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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="description"
content="Evaluating mathematical reasoning of foundation models in visual contexts">
<meta name="keywords" content="MathVista, Math Vista">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title> LEXam: Benchmarking Legal Reasoning on 340 Law Exams</title>
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<h1 class="title is-1 publication-title is-bold">
<img src="static/images_lexam/logo.png" style="width:4em;vertical-align: middle" alt="Logo"/>
<span class="mathvista" style="vertical-align: middle">LEXam</span>
</h1>
<h2 class="subtitle is-3 publication-subtitle">
<b><i>LEXam: Benchmarking Legal Reasoning on 340 Law Exams</i></b>
<!-- <br> -->
<!-- with GPT-4V, Bard, and Other Large Multimodal Models -->
</h2>
<div class="is-size-5 publication-authors">
<span class="author-block">
<a href="https://www.linkedin.com/in/yu-fan-17b699198/">
<b>Yu Fan*</b></a><sup style="color:#6fbf73;">E</sup>,</span>
<span class="author-block">
<a href="https://edisonni-hku.github.io/">
<b>Jingwei Ni*</b></a><sup style="color:#6fbf73;">E</sup><sup style="color:#010101;">,</sup><sup style="color:#ed4b82;">Z</sup>,</span>
<span class="author-block">
<a href="https://lawecon.ethz.ch/group/scientific-team/merane.html">
<b>Jakob Merane*</b></a><sup style="color:#6fbf73;">E</sup><sup style="color:#010101;">,</sup><sup style="color:#9400D3">L</sup><sup style="color:#010101;">,</sup><sup style="color:#ffac33">M</sup>,</span>
<span class="author-block">
<a href="https://www.linkedin.com/in/yang-tian-6b068b19b/">
<b>Yang Tian*</a></b><sup style="color:#ed4b82">Z</sup>,</span>
<span class="author-block"><a href="https://www.ius.uzh.ch/de/staff/professorships/alphabetical/hermstruewer/team/person.html">Yoan Hermstrüwer</a><sup style="color:#ed4b82">Z</sup><sup style="color:#010101;">,</sup><sup style="color:#ffac33">M</sup>,</span>
<span class="author-block"><a href="https://eleanor-h.github.io/">Yinya Huang</a><sup style="color:#6fbf73;">E</sup>,</span>
<span class="author-block"><a href="https://www.mubasharaakhtar.com/">Mubashara Akhtar</a><sup style="color:#6fbf73;">E</sup>,</span>
<span class="author-block">
<a href="https://www.linkedin.com/in/etienne-salimbeni-595221170/">Etienne Salimbeni</a><sup style="color:#033cfb">O</sup>,
</span>
<span class="author-block">
<a href="https://orcid.org/0000-0003-4733-6049">Florian Geering</a><sup style="color:#ed4b82">Z</sup>,
</span>
<span class="author-block">
<a href="https://www.linkedin.com/in/oliver-dreyer-092ba6133/?originalSubdomain=ch">Oliver Dreyer</a><sup style="color:#68d8fb">S</sup>,
</span>
<span class="author-block">
<a href="https://www.linkedin.com/in/daniel-brunner-06668937/?originalSubdomain=ch">Daniel Brunner</a><sup style="color:#f41c1c">C</sup>,
</span>
<!-- <br> -->
<span class="author-block">
<a href="https://www.df.uzh.ch/en/people/professor/leippold.html">Markus Leippold</a><sup style="color:#ed4b82">Z</sup>,
</span>
<span class="author-block">
<a href="https://www.linkedin.com/in/mrinmaya-sachan-80b08183/">Mrinmaya Sachan</a><sup style="color:#6fbf73;">E</sup>,
</span>
<span class="author-block">
<a href="https://laweconbusiness.ethz.ch/group/professor/stremitzer.html">Alexander Stremitzer</a><sup style="color:#6fbf73;">E</sup>,
</span>
<span class="author-block">
<a href="https://www.coll.mpg.de/engel">Christoph Engel</a><sup style="color:#ffac33">M</sup>,
</span>
<span class="author-block">
<a href="https://elliottash.com/">Elliott Ash</a><sup style="color:#6fbf73;">E</sup>,
</span>
<span class="author-block">
<a href="https://niklaus.ai/">Joel Niklaus</a><sup style="color:#6f68fb">N</sup>
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><sup style="color:#6fbf73;">E</sup>ETH Zurich,</span>
