开始训练\n",
    "    num_epochs = 10\n",
    "    for epoch in range(num_epochs):\n",
    "        epoch_loss = train_one_epoch(model, train_dataloader, optimizer, criterion)\n",
    "        print(f'Epoch {epoch + 1}/{num_epochs}, Loss: {epoch_loss:.4f}')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def generate_latex(model, image, tokenizer, max_length=100):\n",
    "    model.eval()\n",
    "    with torch.no_grad():\n",
    "        image = image.unsqueeze(0).to(device)\n",
    "        encoder_outputs = model.encoder(image)\n",
    "\n",
    "        tgt_input = torch.tensor([[tokenizer.cls_token_id]], device=device)\n",
    "\n",
    "        for _ in range(max_length):\n",
    "            tgt_mask = nn.Transformer.generate_square_subsequent_mask(tgt_input.size(1)).to(device)\n",
    "            output = model.decoder(encoder_outputs, tgt_input, tgt_mask)\n",
    "            \n",
    "            next_token = output.argmax(2)[:, -1].unsqueeze(0)\n",
    "            tgt_input = torch.cat([tgt_input, next_token], dim=1)\n",
    "\n",
    "            if next_token.item() == tokenizer.sep_token_id:\n",
    "                break\n",
    "\n",
    "    return tokenizer.decode(tgt_input.squeeze().tolist(), skip_special_tokens=True)\n",
    "\n",
    "# 使用 BertTokenizer 进行示例推理\n",
    "tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')\n",
    "test_image, _ = train_dataset[0]  # 获取测试图像\n",
    "\n",
    "latex_expr = generate_latex(model, test_image, tokenizer)\n",
    "print(latex_expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def save_to_ig_format(latex_expr, output_path):\n",
    "    with open(output_path, 'w', encoding='utf-8') as f:\n",
    "        f.write(latex_expr)\n",
    "\n",
    "# 示例保存\n",
    "output_path = 'output/formula.ig'\n",
    "save_to_ig_format(latex_expr, output_path)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Older Version"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\" 构建 LaTeX 表达式词汇表与编码函数 \"\n",
    "# 构建词汇表\n",
    "vocab = defaultdict(int)\n",
    "\n",
    "# 确保遍历 LaTeX 表达式所在的列\n",
    "for _, row in dataset.data.iterrows():\n",
    "    latex = row['latex']  # 获取 'latex' 列\n",
    "    for token in latex.split():\n",
    "        vocab[token] += 1\n",
    "\n",
    "# 分配索引：<pad>, <bos>, <eos> 分别为填充、起始和结束标记\n",
    "vocab = {token: idx + 3 for idx, (token, _) in enumerate(sorted(vocab.items(), key=lambda x: -x[1]))}\n",
    "vocab['<pad>'] = 0\n",
    "vocab['<bos>'] = 1\n",
    "vocab['<eos>'] = 2\n",
    "\n",
    "# 构建反向词汇表\n",
    "idx2vocab = {idx: token for token, idx in vocab.items()}\n",
    "\n",
    "# 定义编码与解码函数\n",
    "def encode_latex(latex):\n",
    "    return [vocab['<bos>']] + [vocab.get(token, vocab['<pad>']) for token in latex.split()] + [vocab['<eos>']]\n",
    "\n",
    "def decode_latex(indices):\n",
    "    return ' '.join(idx2vocab[idx] for idx in indices if idx not in [0, 1, 2])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. 模型架构"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\" 定义 ViT + Transformer 模型 \"\n",
    "import torch\n",
    "import torch.nn as nn\n",
    "from transformers import ViTModel\n",
    "\n",
    "class ViTTransformerModel(nn.Module):\n",
    "    def __init__(self, vit_model_name='google/vit-base-patch16-224', decoder_vocab_size=3000, d_model=768, num_decoder_layers=6):\n",
    "        super(ViTTransformerModel, self).__init__()\n",
    "        \n",
    "        self.vit = ViTModel.from_pretrained(vit_model_name)\n",
    "        self.d_model = d_model\n",
    "        \n",
    "        decoder_layer = nn.TransformerDecoderLayer(d_model=d_model, nhead=8)\n",
    "        self.transformer_decoder = nn.TransformerDecoder(decoder_layer, num_layers=num_decoder_layers)\n",
    "        \n",
