diff --git a/graphs/dependency.dot b/graphs/dependency.dot
index 80a62a0..4dbe0fb 100644
--- a/graphs/dependency.dot
+++ b/graphs/dependency.dot
@@ -83,6 +83,8 @@ digraph theorem_deps {
   nplug_fv_mono [label="nplug_fv_mono", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nplug_fv_mono theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   nplug_fv_upper [label="nplug_fv_upper", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nplug_fv_upper theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   nred_core_fv [label="nred_core_fv", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nred_core_fv theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  nplug_esub_fv [label="nplug_esub_fv", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nplug_esub_fv theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  nred_fv [label="nred_fv", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nred_fv theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   par_red1_iff [label="par_red1_iff", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   red1_in_par [label="red1_in_par", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   par_context [label="par_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
@@ -194,6 +196,13 @@ digraph theorem_deps {
   in_filter_neq -> nred_core_fv;
   nplug_fv_mono -> nred_core_fv;
   nplug_fv_upper -> nred_core_fv;
+  filter_neq_in -> nplug_esub_fv;
+  in_filter_neq -> nplug_esub_fv;
+  filter_neq_in -> nred_fv;
+  in_filter_neq -> nred_fv;
+  nplug_esub_fv -> nred_fv;
+  nplug_fv_mono -> nred_fv;
+  nplug_fv_upper -> nred_fv;
   par_red1_iff -> par_to_red1_or_eq;
   par_red1_iff -> red1_or_eq_to_par;
   has_red_spec -> check_term_true;
diff --git a/graphs/dependency.png b/graphs/dependency.png
index f66b2a0..89c8c95 100644
Binary files a/graphs/dependency.png and b/graphs/dependency.png differ
diff --git a/graphs/dependency.svg b/graphs/dependency.svg
index bee9854..466e642 100644
--- a/graphs/dependency.svg
+++ b/graphs/dependency.svg
@@ -4,19 +4,19 @@
 <!-- Generated by graphviz version 2.42.4 (0)
  -->
 <!-- Title: theorem_deps Pages: 1 -->
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- viewBox="0.00 0.00 1296.27 3882.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
-<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 3878)">
+<svg width="1296pt" height="3950pt"
+ viewBox="0.00 0.00 1296.27 3950.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 3946)">
 <title>theorem_deps</title>
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-3878 1292.27,-3878 1292.27,4 -4,4"/>
-<text text-anchor="start" x="407.63" y="-3861.4" font-family="Inter" font-weight="bold" font-size="12.00">Milner&#39;s Lambda&#45;Calculus with Partial Substitutions &#45; theorem dependency (M4)</text>
-<text text-anchor="start" x="571.13" y="-3851.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
-<text text-anchor="start" x="601.13" y="-3851.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="608.13" y="-3851.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
-<text text-anchor="start" x="635.13" y="-3851.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="642.13" y="-3851.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
-<text text-anchor="start" x="676.13" y="-3851.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="683.13" y="-3851.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
+<polygon fill="white" stroke="transparent" points="-4,4 -4,-3946 1292.27,-3946 1292.27,4 -4,4"/>
+<text text-anchor="start" x="407.63" y="-3929.4" font-family="Inter" font-weight="bold" font-size="12.00">Milner&#39;s Lambda&#45;Calculus with Partial Substitutions &#45; theorem dependency (M4)</text>
+<text text-anchor="start" x="571.13" y="-3919.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
+<text text-anchor="start" x="601.13" y="-3919.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="608.13" y="-3919.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
+<text text-anchor="start" x="635.13" y="-3919.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="642.13" y="-3919.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
+<text text-anchor="start" x="676.13" y="-3919.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="683.13" y="-3919.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
 <!-- fvs_lift -->
 <g id="node1" class="node">
 <title>fvs_lift</title>
@@ -138,8 +138,8 @@
 <g id="node6" class="node">
 <title>open_rec_occurs_false</title>
 <g id="a_node6"><a xlink:title="open_rec_occurs_false theory/Binding.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-239" rx="65.48" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-236.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_occurs_false</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-238" rx="65.48" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-235.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_occurs_false</text>