<span class="author-block"><sup style="color:#ed4b82">Z</sup>University of Zurich,</span>
<span class="author-block"><sup style="color:#9400D3">L</sup>University of Lausanne,</span><br>
<span class="author-block"><sup style="color:#ffac33">M</sup>Max Planck Institute for Research on Collective Goods</span><br>
<span class="author-block"><sup style="color:#f41c1c">C</sup>Swiss Federal Supreme Court</span>
<span class="author-block"><sup style="color:#033cfb">O</sup>Omnilex</span>
<span class="author-block"><sup style="color:#68d8fb">S</sup>University of St. Gallen</span>
<span class="author-block"><sup style="color:#6f68fb">N</sup>Niklaus.ai</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><sup>*</sup>Equal Contribution</span>
</div>
<div class="column has-text-centered">
<div class="publication-links">
<!-- PDF Link. -->
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<a href="https://huggingface.co/datasets/LEXam-Benchmark/LEXam"
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<p style="font-size:18px">🤗</p>
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</span>
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<a href="https://arxiv.org/abs/2505.12864"
class="external-link button is-normal is-rounded is-dark">
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<span>arXiv</span>
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class="external-link button is-normal is-rounded is-dark">
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<p style="font-size:18px">🏆</p>
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<span>Leaderboard</span>
</a>
</span>
<span class="link-block">
<a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5265144"
class="external-link button is-normal is-rounded is-dark">
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<i class="ai ai-ssrn"></i>
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<span>SSRN</span>
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<a href="https://github.com/LEXam-Benchmark/LEXam"
class="external-link button is-normal is-rounded is-dark">
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<span>Code</span>
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<!-- Abstract. -->
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<h2 class="title is-3">Introduction</h2>
<div class="content has-text-justified">
<p>
Long-form legal reasoning remains a key challenge for large language models (LLMs) in spite of recent advances in test-time scaling. To address this, we introduce <img src="static/images_lexam/logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/><span class="mathvista">LEXam</span>, a novel benchmark derived from 340 law exams spanning 116 law school courses across a range of subjects and degree levels. The dataset comprises 7,537 law exam questions in English and German. It includes both long-form, open-ended questions and multiple-choice questions with varying numbers of options. Besides reference answers, the open questions are also accompanied by explicit guidance outlining the expected legal reasoning approach such as issue spotting, rule recall, or rule application. Our evaluation on both open-ended and multiple-choice questions present significant challenges for current LLMs; in particular, they notably struggle with open questions that require structured, multi-step legal reasoning. Moreover, our results underscore the effectiveness of the dataset in differentiating between models with varying capabilities. Deploying an ensemble LLM-as-a-Judge paradigm with rigorous human expert validation, we demonstrate how model-generated reasoning steps can be evaluated consistently and accurately, closely aligning with human expert assessments. Our evaluation setup provides a scalable method to assess legal reasoning quality beyond simple accuracy metrics.