    "        self.embedding = nn.Embedding(decoder_vocab_size, d_model)\n",
    "        self.fc_out = nn.Linear(d_model, decoder_vocab_size)\n",
    "        self.positional_encoding = nn.Parameter(torch.zeros(1, 500, d_model))\n",
    "\n",
    "    def forward(self, images, target_seq, target_mask):\n",
    "        encoded_features = self.vit(images).last_hidden_state\n",
    "        memory = encoded_features.permute(1, 0, 2)\n",
    "        \n",
    "        tgt_embedded = self.embedding(target_seq) + self.positional_encoding[:, :target_seq.size(1), :]\n",
    "        tgt_embedded = tgt_embedded.permute(1, 0, 2)\n",
    "        \n",
    "        output = self.transformer_decoder(tgt_embedded, memory, tgt_mask=target_mask)\n",
    "        output = self.fc_out(output)\n",
    "        \n",
    "        return output.permute(1, 0, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\" 训练模型 \"\n",
    "model = ViTTransformerModel(decoder_vocab_size=len(vocab))\n",
    "criterion = nn.CrossEntropyLoss(ignore_index=0)\n",
    "optimizer = optim.AdamW(model.parameters(), lr=1e-4)\n",
    "\n",
    "def generate_square_subsequent_mask(sz):\n",
    "    mask = (torch.triu(torch.ones(sz, sz)) == 1).transpose(0, 1)\n",
    "    mask = mask.float().masked_fill(mask == 0, float('-inf')).masked_fill(mask == 1, float(0.0))\n",
    "    return mask\n",
    "\n",
    "num_epochs = 10\n",
    "for epoch in range(num_epochs):\n",
    "    model.train()\n",
    "    total_loss = 0\n",
    "    for images, target_seq in dataloader:\n",
    "        optimizer.zero_grad()\n",
    "        \n",
    "        target_input = torch.tensor([([vocab['<bos>']] + encode_latex(seq)[:-1]) for seq in target_seq], dtype=torch.long)\n",
    "        target_output = torch.tensor([encode_latex(seq) + [vocab['<eos>']] for seq in target_seq], dtype=torch.long)\n",
    "        \n",
    "        target_mask = generate_square_subsequent_mask(target_input.size(1)).to(images.device)\n",
    "        outputs = model(images, target_input, target_mask)\n",
    "        \n",
    "        loss = criterion(outputs.view(-1, outputs.size(-1)), target_output.view(-1))\n",
    "        loss.backward()\n",
    "        optimizer.step()\n",
    "        \n",
    "        total_loss += loss.item()\n",
    "    \n",
    "    print(f\"Epoch [{epoch + 1}/{num_epochs}], Loss: {total_loss / len(dataloader)}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 推理"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def predict_latex(model, image, start_token, end_token, max_length=500):\n",
    "    model.eval()\n",
    "    target_seq = torch.LongTensor([[start_token]]).to(image.device)\n",
    "    \n",
    "    with torch.no_grad():\n",
    "        for _ in range(max_length):\n",
    "            target_mask = generate_square_subsequent_mask(target_seq.size(1)).to(image.device)\n",
    "            output = model(image, target_seq, target_mask)\n",
    "            \n",
    "            next_token = output.argmax(-1)[:, -1]\n",
    "            target_seq = torch.cat([target_seq, next_token.unsqueeze(1)], dim=1)\n",
    "            \n",
    "            if next_token.item() == end_token:\n",
    "                break\n",
    "                \n",
    "    return decode_latex(target_seq.squeeze().tolist())\n",
    "\n",
    "# Example usage for one image\n",
    "image, _ = dataset[0]\n",
    "image = image.unsqueeze(0)  # Add batch dimension\n",
    "predicted_latex = predict_latex(model, image, vocab['<bos>'], vocab['<eos>'])\n",
    "print(predicted_latex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pickle\n",
    "import os\n",
    "import cv2\n",
    "\n",
    "def extract_img(pkl_file: str, save_dir: str):\n",
    "    \"\"\"\n",