 </a>
 </g>
 </g>
@@ -155,8 +155,8 @@
 <!-- open_rec_occurs_false&#45;&gt;open_rec_zplug_lift -->
 <g id="edge24" class="edge">
 <title>open_rec_occurs_false&#45;&gt;open_rec_zplug_lift</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M125.17,-246.17C143.16,-248.36 162.92,-250.76 180.85,-252.94"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="180.65,-255.03 186.86,-253.67 181.15,-250.86 180.65,-255.03"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M124.69,-245.45C142.88,-247.77 162.92,-250.32 181.07,-252.63"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="180.93,-254.73 187.15,-253.41 181.46,-250.57 180.93,-254.73"/>
 </g>
 <!-- full_comp_aux -->
 <g id="node54" class="node">
@@ -170,38 +170,38 @@
 <!-- open_rec_occurs_false&#45;&gt;full_comp_aux -->
 <g id="edge27" class="edge">
 <title>open_rec_occurs_false&#45;&gt;full_comp_aux</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M83.93,-256.56C111.5,-283.74 162.71,-333.39 173.48,-338 233.83,-363.85 311,-363.11 359.92,-359"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="360.25,-361.08 366.04,-358.46 359.87,-356.9 360.25,-361.08"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M87.92,-255.13C100.64,-265.45 116.85,-279.06 130.48,-292 150.78,-311.26 148.18,-326.04 173.48,-338 233.2,-366.22 311.19,-364.72 360.39,-359.82"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="360.79,-361.89 366.54,-359.18 360.35,-357.71 360.79,-361.89"/>
 </g>
 <!-- subst_notin -->
-<g id="node117" class="node">
+<g id="node119" class="node">
 <title>subst_notin</title>
-<g id="a_node117"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node119"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-18" rx="37.56" ry="18"/>
 <text text-anchor="middle" x="239.88" y="-15.8" font-family="Inter" font-size="9.00" fill="#222222">subst_notin</text>
 </a>
 </g>
 </g>
 <!-- open_rec_occurs_false&#45;&gt;subst_notin -->
-<g id="edge76" class="edge">
+<g id="edge83" class="edge">
 <title>open_rec_occurs_false&#45;&gt;subst_notin</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M71.52,-221.02C83.51,-183.14 116.28,-94.5 173.48,-44 181.14,-37.24 190.78,-32.12 200.28,-28.29"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="201.29,-30.15 206.16,-26.07 199.81,-26.23 201.29,-30.15"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M71.64,-219.82C83.75,-181.98 116.59,-94.11 173.48,-44 181.15,-37.25 190.79,-32.13 200.29,-28.3"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="201.3,-30.17 206.17,-26.08 199.82,-26.24 201.3,-30.17"/>
 </g>
 <!-- full_comp_aux_sub -->
-<g id="node131" class="node">
+<g id="node133" class="node">
 <title>full_comp_aux_sub</title>
-<g id="a_node131"><a xlink:title="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node133"><a xlink:title="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-217" rx="54.7" ry="18"/>
 <text text-anchor="middle" x="408.74" y="-214.8" font-family="Inter" font-size="9.00" fill="#222222">full_comp_aux_sub</text>
 </a>
 </g>
 </g>
 <!-- open_rec_occurs_false&#45;&gt;full_comp_aux_sub -->
-<g id="edge86" class="edge">
+<g id="edge93" class="edge">
 <title>open_rec_occurs_false&#45;&gt;full_comp_aux_sub</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M128.88,-234.96C191.52,-230.93 287.14,-224.77 348.54,-220.81"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="348.89,-222.9 354.75,-220.41 348.62,-218.7 348.89,-222.9"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M129.27,-234.12C191.88,-230.27 287.14,-224.42 348.41,-220.65"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="348.74,-222.73 354.6,-220.27 348.48,-218.54 348.74,-222.73"/>
 </g>
 <!-- close_rec_notin -->
 <g id="node7" class="node">
@@ -213,7 +213,7 @@
 </g>
 </g>
 <!-- close_rec_notin&#45;&gt;subst_notin -->
-<g id="edge75" class="edge">
+<g id="edge82" class="edge">
 <title>close_rec_notin&#45;&gt;subst_notin</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M113.41,-18C139.18,-18 170.86,-18 195.93,-18"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="196.16,-20.1 202.16,-18 196.16,-15.9 196.16,-20.1"/>
@@ -264,16 +264,16 @@
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="373.87,-950.65 380.23,-950.43 375.12,-946.64 373.87,-950.65"/>
 </g>
 <!-- subst_lc_self -->
-<g id="node118" class="node">
+<g id="node120" class="node">
 <title>subst_lc_self</title>
-<g id="a_node118"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node120"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="578.76" cy="-1011" rx="40.33" ry="18"/>
 <text text-anchor="middle" x="578.76" y="-1008.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lc_self</text>
 </a>
 </g>
 </g>
 <!-- subst_fvar_self&#45;&gt;subst_lc_self -->
-<g id="edge77" class="edge">
+<g id="edge84" class="edge">