</p>
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</div>
<!--/ Abstract. -->
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</section>
<!-- Leaderboard: Open Questions -->
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<h1 class="title is-1 mathvista" id="leaderboard">
<img src="static/images_lexam/logo.png" style="width:1em;vertical-align: middle" alt="Logo"/>
<span class="mathvista" style="vertical-align: middle">Leaderboard on LEXam – Open Questions</span>
</h1>
</div>
</section>
<section class="section">
<div class="container">
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<table class="table is-bordered is-striped is-narrow is-hoverable is-fullwidth">
<thead>
<tr>
<th>#</th>
<th>LLMs</th>
<th>Judge Scores on Open Questions</th>
</tr>
</thead>
<tbody>
<tr><td><strong>1</strong></td><td><strong><span style="color:red;">GPT-5🥇</span></strong></td><td><strong>70.20</strong></td></tr>
<tr><td><strong>2</strong></td><td><strong><span style="color:red;">Gemini-2.5-Pro🥈</span></strong></td><td>67.40</td></tr>
<tr><td><strong>3</strong></td><td><strong><span style="color:red;">Claude-3.7-Sonnet🥉</span></strong></td><td>62.86</td></tr>
<tr><td><strong>4</strong></td><td><strong>Claude-4.5-Sonnet</strong></td><td>62.76</td></tr>
<tr><td><strong>5</strong></td><td><strong>GPT-5-mini</strong></td><td>60.32</td></tr>
<tr><td><strong>6</strong></td><td><strong>GPT-4.1</strong></td><td>57.50</td></tr>
<tr><td><strong>7</strong></td><td><strong>DeepSeek-V3.2-Exp</strong></td><td>57.42</td></tr>
<tr><td><strong>8</strong></td><td><strong>GPT-4o</strong></td><td>56.93</td></tr>
<tr><td><strong>9</strong></td><td><strong>DeepSeek-V3.2-reasoner</strong></td><td>56.53</td></tr>
<tr><td><strong>10</strong></td><td><strong>DeepSeek-V3.2-chat</strong></td><td>55.99</td></tr>
<tr><td><strong>11</strong></td><td><strong>DeepSeek-R1</strong></td><td>55.91</td></tr>
<tr><td><strong>12</strong></td><td><strong>Gemini-3-Pro-preview</strong></td><td>55.38</td></tr>
<tr><td><strong>13</strong></td><td><strong>GPT-4.1-mini</strong></td><td>54.58</td></tr>
<tr><td><strong>14</strong></td><td><strong>DeepSeek-V3</strong></td><td>52.53</td></tr>
<tr><td><strong>15</strong></td><td><strong>GPT-OSS-120B</strong></td><td>51.74</td></tr>
<tr><td><strong>16</strong></td><td><strong>O3-mini</strong></td><td>48.13</td></tr>
<tr><td><strong>17</strong></td><td><strong>Llama-4-Maverick</strong></td><td>47.25</td></tr>
<tr><td><strong>18</strong></td><td><strong>Qwen3-235B</strong></td><td>47.25</td></tr>
<tr><td><strong>19</strong></td><td><strong>QwQ-32B</strong></td><td>44.36</td></tr>
<tr><td><strong>20</strong></td><td><strong>GPT-4.1-nano</strong></td><td>43.68</td></tr>
<tr><td><strong>21</strong></td><td><strong>Qwen3-Next</strong></td><td>43.37</td></tr>
<tr><td><strong>22</strong></td><td><strong>Llama-3.1-405B-it</strong></td><td>43.14</td></tr>
<tr><td><strong>23</strong></td><td><strong>GPT-4o-mini</strong></td><td>42.55</td></tr>
<tr><td><strong>24</strong></td><td><strong>Gemma-3-12B-it</strong></td><td>41.29</td></tr>
<tr><td><strong>25</strong></td><td><strong>Llama-3.3-70B-it</strong></td><td>41.27</td></tr>
<tr><td><strong>26</strong></td><td><strong>Qwen3-32B</strong></td><td>40.00</td></tr>
<tr><td><strong>27</strong></td><td><strong>Phi-4</strong></td><td>38.54</td></tr>
<tr><td><strong>28</strong></td><td><strong>Apertus-70B</strong></td><td>34.70</td></tr>
<tr><td><strong>29</strong></td><td><strong>GPT-OSS-20B</strong></td><td>32.12</td></tr>
<tr><td><strong>30</strong></td><td><strong>Gemma-2-9B-it</strong></td><td>27.41</td></tr>
<tr><td><strong>31</strong></td><td><strong>GPT-5-nano</strong></td><td>27.25</td></tr>
