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<text text-anchor="start" x="236.65" y="-909.4" font-family="Inter" font-weight="bold" font-size="12.00">Milner&#39;s Lambda&#45;Calculus with Partial Substitutions &#45; theorem dependency (M2)</text>
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<title>occurs_count_zero&#45;&gt;occurs_count_pos</title>
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<g id="edge12" class="edge">
<title>occurs_lift_self&#45;&gt;open_rec_zplug_lift</title>
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<g id="node20" class="node">
<title>zdecs_nonempty</title>
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<text text-anchor="middle" x="239.88" y="-171.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_nonempty</text>
</a>
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</g>
<!-- zdecs_nonempty&#45;&gt;full_comp_aux -->
<g id="edge18" class="edge">
<title>zdecs_nonempty&#45;&gt;full_comp_aux</title>
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<!-- occurs_count_zfill_plug&#45;&gt;full_comp_aux -->
<g id="edge15" class="edge">
<title>occurs_count_zfill_plug&#45;&gt;full_comp_aux</title>
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<g id="edge17" class="edge">
<title>open_rec_zplug_lift&#45;&gt;full_comp_aux</title>
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<g id="node24" class="node">
<title>lemma_2_2_full_comp</title>
<g id="a_node24"><a xlink:title="lemma_2_2_full_comp (section 2.2) theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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</g>
<!-- full_comp_aux&#45;&gt;lemma_2_2_full_comp -->
<g id="edge20" class="edge">
<title>full_comp_aux&#45;&gt;lemma_2_2_full_comp</title>
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<g id="edge27" class="edge">
<title>lemma_2_2_full_comp&#45;&gt;lemma_2_3_beta_sim</title>
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<!-- red1_root_fv -->
<g id="node27" class="node">
<title>red1_root_fv</title>
<g id="a_node27"><a xlink:title="red1_root_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-563" rx="38.87" ry="18"/>
<text text-anchor="middle" x="408.74" y="-560.8" font-family="Inter" font-size="9.00" fill="#222222">red1_root_fv</text>
</a>
</g>
</g>
<!-- fvs_zplug_lift&#45;&gt;red1_root_fv -->
<g id="edge22" class="edge">
<title>fvs_zplug_lift&#45;&gt;red1_root_fv</title>
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<!-- plug_fv_mono -->
<g id="node26" class="node">
<title>plug_fv_mono</title>
<g id="a_node26"><a xlink:title="plug_fv_mono theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-615" rx="42.95" ry="18"/>
<text text-anchor="middle" x="408.74" y="-612.8" font-family="Inter" font-size="9.00" fill="#222222">plug_fv_mono</text>
</a>
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</g>
<!-- red1r_fv -->
<g id="node28" class="node">
<title>red1r_fv</title>
<g id="a_node28"><a xlink:title="red1r_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-563" rx="28.74" ry="18"/>
<text text-anchor="middle" x="565.21" y="-560.8" font-family="Inter" font-size="9.00" fill="#222222">red1r_fv</text>
</a>
</g>
</g>
<!-- plug_fv_mono&#45;&gt;red1r_fv -->
<g id="edge23" class="edge">
<title>plug_fv_mono&#45;&gt;red1r_fv</title>
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<!-- red1_root_fv&#45;&gt;red1r_fv -->
<g id="edge24" class="edge">
<title>red1_root_fv&#45;&gt;red1r_fv</title>
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<g id="node29" class="node">
<title>lemma_2_1_fv_preserved</title>
<g id="a_node29"><a xlink:title="lemma_2_1_fv_preserved (section 2.1) theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="722.21" y="-534.8" font-family="Inter" font-size="9.00" fill="#222222">lemma_2_1_fv_preserved</text>
</a>
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</g>
<!-- red1r_fv&#45;&gt;lemma_2_1_fv_preserved -->
<g id="edge25" class="edge">
<title>red1r_fv&#45;&gt;lemma_2_1_fv_preserved</title>
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<!-- corpus_fv_preserved -->
<g id="node50" class="node">
<title>corpus_fv_preserved</title>
<g id="a_node50"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="885.25" y="-478.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_fv_preserved</text>
</a>
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</g>
<!-- lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved -->
<g id="edge46" class="edge">
<title>lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved</title>
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<!-- zdecs_sound -->
<g id="node31" class="node">
<title>zdecs_sound</title>
<g id="a_node31"><a xlink:title="zdecs_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-260" rx="40.98" ry="18"/>
<text text-anchor="middle" x="239.88" y="-257.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_sound</text>
</a>
</g>
</g>
<!-- zdecs_sound&#45;&gt;full_comp_aux -->
<g id="edge19" class="edge">
<title>zdecs_sound&#45;&gt;full_comp_aux</title>
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<!-- root_steps_sound -->
<g id="node34" class="node">
<title>root_steps_sound</title>