    "    Extracts images from a pickle file and saves them as image files in the specified directory.\n",
    "    \n",
    "    Parameters:\n",
    "        pkl_file (str): Path to the pickle file containing image data.\n",
    "        save_dir (str): Directory where the extracted images will be saved.\n",
    "    \"\"\"\n",
    "    # Load the pickle file\n",
    "    with open(pkl_file, 'rb') as f:\n",
    "        data = pickle.load(f)\n",
    "    \n",
    "    # Ensure the output directory exists\n",
    "    os.makedirs(save_dir, exist_ok=True)\n",
    "    \n",
    "    # Iterate through the dictionary items\n",
    "    for image_name, image_array in data.items():\n",
    "        # Ensure image_name has a proper extension\n",
    "        if not image_name.lower().endswith(('.png', '.jpg', '.jpeg')):\n",
    "            image_name += '.png'  # Default to .png if no extension\n",
    "        \n",
    "        # Construct the full path for saving the image\n",
    "        output_path = os.path.join(save_dir, image_name)\n",
    "        \n",
    "        # Convert image_array to an appropriate data type if needed\n",
    "        if image_array.dtype != 'uint8':\n",
    "            image_array = image_array.astype('uint8')\n",
    "        \n",
    "        # Save the image using OpenCV\n",
    "        success = cv2.imwrite(output_path, image_array)\n",
    "        if success:\n",
    "            print(f\"Saved image: {output_path}\")\n",
    "        else:\n",
    "            print(f\"Failed to save image: {output_path}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\" Merge image and labels to a csv \"\n",
    "def merge_hmer_or_crohme(images: str, caption: str, extract_img_dir: str, save_file: str):\n",
    "    \"\"\"\n",
    "    Optimized function to read images from a pickle file and labels from a text file,\n",
    "    then saves the matched image paths with their corresponding LaTeX labels into a CSV file.\n",
    "\n",
    "    Parameters:\n",
    "        images (str): Path to the pickle (.pkl) file containing image paths.\n",
    "        caption (str): Path to the caption text file containing image-label pairs.\n",
    "        save_file (str): Path to the output CSV file where the results will be saved.\n",
    "    \"\"\"\n",
    "    \n",
    "    # Load the image paths from the pickle file\n",
    "    with open(images, 'rb') as f:\n",
    "        image_list = pickle.load(f)\n",
    "    \"\"\"\n",
    "    Load image paths into `image_list` from the pickle file.\n",
    "    \"\"\"\n",
    "\n",
    "    # Create a dictionary for fast lookup: key is the image filename, value is the full path\n",
    "    image_dict = {os.path.basename(img_path): img_path for img_path in image_list}\n",
    "    \"\"\"\n",
    "    The `image_dict` allows for O(1) average-time complexity lookups for matching images.\n",
    "    \"\"\"\n",
    "\n",
    "    # Read the caption file\n",
    "    with open(caption, 'r', encoding='utf-8') as f:\n",
    "        data = f.readlines()\n",
    "    \"\"\"\n",
    "    Read all lines from the caption.txt file.\n",
    "    \"\"\"\n",
    "    \n",
    "    img_labels = []\n",
    "\n",
    "    # Process each line in the caption file\n",
    "    for line in data:\n",
    "        \"\"\"\n",
    "        Loop through each line in the caption file.\n",
    "        \"\"\"\n",
    "        \n",
    "        # Strip any leading/trailing whitespace/newlines\n",
    "        line = line.strip()\n",
    "        \n",
    "        # Check for tab separation and split accordingly\n",
    "        if '\\t' in line:\n",
    "            image_name, label = line.split('\\t', 1)  # Split by the first tab\n",
    "        else:\n",
    "            # Skip if improperly formatted\n",