<tr><td><strong>32</strong></td><td><strong>EuroLLM-9B-it</strong></td><td>22.95</td></tr>
<tr><td><strong>33</strong></td><td><strong>Apertus-8B</strong></td><td>22.44</td></tr>
<tr><td><strong>34</strong></td><td><strong>Qwen-2.5-7B-it</strong></td><td>16.67</td></tr>
<tr><td><strong>35</strong></td><td><strong>Ministral-8B-it</strong></td><td>14.88</td></tr>
<tr><td><strong>36</strong></td><td><strong>Llama-3.1-8B-it</strong></td><td>10.00</td></tr>
</tbody>
</table>
</div>
</div>
</div>
</section>
<!-- Leaderboard: Multiple-Choice Questions -->
<section class="hero is-light is-small">
<div class="hero-body has-text-centered">
<h1 class="title is-1 mathvista" id="leaderboard-mcq">
<img src="static/images_lexam/logo.png" style="width:1em;vertical-align: middle" alt="Logo"/>
<span class="mathvista" style="vertical-align: middle">Leaderboard on LEXam – Multiple-Choice Questions</span>
</h1>
</div>
</section>
<section class="section">
<div class="container">
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<table class="table is-bordered is-striped is-narrow is-hoverable is-fullwidth">
<thead>
<tr>
<th>#</th>
<th>LLMs</th>
<th>Accuracy on Multiple-Choice Questions</th>
</tr>
</thead>
<tbody>
<tr><td><strong>1</strong></td><td><strong><span style="color:red;">GPT-5🥇</span></strong></td><td><strong>62.65</strong></td></tr>
<tr><td><strong>2</strong></td><td><strong><span style="color:red;">Claude-4.5-Sonnet🥈</span></strong></td><td>58.01</td></tr>
<tr><td><strong>3</strong></td><td><strong><span style="color:red;">Claude-3.7-Sonnet🥉</span></strong></td><td>57.23</td></tr>
<tr><td><strong>4</strong></td><td><strong>Gemini-2.5-Pro</strong></td><td>55.72</td></tr>
<tr><td><strong>5</strong></td><td><strong>GPT-5-mini</strong></td><td>54.82</td></tr>
<tr><td><strong>6</strong></td><td><strong>GPT-4.1</strong></td><td>54.40</td></tr>
<tr><td><strong>7</strong></td><td><strong>GPT-4o</strong></td><td>53.13</td></tr>
<tr><td><strong>8</strong></td><td><strong>DeepSeek-V3.2-Exp</strong></td><td>53.07</td></tr>
<tr><td><strong>9</strong></td><td><strong>DeepSeek-R1</strong></td><td>52.41</td></tr>
<tr><td><strong>10</strong></td><td><strong>Llama-4-Maverick</strong></td><td>49.10</td></tr>
<tr><td><strong>11</strong></td><td><strong>GPT-4.1-mini</strong></td><td>48.49</td></tr>
<tr><td><strong>12</strong></td><td><strong>Qwen3-235B</strong></td><td>48.19</td></tr>
<tr><td><strong>13</strong></td><td><strong>QwQ-32B</strong></td><td>47.83</td></tr>
<tr><td><strong>14</strong></td><td><strong>GPT-OSS-120B</strong></td><td>47.71</td></tr>
<tr><td><strong>15</strong></td><td><strong>GPT-5-nano</strong></td><td>47.11</td></tr>
<tr><td><strong>16</strong></td><td><strong>DeepSeek-V3</strong></td><td>46.57</td></tr>
<tr><td><strong>17</strong></td><td><strong>Qwen3-32B</strong></td><td>45.30</td></tr>
<tr><td><strong>18</strong></td><td><strong>O3-mini</strong></td><td>44.22</td></tr>
<tr><td><strong>19</strong></td><td><strong>Qwen3-Next</strong></td><td>43.31</td></tr>
<tr><td><strong>20</strong></td><td><strong>Llama-3.1-405B-it</strong></td><td>43.19</td></tr>
<tr><td><strong>21</strong></td><td><strong>GPT-4o-mini</strong></td><td>40.96</td></tr>
<tr><td><strong>22</strong></td><td><strong>GPT-OSS-20B</strong></td><td>40.78</td></tr>
<tr><td><strong>23</strong></td><td><strong>Phi-4</strong></td><td>40.66</td></tr>
<tr><td><strong>24</strong></td><td><strong>GPT-4.1-nano</strong></td><td>39.22</td></tr>
<tr><td><strong>25</strong></td><td><strong>Gemma-3-12B-it</strong></td><td>29.94</td></tr>
<tr><td><strong>26</strong></td><td><strong>Qwen-2.5-7B-it</strong></td><td>29.28</td></tr>
<tr><td><strong>27</strong></td><td><strong>Llama-3.3-70B-it</strong></td><td>28.19</td></tr>
