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/usr/share/sphinx_rtd_theme/searchbox.html is in sphinx-rtd-theme-common 0.2.4-1.

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{%- if builder != 'singlehtml' %}
<div role="search">
  <form id="rtd-search-form" class="wy-form" action="{{ pathto('search') }}" method="get">
    <input type="text" name="q" placeholder="{{ _('Search docs') }}" />
    <input type="hidden" name="check_keywords" value="yes" />
    <input type="hidden" name="area" value="default" />
  </form>
</div>
{%- endif %}