 <title>subst_fvar_self&#45;&gt;subst_lc_self</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M454.76,-1011C478.68,-1011 508.01,-1011 532.03,-1011"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="532.33,-1013.1 538.33,-1011 532.33,-1008.9 532.33,-1013.1"/>
@@ -288,16 +288,16 @@
 </g>
 </g>
 <!-- subst_other -->
-<g id="node119" class="node">
+<g id="node121" class="node">
 <title>subst_other</title>
-<g id="a_node119"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node121"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1115" rx="37.56" ry="18"/>
 <text text-anchor="middle" x="239.88" y="-1112.8" font-family="Inter" font-size="9.00" fill="#222222">subst_other</text>
 </a>
 </g>
 </g>
 <!-- subst_fvar_other&#45;&gt;subst_other -->
-<g id="edge78" class="edge">
+<g id="edge85" class="edge">
 <title>subst_fvar_other&#45;&gt;subst_other</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M115.25,-1115C140.7,-1115 171.52,-1115 196.02,-1115"/>
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@@ -330,16 +330,16 @@
 </g>
 </g>
 <!-- star_red_sub_full_comp -->
-<g id="node133" class="node">
+<g id="node135" class="node">
 <title>star_red_sub_full_comp</title>
-<g id="a_node133"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node135"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-223" rx="65.64" ry="18"/>
 <text text-anchor="middle" x="755.12" y="-220.8" font-family="Inter" font-size="9.00" fill="#222222">star_red_sub_full_comp</text>
 </a>
 </g>
 </g>
 <!-- Plus_to_Star&#45;&gt;star_red_sub_full_comp -->
-<g id="edge93" class="edge">
+<g id="edge100" class="edge">
 <title>Plus_to_Star&#45;&gt;star_red_sub_full_comp</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M616.31,-243.55C637.94,-240.32 665.94,-236.15 691.05,-232.41"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="691.4,-234.48 697.02,-231.51 690.78,-230.32 691.4,-234.48"/>
@@ -693,7 +693,7 @@
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 </g>
 <!-- occurs_count_zero&#45;&gt;full_comp_aux_sub -->
-<g id="edge84" class="edge">
+<g id="edge91" class="edge">
 <title>occurs_count_zero&#45;&gt;full_comp_aux_sub</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M277.83,-135.06C287.29,-138.84 297.31,-143.25 306.27,-148 333.16,-162.27 361.65,-182.1 381.57,-196.85"/>
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@@ -726,22 +726,22 @@
 <g id="node50" class="node">
 <title>occurs_lift_self</title>
 <g id="a_node50"><a xlink:title="occurs_lift_self theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-329" rx="46.38" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-326.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_lift_self</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-318" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-315.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_lift_self</text>
 </a>
 </g>
 </g>
 <!-- occurs_lift_self&#45;&gt;occurs_count_zfill_plug -->
 <g id="edge22" class="edge">
 <title>occurs_lift_self&#45;&gt;occurs_count_zfill_plug</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M98.31,-341.82C125.82,-352.81 165.71,-368.76 195.97,-380.85"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="195.54,-382.94 201.89,-383.22 197.1,-379.04 195.54,-382.94"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M95.8,-331.84C117.33,-341.91 147.19,-355.84 173.48,-368 181.86,-371.88 190.86,-376.02 199.41,-379.94"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="198.92,-382.03 205.25,-382.62 200.68,-378.21 198.92,-382.03"/>
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 <!-- occurs_lift_self&#45;&gt;open_rec_zplug_lift -->
 <g id="edge23" class="edge">
 <title>occurs_lift_self&#45;&gt;open_rec_zplug_lift</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M98.14,-316.18C119.46,-307.62 148.15,-296.12 173.48,-286 181.26,-282.9 189.56,-279.59 197.54,-276.41"/>
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@@ -759,7 +759,7 @@
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 </g>
 <!-- zdecs_nonempty&#45;&gt;full_comp_aux_sub -->
-<g id="edge90" class="edge">
+<g id="edge97" class="edge">
 <title>zdecs_nonempty&#45;&gt;full_comp_aux_sub</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M276.81,-299.35C286.56,-295.47 296.99,-290.92 306.27,-286 333.16,-271.73 361.65,-251.9 381.57,-237.15"/>
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@@ -771,7 +771,7 @@
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 <!-- occurs_count_zfill_plug&#45;&gt;full_comp_aux_sub -->
-<g id="edge85" class="edge">
+<g id="edge92" class="edge">
 <title>occurs_count_zfill_plug&#45;&gt;full_comp_aux_sub</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M260.97,-380.79C274.24,-369.13 291.79,-353.14 306.27,-338 337.71,-305.12 371.03,-264.03 390.57,-239.18"/>
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@@ -783,7 +783,7 @@
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 <!-- open_rec_zplug_lift&#45;&gt;full_comp_aux_sub -->