<tr><td><strong>28</strong></td><td><strong>Ministral-8B-it</strong></td><td>26.27</td></tr>
<tr><td><strong>29</strong></td><td><strong>Gemma-2-9B-it</strong></td><td>25.36</td></tr>
<tr><td><strong>30</strong></td><td><strong>Llama-3.1-8B-it</strong></td><td>24.04</td></tr>
<tr><td><strong>31</strong></td><td><strong>EuroLLM-9B-it</strong></td><td>23.31</td></tr>
</tbody>
</table>
</div>
</div>
</div>
</section>
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<section class="hero is-light is-small">
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<!-- <h1 class="title is-1 mathvista"><img src="static/images/mathverse.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>MathVista Dataset</h1> -->
<h1 class="title is-1 mathvista">
<img src="static/images_lexam/logo.png" style="width:1em;vertical-align: middle" alt="Logo"/>
<span class="mathvista" style="vertical-align: middle">LEXam Dataset</span>
</h1>
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</section>
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<h2 class="title is-3">Overview</h2>
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<div class="content has-text-centered">
<img src="static/images_lexam/collection_of_plots_new.png" alt="FVEL" width="100%"/>
<p> <img src="static/images_lexam/logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">LEXam</span> data generation pipeline. </p>
</div>
</div>
<div class="content has-text-justified">
<!-- <p>
<img src="static/images_fvel/fveler_logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">FVELer</span> contains transformed Isabelle theories and lemmas from C codes that support the FVEL environment for C code verification. It has two main components:
<ul>
<li><b>Theories dependencies.</b> A resource for dependencies among theories, lemmas, and c code specified by SeL4 verification. These data provide the ground-truth seL4 premises for proving the current lemma and enable a model to retrieve related statements or proof context at both the training and testing stages.</li>
<li><b>Lemmas from theories with their Isabelle proof states.</b> The step-wise lemmas with multiple proof states that support the Isabelle proving process in FVEL. These data on the one hand enhance LLMs with search-based/step-wise ATP while interacting with FVEL, and on the other hand, provide a benchmark for interactive formal verification.</li>
</ul>
</p> -->
<!-- <p>
You can download the dataset on <a href="https://github.com/FVELER/FVELerExtraction/blob/main/FVELer.zip" target="_blank">GitHub</a>.
</p> -->
</div>
</div>
</div>
<span style="margin:25pt">
</span>
<!-- <div class="columns is-centered m-6"> -->
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">Dataset Statistics</h2>
<div class="content has-text-justified">
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_lexam/combined_distribution_subplots_svg.png" alt="FVEL" width="120%"/>
<p> The statistics of <img src="static/images_lexam/logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">LEXam</span> dataset. </p>
</div>
</div>
<!-- <p>
<img src="static/images_fvel/fveler_logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">FVELer</span> contains contains <b>758 theories, 29,125 lemmas, and 200,646 proof steps</b> in total with in-depth dependencies. We randomly split <span class="mathvista">FVELer</span> according to lemmas, resulting in a training set, a validation set, a test set, and an especially selected test-hard set. The test-hard set is selected from those lemmas in the three sessions <b>SysInit</b>, <b>SysInitExamples</b>, <b>LibTest</b>. Such lemmas are in higher depths in the dependency relationship, therefore they have less <i>import</i> relationships by other theories.