    "            continue\n",
    "        \n",
    "        # Use the dictionary to find the matching image path\n",
    "        if image_name in image_dict:\n",
    "            # image_path = (extract_img_dir + '/' + image_dict[image_name])  # if extract_img_dir not None else image_dict[image_name]\n",
    "            image_path = os.path.join(extract_img_dir, image_dict[image_name] + '.png')\n",
    "            print(image_path)\n",
    "            if os.path.isfile(image_path):\n",
    "                img_labels.append([image_path, label])\n",
    "            else:\n",
    "                print(f\"Warning: Image {image_name} not found in directory {extract_img_dir}. Skipping.\")\n",
    "        else:\n",
    "            print(f\"Warning: Image {image_name} not found in the pickle file. Skipping.\")\n",
    "    \"\"\"\n",
    "    The loop now efficiently finds image paths using dictionary lookups instead of iterating through a list.\n",
    "    \"\"\"\n",
    "\n",
    "    # Create a DataFrame from the list of image-label pairs\n",
    "    df = pd.DataFrame(img_labels, columns=['Image Path', 'Label'])\n",
    "\n",
    "    # Save the DataFrame to a CSV file without index\n",
    "    df.to_csv(save_file, index=False)\n",
    "    print(f\"CSV file saved as {save_file}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Extracting images...\n"
     ]
    }
   ],
   "source": [
    "training_img_pkl_dir = 'dataset/crohme/train/images.pkl'\n",
    "train_img_base_dir = 'dataset/crohme/train/extracted_img'\n",
    "caption_dir = 'dataset/crohme/train/caption.txt'\n",
    "mapping_csv = 'dataset/crohme/train/crohme_labels.csv'\n",
    "\n",
    "# extract_img(pkl_file=training_img_pkl_dir, save_dir=train_img_base_dir)\n",
    "# train_img_base_dir is empty\n",
    "if not os.path.exists(train_img_base_dir) or os.path.exists(train_img_base_dir) and not os.listdir(train_img_base_dir):\n",
    "    print('Extracting images...')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "dataset/crohme/train/extracted_img/200924-1331-216.png\n",
      "dataset/crohme/train/extracted_img/200923-131-185.png\n",
      "dataset/crohme/train/extracted_img/200923-1553-117.png\n",
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      "dataset/crohme/train/extracted_img/94_Fabricio.png\n",
      "dataset/crohme/train/extracted_img/formulaire011-equation019.png\n",
      "dataset/crohme/train/extracted_img/111_hirata.png\n",
      "dataset/crohme/train/extracted_img/formulaire010-equation032.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_14_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire019-equation002.png\n",
      "dataset/crohme/train/extracted_img/formulaire016-equation004.png\n",
      "dataset/crohme/train/extracted_img/formulaire009-equation033.png\n",
      "dataset/crohme/train/extracted_img/TrainData2_24_sub_98.png\n",
      "dataset/crohme/train/extracted_img/MfrDB1508.png\n",
      "dataset/crohme/train/extracted_img/formulaire038-equation063.png\n",
      "dataset/crohme/train/extracted_img/formulaire001-equation017.png\n",
      "dataset/crohme/train/extracted_img/formulaire028-equation074.png\n",
      "dataset/crohme/train/extracted_img/formulaire028-equation057.png\n",
      "dataset/crohme/train/extracted_img/formulaire009-equation067.png\n",
      "dataset/crohme/train/extracted_img/formulaire022-equation042.png\n",
      "dataset/crohme/train/extracted_img/formulaire032-equation018.png\n",
      "dataset/crohme/train/extracted_img/200922-1017-79.png\n",
      "dataset/crohme/train/extracted_img/111_miguel.png\n",
      "dataset/crohme/train/extracted_img/formulaire033-equation052.png\n",
      "dataset/crohme/train/extracted_img/92_miguel.png\n",