-<g id="edge87" class="edge">
+<g id="edge94" class="edge">
 <title>open_rec_zplug_lift&#45;&gt;full_comp_aux_sub</title>
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+<g id="a_node143"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
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+<g id="edge108" class="edge">
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+<text text-anchor="middle" x="65.24" y="-2724.8" font-family="Inter" font-size="9.00" fill="#222222">in_filter_neq</text>
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+<text text-anchor="middle" x="239.88" y="-2672.8" font-family="Inter" font-size="9.00" fill="#222222">nplug_fv_mono</text>
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-<text text-anchor="middle" x="408.74" y="-2750.8" font-family="Inter" font-size="9.00" fill="#222222">nred_core_fv</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-2697" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-2694.8" font-family="Inter" font-size="9.00" fill="#222222">nred_core_fv</text>
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+<!-- nplug_esub_fv -->
+<g id="node77" class="node">
+<title>nplug_esub_fv</title>
+<g id="a_node77"><a xlink:title="nplug_esub_fv theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2813" rx="43.76" ry="18"/>
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+</a>
+</g>
+</g>
+<!-- in_filter_neq&#45;&gt;nplug_esub_fv -->
+<g id="edge48" class="edge">
+<title>in_filter_neq&#45;&gt;nplug_esub_fv</title>
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+<!-- nred_fv -->
+<g id="node78" class="node">
+<title>nred_fv</title>
+<g id="a_node78"><a xlink:title="nred_fv theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
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+<text text-anchor="middle" x="408.74" y="-2767.8" font-family="Inter" font-size="9.00" fill="#222222">nred_fv</text>
+</a>
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+</g>
+<!-- in_filter_neq&#45;&gt;nred_fv -->
+<g id="edge50" class="edge">
+<title>in_filter_neq&#45;&gt;nred_fv</title>
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+<text text-anchor="middle" x="65.24" y="-2810.8" font-family="Inter" font-size="9.00" fill="#222222">filter_neq_in</text>
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+<!-- filter_neq_in&#45;&gt;nplug_esub_fv -->
+<g id="edge47" class="edge">
+<title>filter_neq_in&#45;&gt;nplug_esub_fv</title>
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+<!-- filter_neq_in&#45;&gt;nred_fv -->
+<g id="edge49" class="edge">
+<title>filter_neq_in&#45;&gt;nred_fv</title>
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+<!-- nplug_fv_mono&#45;&gt;nred_fv -->
+<g id="edge52" class="edge">
+<title>nplug_fv_mono&#45;&gt;nred_fv</title>
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+<!-- nplug_fv_upper&#45;&gt;nred_fv -->
+<g id="edge53" class="edge">
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+<!-- nplug_esub_fv&#45;&gt;nred_fv -->
+<g id="edge51" class="edge">
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 <title>par_red1_iff&#45;&gt;red1_or_eq_to_par</title>
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 <title>red1_in_par</title>
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 </a>
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 <title>par_context</title>
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 </a>
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 </a>
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 </a>
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 <title>check_term_true</title>
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 <title>check_all_true</title>
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 </a>
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 <!-- check_term_true&#45;&gt;check_all_true -->
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 <title>check_term_true&#45;&gt;check_all_true</title>
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 <title>gen_sound</title>
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 </a>
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 </g>
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+<g id="node89" class="node">
 <title>gen_complete</title>
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 </a>
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 <title>zdecs_sound</title>
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+<g id="edge98" class="edge">
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 </a>
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+<g id="node95" class="node">
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 </a>
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 <title>steps_sound</title>
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 </a>
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 <title>positions_at&#45;&gt;steps_sound</title>