</p>
<ul><li> Figures below shows the distribution of extracted theories and lemmas by it's depth. where <b>depth</b> is the degree of the theory dependency graph by the <i>import</i> dependency relationship among the theory files.</li></ul> -->
<!-- <div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fveler_dastaset/theorem_depth.png" alt="FVEL" width="100%"/>
<p> Distribution of dependency by theory </p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fveler_dastaset/lemma_depth.png" alt="FVEL" width="100%"/>
<p> Distribution of dependency by lemma. </p>
</div>
</div> -->
<!-- <ul><li> Figure below demonstrates the distribution of intermediate proof steps of the 29,125 lemmas. It is indicated that the number of proof steps are dramatically different amount the lemmas. 12,089 out of the 29,125 lemmas can be proved via one proof step. Proof steps between 2 and 10 there are 12,957 lemmas. Therefore, over 85% of the lemmas in FVELER can be proved within 10 steps. Moreover, 28,954 out of the 29,125 lemmas can be proved within 100 steps </li></ul>
-->
<!-- <div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fveler_dastaset/lemma_step.pdf.png" alt="FVEL" width="100%"/>
<p> The <span class="mathvista">FVELer</span> lemma distribution over step intervals. </p>
</div>
</div> -->
</div>
</div>
</div>
<!-- </div> -->
<div class="columns is-centered m-6">
<div class="column is-full has-text-centered content">
<h2 class="title is-3">Dataset Examples</h2>
<!-- <p>Examples in <img src="static/images_fvel/fveler_logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">FVELer</span></p> -->
<!-- <p align="justify">
<ul align="justify">
<li><b>sel4_extraction/</b> is a folder that has the same structure as the sel4 verification project (l4v).
Each file is the extracted step-wise proof state of the corresponding l4v theory files.
For example, “sel4_extraction/proof/invariant-abstract/AInvs.json” is the
proof state of the file l4v/proof/invariant-abstract/AInvs.thy.
</li>
<li><b>dataset_lemma_split.json</b> contains all lemmas proof steps and states, and splits them into the train, val, test, and test-hard set.</li>
<li><b>sel4_thy_info.json</b> contains information of all theory files, including their names, dependency relations, and lemmas.</li>
<li><b>sel4_session_info.json</b> contains all session information, including dependent sessions, theories, and directories.</li>
</ul>
</p> -->
<div id="results-carousel" class="carousel results-carousel">
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_lexam/questions_pic.png" alt="" width="100%"/>
<p><img src="static/images_lexam/logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">LEXam</span> sample open and multiple-choice question.</p>
</div>
</div>
<!--
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<div class="content has-text-centered">
<img src="static/images_fvel/cases_dataset/C.1.3_sel4_thy_info_json.png" alt="" width="70%"/>
<p><b>sel4_thy_info.json</b> example.</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/cases_dataset/C.1.4_sel4_session_info_json.png" alt="" width="70%"/>
<p><b>sel4_session_info.json</b> example.</p>
</div>
</div> -->
</section>
<!-- LEADERBAORD SECTION -->
<!-- <section class="hero is-light is-small">
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<h1 class="title is-1 mathvista">
<img src="static/images_lexam/logo.png" style="width:1em;vertical-align: middle" alt="Logo"/>
<span class="mathvista" style="vertical-align: middle">LEXam Leaderboard</span>
</h1>
</div>
</section>
<span style="margin:25pt">
</span> -->
<!-- <div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<h2 class="title is-3">Dataset statistics</h2>
<div class="content has-text-justified">
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fvel_env/fevl_table1.png" alt="algebraic reasoning" width="60%"/>
<p> The statistics of <img src="static/images_fvel/fveler_logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">FVELer</span> dataset. </p>
</div>
</div>
<p>
<img src="static/images_fvel/fveler_logo.png" style="width:2.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">FVELer</span> contains contains <b>758 theories, 29,125 lemmas, and 200,646 proof steps</b> in total with in-depth dependencies. We randomly split <span class="mathvista">FVELer</span> according to lemmas, resulting in a training set, a validation set, a test set, and an especially selected test-hard set. The test-hard set is selected from those lemmas in the three sessions <b>SysInit</b>, <b>SysInitExamples</b>, <b>LibTest</b>. Such lemmas are in higher depths in the dependency relationship, therefore they have less <i>import</i> relationships by other theories.