      "dataset/crohme/train/extracted_img/formulaire024-equation006.png\n",
      "dataset/crohme/train/extracted_img/124_caue.png\n",
      "dataset/crohme/train/extracted_img/131_maira.png\n",
      "dataset/crohme/train/extracted_img/formulaire018-equation016.png\n",
      "dataset/crohme/train/extracted_img/124_Fabricio.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_24_sub_38.png\n",
      "dataset/crohme/train/extracted_img/105_user0.png\n",
      "dataset/crohme/train/extracted_img/formulaire033-equation059.png\n",
      "dataset/crohme/train/extracted_img/formulaire017-equation001.png\n",
      "dataset/crohme/train/extracted_img/formulaire008-equation071.png\n",
      "dataset/crohme/train/extracted_img/formulaire013-equation027.png\n",
      "dataset/crohme/train/extracted_img/formulaire029-equation057.png\n",
      "dataset/crohme/train/extracted_img/TrainData2_23_sub_51.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_1_sub_38.png\n",
      "dataset/crohme/train/extracted_img/TrainData2_16_sub_13.png\n",
      "dataset/crohme/train/extracted_img/105_herbert.png\n",
      "dataset/crohme/train/extracted_img/94_bruno.png\n",
      "dataset/crohme/train/extracted_img/94_carlos.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_28_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire014-equation067.png\n",
      "dataset/crohme/train/extracted_img/MfrDB2383.png\n",
      "dataset/crohme/train/extracted_img/105_carlos.png\n",
      "dataset/crohme/train/extracted_img/formulaire024-equation053.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_6_sub_38.png\n",
      "dataset/crohme/train/extracted_img/92_alfonso.png\n",
      "dataset/crohme/train/extracted_img/formulaire019-equation058.png\n",
      "dataset/crohme/train/extracted_img/formulaire019-equation006.png\n",
      "dataset/crohme/train/extracted_img/MfrDB3220.png\n",
      "dataset/crohme/train/extracted_img/105_rosario.png\n",
      "dataset/crohme/train/extracted_img/124_maira.png\n",
      "dataset/crohme/train/extracted_img/98_silas.png\n",
      "dataset/crohme/train/extracted_img/formulaire012-equation068.png\n",
      "dataset/crohme/train/extracted_img/formulaire017-equation053.png\n",
      "dataset/crohme/train/extracted_img/105_lucelia.png\n",
      "dataset/crohme/train/extracted_img/200924-1312-84.png\n",
      "dataset/crohme/train/extracted_img/formulaire011-equation029.png\n",
      "dataset/crohme/train/extracted_img/formulaire026-equation019.png\n",
      "dataset/crohme/train/extracted_img/TrainData2_8_sub_13.png\n",
      "dataset/crohme/train/extracted_img/124_leo.png\n",
      "dataset/crohme/train/extracted_img/formulaire020-equation014.png\n",
      "dataset/crohme/train/extracted_img/formulaire025-equation058.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_0_sub_38.png\n",
      "dataset/crohme/train/extracted_img/81_miguel.png\n",
      "dataset/crohme/train/extracted_img/formulaire023-equation007.png\n",
      "dataset/crohme/train/extracted_img/formulaire009-equation068.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_3_sub_38.png\n",
      "dataset/crohme/train/extracted_img/105_mijail.png\n",
      "dataset/crohme/train/extracted_img/TrainData2_1_sub_13.png\n",
      "dataset/crohme/train/extracted_img/formulaire028-equation041.png\n",
      "dataset/crohme/train/extracted_img/formulaire018-equation034.png\n",
      "dataset/crohme/train/extracted_img/94_alfonso.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_26_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire013-equation024.png\n",
      "dataset/crohme/train/extracted_img/formulaire005-equation062.png\n",
      "dataset/crohme/train/extracted_img/formulaire003-equation074.png\n",
      "dataset/crohme/train/extracted_img/formulaire024-equation013.png\n",