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+<g id="edge72" class="edge">
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 <title>steps_sound_red1</title>
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 </a>
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+<g id="edge67" class="edge">
 <title>steps_sound&#45;&gt;steps_sound_red1</title>
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 <title>corpus_sound</title>
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-<g id="edge98" class="edge">
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 <title>steps_sound_red1&#45;&gt;check_term_true</title>
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+<g id="a_node114"><a xlink:title="close_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3427" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3424.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_lam</text>
 </a>
 </g>
 </g>
 <!-- close_rec_esub -->
-<g id="node113" class="node">
+<g id="node115" class="node">
 <title>close_rec_esub</title>
-<g id="a_node113"><a xlink:title="close_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3411" rx="46.53" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3408.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
+<g id="a_node115"><a xlink:title="close_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3479" rx="46.53" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3476.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
 </a>
 </g>
 </g>
 <!-- subst_app -->
-<g id="node114" class="node">
+<g id="node116" class="node">
 <title>subst_app</title>
-<g id="a_node114"><a xlink:title="subst_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3463" rx="34.13" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3460.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
+<g id="a_node116"><a xlink:title="subst_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3531" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3528.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
 </a>
 </g>
 </g>
 <!-- subst_lam -->
-<g id="node115" class="node">
+<g id="node117" class="node">
 <title>subst_lam</title>
-<g id="a_node115"><a xlink:title="subst_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3515" rx="34.13" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3512.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3583" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3580.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
 </a>
 </g>
 </g>
 <!-- subst_esub -->
-<g id="node116" class="node">
+<g id="node118" class="node">
 <title>subst_esub</title>
-<g id="a_node116"><a xlink:title="subst_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3567" rx="36.25" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3564.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
+<g id="a_node118"><a xlink:title="subst_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3635" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3632.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
 </a>
 </g>
 </g>
 <!-- open_rec_lc_atom -->
-<g id="node120" class="node">
+<g id="node122" class="node">
 <title>open_rec_lc_atom</title>
-<g id="a_node120"><a xlink:title="open_rec_lc_atom theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3619" rx="53.89" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3616.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
+<g id="a_node122"><a xlink:title="open_rec_lc_atom theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3687" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3684.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
 </a>
 </g>
 </g>
 <!-- open_rec_lc -->
-<g id="node121" class="node">
+<g id="node123" class="node">
 <title>open_rec_lc</title>
-<g id="a_node121"><a xlink:title="open_rec_lc theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3619" rx="39.02" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-3616.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
+<g id="a_node123"><a xlink:title="open_rec_lc theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3687" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-3684.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
 </a>
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 <!-- open_rec_lc_atom&#45;&gt;open_rec_lc -->
-<g id="edge79" class="edge">
+<g id="edge86" class="edge">
 <title>open_rec_lc_atom&#45;&gt;open_rec_lc</title>
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 <!-- red_sub_root_to_red1 -->
-<g id="node122" class="node">
+<g id="node124" class="node">
 <title>red_sub_root_to_red1</title>
-<g id="a_node122"><a xlink:title="red_sub_root_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3671" rx="60.09" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3668.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_root_to_red1</text>