</p>
<ul><li> Figures below shows the distribution of extracted theories and lemmas by it's depth. where <b>depth</b> is the degree of the theory dependency graph by the <i>import</i> dependency relationship among the theory files.</li></ul>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fveler_dastaset/theorem_depth.png" alt="algebraic reasoning" width="100%"/>
<p> Distribution of dependency by theory </p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fveler_dastaset/lemma_depth.png" alt="arithmetic reasoning" width="100%"/>
<p> Distribution of dependency by lemma. </p>
</div>
</div>
<ul><li> Figure below demonstrates the distribution of intermediate proof steps of the 29,125 lemmas. It is indicated that the number of proof steps are dramatically different amount the lemmas. 12,089 out of the 29,125 lemmas can be proved via one proof step. Proof steps between 2 and 10 there are 12,957 lemmas. Therefore, over 85% of the lemmas in FVELER can be proved within 10 steps. Moreover, 28,954 out of the 29,125 lemmas can be proved within 100 steps </li></ul>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/overview_fveler_dastaset/lemma_step.pdf.png" alt="arithmetic reasoning" width="100%"/>
<p> The <span class="mathvista">FVELer</span> lemma distribution over step intervals. </p>
</div>
</div>
</div>
</div>
</div> -->
<!--
<div class="columns is-centered m-6">
<div class="column is-full has-text-centered content">
<h2 class="title is-3">Dataset Examples</h2>
<p align="justify">
<ul align="justify">
<li><b>sel4_extraction/</b> is a folder that has the same structure as the sel4 verification project (l4v).
Each file is the extracted step-wise proof state of the corresponding l4v theory files.
For example, “sel4_extraction/proof/invariant-abstract/AInvs.json” is the
proof state of the file l4v/proof/invariant-abstract/AInvs.thy.
</li>
<li><b>dataset_lemma_split.json</b> contains all lemmas proof steps and states, and splits them into the train, val, test, and test-hard set.</li>
<li><b>sel4_thy_info.json</b> contains information of all theory files, including their names, dependency relations, and lemmas.</li>
<li><b>sel4_session_info.json</b> contains all session information, including dependent sessions, theories, and directories.</li>
</ul>
</p>
<div id="results-carousel" class="carousel results-carousel">
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/cases_dataset/C.1.1_sel4_extraction.png" alt="" width="70%"/>
<p><b>sel4_extraction/</b> folder example.</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/cases_dataset/C.1.2_dataset_lemma_split_json.png" alt="" width="70%"/>
<p><b>dataset_lemma_split.json</b> example.</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/cases_dataset/C.1.3_sel4_thy_info_json.png" alt="" width="70%"/>
<p><b>sel4_thy_info.json</b> example.</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images_fvel/cases_dataset/C.1.4_sel4_session_info_json.png" alt="" width="70%"/>
<p><b>sel4_session_info.json</b> example.</p>
</div>
</div> -->
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<!-- RESULTS SECTION -->
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<h2 class="title is-3 has-text-centered">BibTeX</h2>
<pre><code>
@article{fan2025lexam,
title={LEXam: Benchmarking Legal Reasoning on 340 Law Exams},
author={Fan, Yu and Ni, Jingwei and Merane, Jakob and Tian, Yang and Hermstrüwer, Yoan and Huang, Yinya and Akhtar, Mubashara and Salimbeni, Etienne and Geering, Florian and Dreyer, Oliver and Brunner, Daniel and Leippold, Markus and Sachan, Mrinmaya and Stremitzer, Alexander and Engel, Christoph and Ash, Elliott and Niklaus, Joel},
journal={arXiv preprint arXiv:2505.12864},
year={2025}
}
</code></pre>
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