      "dataset/crohme/train/extracted_img/formulaire026-equation016.png\n",
      "dataset/crohme/train/extracted_img/105_Fabricio.png\n",
      "dataset/crohme/train/extracted_img/105_hirata.png\n",
      "dataset/crohme/train/extracted_img/105_caue.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_15_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire007-equation073.png\n",
      "dataset/crohme/train/extracted_img/formulaire012-equation044.png\n",
      "dataset/crohme/train/extracted_img/105_leissi.png\n",
      "dataset/crohme/train/extracted_img/105_miguel.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_22_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire018-equation056.png\n",
      "dataset/crohme/train/extracted_img/105_leo.png\n",
      "dataset/crohme/train/extracted_img/105_Frank.png\n",
      "dataset/crohme/train/extracted_img/94_fujita.png\n",
      "dataset/crohme/train/extracted_img/formulaire008-equation011.png\n",
      "dataset/crohme/train/extracted_img/105_edwin.png\n",
      "dataset/crohme/train/extracted_img/formulaire020-equation056.png\n",
      "dataset/crohme/train/extracted_img/formulaire009-equation014.png\n",
      "dataset/crohme/train/extracted_img/formulaire006-equation024.png\n",
      "dataset/crohme/train/extracted_img/formulaire025-equation003.png\n",
      "dataset/crohme/train/extracted_img/formulaire010-equation061.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_5_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire023-equation069.png\n",
      "dataset/crohme/train/extracted_img/formulaire023-equation031.png\n",
      "dataset/crohme/train/extracted_img/formulaire032-equation014.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_10_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire018-equation033.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_8_sub_38.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_21_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire030-equation023.png\n",
      "dataset/crohme/train/extracted_img/105_alfonso.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_30_sub_38.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_4_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire031-equation031.png\n",
      "dataset/crohme/train/extracted_img/105_fujita.png\n",
      "dataset/crohme/train/extracted_img/KME2G3_25_sub_38.png\n",
      "dataset/crohme/train/extracted_img/formulaire008-equation034.png\n",
      "dataset/crohme/train/extracted_img/formulaire014-equation037.png\n",
      "dataset/crohme/train/extracted_img/formulaire026-equation049.png\n",
      "dataset/crohme/train/extracted_img/formulaire009-equation046.png\n",
      "dataset/crohme/train/extracted_img/formulaire029-equation062.png\n",
      "dataset/crohme/train/extracted_img/formulaire013-equation038.png\n",
      "dataset/crohme/train/extracted_img/105_danilo.png\n",
      "dataset/crohme/train/extracted_img/formulaire035-equation053.png\n",
      "dataset/crohme/train/extracted_img/MfrDB3493.png\n",
      "dataset/crohme/train/extracted_img/formulaire021-equation072.png\n",
      "dataset/crohme/train/extracted_img/formulaire013-equation046.png\n",
      "dataset/crohme/train/extracted_img/formulaire018-equation059.png\n",
      "dataset/crohme/train/extracted_img/formulaire026-equation008.png\n",
      "dataset/crohme/train/extracted_img/formulaire027-equation007.png\n",
      "CSV file saved as dataset/crohme/train/crohme_labels.csv\n"
     ]
    }
   ],
   "source": [
    "merge_hmer_or_crohme(images=training_img_pkl_dir, caption=caption_dir, extract_img_dir=train_img_base_dir, save_file=mapping_csv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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