+<g id="a_node124"><a xlink:title="red_sub_root_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3739" rx="60.09" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3736.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_root_to_red1</text>
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 <!-- red_sub_to_red1 -->
-<g id="node123" class="node">
+<g id="node125" class="node">
 <title>red_sub_to_red1</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3671" rx="47.84" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-3668.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_to_red1</text>
+<g id="a_node125"><a xlink:title="red_sub_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3739" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-3736.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_to_red1</text>
 </a>
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 </g>
 <!-- red_sub_root_to_red1&#45;&gt;red_sub_to_red1 -->
-<g id="edge80" class="edge">
+<g id="edge87" class="edge">
 <title>red_sub_root_to_red1&#45;&gt;red_sub_to_red1</title>
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 <!-- red_sub_context -->
-<g id="node124" class="node">
+<g id="node126" class="node">
 <title>red_sub_context</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3723" rx="49.15" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3720.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_context</text>
+<g id="a_node126"><a xlink:title="red_sub_context theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3791" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3788.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_context</text>
 </a>
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 <!-- Star_red_sub_context -->
-<g id="node130" class="node">
+<g id="node132" class="node">
 <title>Star_red_sub_context</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3723" rx="61.55" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-3720.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red_sub_context</text>
+<g id="a_node132"><a xlink:title="Star_red_sub_context theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3791" rx="61.55" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-3788.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red_sub_context</text>
 </a>
 </g>
 </g>
 <!-- red_sub_context&#45;&gt;Star_red_sub_context -->
-<g id="edge83" class="edge">
+<g id="edge90" class="edge">
 <title>red_sub_context&#45;&gt;Star_red_sub_context</title>
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 <!-- red_sub_gc -->
-<g id="node125" class="node">
+<g id="node127" class="node">
 <title>red_sub_gc</title>
-<g id="a_node125"><a xlink:title="red_sub_gc theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node127"><a xlink:title="red_sub_gc theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-70" rx="36.25" ry="18"/>
 <text text-anchor="middle" x="239.88" y="-67.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_gc</text>
 </a>
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 <!-- red_sub_gc&#45;&gt;full_comp_aux_sub -->
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+<g id="edge95" class="edge">
 <title>red_sub_gc&#45;&gt;full_comp_aux_sub</title>
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 <!-- red_gc_to_red_sub -->
-<g id="node137" class="node">
+<g id="node139" class="node">
 <title>red_gc_to_red_sub</title>
-<g id="a_node137"><a xlink:title="red_gc_to_red_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node139"><a xlink:title="red_gc_to_red_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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 <!-- red_sub_gc&#45;&gt;red_gc_to_red_sub -->
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+<g id="edge104" class="edge">
 <title>red_sub_gc&#45;&gt;red_gc_to_red_sub</title>
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 <!-- red_sub_r -->
-<g id="node126" class="node">
+<g id="node128" class="node">
 <title>red_sub_r</title>
-<g id="a_node126"><a xlink:title="red_sub_r theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node128"><a xlink:title="red_sub_r theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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 <text text-anchor="middle" x="239.88" y="-171.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_r</text>
 </a>
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 </g>
 <!-- red_sub_r&#45;&gt;full_comp_aux_sub -->
-<g id="edge89" class="edge">
+<g id="edge96" class="edge">
 <title>red_sub_r&#45;&gt;full_comp_aux_sub</title>
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 <!-- occurs_zfill -->
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+<g id="node129" class="node">