<g id="a_node34"><a xlink:title="root_steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-407" rx="51.27" ry="18"/>
<text text-anchor="middle" x="408.74" y="-404.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_sound</text>
</a>
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</g>
<!-- zdecs_sound&#45;&gt;root_steps_sound -->
<g id="edge28" class="edge">
<title>zdecs_sound&#45;&gt;root_steps_sound</title>
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<!-- zdecs_complete -->
<g id="node32" class="node">
<title>zdecs_complete</title>
<g id="a_node32"><a xlink:title="zdecs_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="239.88" y="-456.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_complete</text>
</a>
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</g>
<!-- root_steps_complete -->
<g id="node35" class="node">
<title>root_steps_complete</title>
<g id="a_node35"><a xlink:title="root_steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="408.74" y="-456.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_complete</text>
</a>
</g>
</g>
<!-- zdecs_complete&#45;&gt;root_steps_complete -->
<g id="edge29" class="edge">
<title>zdecs_complete&#45;&gt;root_steps_complete</title>
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<g id="node33" class="node">
<title>positions_at</title>
<g id="a_node33"><a xlink:title="positions_at theory/Reduction.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-511" rx="38.21" ry="18"/>
<text text-anchor="middle" x="408.74" y="-508.8" font-family="Inter" font-size="9.00" fill="#222222">positions_at</text>
</a>
</g>
</g>
<!-- steps_sound -->
<g id="node36" class="node">
<title>steps_sound</title>
<g id="a_node36"><a xlink:title="steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-428" rx="39.02" ry="18"/>
<text text-anchor="middle" x="565.21" y="-425.8" font-family="Inter" font-size="9.00" fill="#222222">steps_sound</text>
</a>
</g>
</g>
<!-- positions_at&#45;&gt;steps_sound -->
<g id="edge30" class="edge">
<title>positions_at&#45;&gt;steps_sound</title>
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<g id="edge31" class="edge">
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<g id="node43" class="node">
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<g id="a_node43"><a xlink:title="steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="565.21" y="-313.8" font-family="Inter" font-size="9.00" fill="#222222">steps_complete</text>
</a>
</g>
</g>
<!-- root_steps_complete&#45;&gt;steps_complete -->
<g id="edge37" class="edge">
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<g id="node37" class="node">
<title>steps_sound_red1</title>
<g id="a_node37"><a xlink:title="steps_sound_red1 theory/Reduction.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="722.21" y="-477.8" font-family="Inter" font-size="9.00" fill="#222222">steps_sound_red1</text>
</a>
</g>
</g>
<!-- steps_sound&#45;&gt;steps_sound_red1 -->
<g id="edge32" class="edge">
<title>steps_sound&#45;&gt;steps_sound_red1</title>
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<!-- has_red_spec -->
<g id="node45" class="node">
<title>has_red_spec</title>
<g id="a_node45"><a xlink:title="has_red_spec theory/Reduction.v https://arxiv.org/html/2312.13270v1">
<ellipse fill="#d9ead3" stroke="#38761d" cx="722.21" cy="-374" rx="42.95" ry="18"/>
<text text-anchor="middle" x="722.21" y="-371.8" font-family="Inter" font-size="9.00" fill="#222222">has_red_spec</text>
</a>
</g>
</g>
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<g id="edge41" class="edge">
<title>steps_sound&#45;&gt;has_red_spec</title>
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<g id="node47" class="node">
<title>corpus_sound</title>
<g id="a_node47"><a xlink:title="corpus_sound theory/Tests.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="722.21" y="-425.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_sound</text>
</a>
</g>
</g>
<!-- steps_sound&#45;&gt;corpus_sound -->
<g id="edge43" class="edge">
<title>steps_sound&#45;&gt;corpus_sound</title>
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<g id="edge47" class="edge">
<title>steps_sound_red1&#45;&gt;corpus_fv_preserved</title>
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<g id="node38" class="node">
<title>positions_hole</title>
<g id="a_node38"><a xlink:title="positions_hole theory/Reduction.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="239.88" y="-309.8" font-family="Inter" font-size="9.00" fill="#222222">positions_hole</text>
</a>
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<g id="node39" class="node">
<title>root_steps_in_steps</title>
<g id="a_node39"><a xlink:title="root_steps_in_steps theory/Reduction.v https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="408.74" y="-300.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_in_steps</text>
</a>
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</g>
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<g id="edge33" class="edge">
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<g id="edge38" class="edge">
<title>root_steps_in_steps&#45;&gt;steps_complete</title>
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