 <title>occurs_zfill</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3775" rx="37.56" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-3772.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_zfill</text>
+<g id="a_node129"><a xlink:title="occurs_zfill theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3843" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3840.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_zfill</text>
 </a>
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 <!-- zfill_occurs0 -->
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+<g id="node130" class="node">
 <title>zfill_occurs0</title>
-<g id="a_node128"><a xlink:title="zfill_occurs0 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3775" rx="40.98" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-3772.8" font-family="Inter" font-size="9.00" fill="#222222">zfill_occurs0</text>
+<g id="a_node130"><a xlink:title="zfill_occurs0 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3843" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-3840.8" font-family="Inter" font-size="9.00" fill="#222222">zfill_occurs0</text>
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 <!-- occurs_zfill&#45;&gt;zfill_occurs0 -->
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+<g id="edge88" class="edge">
 <title>occurs_zfill&#45;&gt;zfill_occurs0</title>
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 <!-- R_Gc_disjoint -->
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+<g id="node131" class="node">
 <title>R_Gc_disjoint</title>
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-<text text-anchor="middle" x="408.74" y="-3772.8" font-family="Inter" font-size="9.00" fill="#222222">R_Gc_disjoint</text>
+<g id="a_node131"><a xlink:title="R_Gc_disjoint theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-3843" rx="41.64" ry="18"/>
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 <!-- zfill_occurs0&#45;&gt;R_Gc_disjoint -->
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+<g id="edge89" class="edge">
 <title>zfill_occurs0&#45;&gt;R_Gc_disjoint</title>
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 <!-- plus_red_sub_full_comp -->
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 <title>plus_red_sub_full_comp</title>
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 <title>full_comp_aux_sub&#45;&gt;plus_red_sub_full_comp</title>
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 <!-- es_count_plug_mono -->
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diff --git a/graphs/index.txt b/graphs/index.txt
index 1bdbd77..fc0b942 100644
--- a/graphs/index.txt
+++ b/graphs/index.txt
@@ -1,8 +1,12 @@
 status	milestone	kind	name	file
+proved	M0	infra	Es_ctx_any	theory/NamedMeta.v
 proved	M0	infra	Es_refl_any	theory/NamedMeta.v
 proved	M0	infra	Es_sym_any	theory/NamedMeta.v
 proved	M0	infra	Es_trans_any	theory/NamedMeta.v
+proved	M0	infra	Plus_mred_of_Plus_red1	theory/Metaterm.v
 proved	M0	infra	eqC_in_Es	theory/NamedMeta.v
+proved	M0	infra	filter_neq_in	theory/NamedMeta.v
+proved	M0	infra	in_filter_neq	theory/NamedMeta.v
 proved	M0	infra	isubst_meta_in	theory/NamedMeta.v
 proved	M0	infra	isubst_meta_notin	theory/NamedMeta.v
 proved	M0	infra	mfvs_close_m	theory/Metaterm.v
@@ -10,10 +14,18 @@ proved	M0	infra	mfvs_lift_m	theory/Metaterm.v
 proved	M0	infra	mfvs_open_m	theory/Metaterm.v
 proved	M0	infra	mred_B_intro	theory/Metaterm.v
 proved	M0	infra	mred_context	theory/Metaterm.v
+proved	M0	infra	mred_full_comp_pure	theory/Metaterm.v
 proved	M0	infra	mred_gc_intro	theory/Metaterm.v
 proved	M0	infra	mred_of_red1	theory/Metaterm.v
+proved	M0	infra	mred_plus_one	theory/Metaterm.v
 proved	M0	infra	mred_r_intro	theory/Metaterm.v
 proved	M0	infra	mred_term_context	theory/Metaterm.v
+proved	M0	infra	nplug_esub_fv	theory/NamedMeta.v
+proved	M0	infra	nplug_fv_mono	theory/NamedMeta.v
+proved	M0	infra	nplug_fv_upper	theory/NamedMeta.v
+proved	M0	infra	nred_R_intro	theory/NamedMeta.v
+proved	M0	infra	nred_core_fv	theory/NamedMeta.v
+proved	M0	infra	nred_fv	theory/NamedMeta.v
 proved	M0	infra	of_trm_close	theory/Metaterm.v
 proved	M0	infra	of_trm_fvs	theory/Metaterm.v
 proved	M0	infra	of_trm_injective	theory/Metaterm.v
diff --git a/graphs/theorems.json b/graphs/theorems.json
index 7536fca..955bfce 100644
--- a/graphs/theorems.json
+++ b/graphs/theorems.json
@@ -850,6 +850,35 @@
         "nplug_fv_upper"
       ]
     },
+    {
+      "id": "nplug_esub_fv",
+      "label": "nplug_esub_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq"
+      ]
+    },
+    {
+      "id": "nred_fv",
+      "label": "nred_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq",
+        "nplug_esub_fv",
+        "nplug_fv_mono",
+        "nplug_fv_upper"
+      ]
+    },
     {
       "id": "par_red1_iff",
       "label": "par_red1_iff",
diff --git a/theory/NamedMeta.v b/theory/NamedMeta.v
index c9ada7e..7461724 100644
--- a/theory/NamedMeta.v
+++ b/theory/NamedMeta.v
@@ -240,3 +240,63 @@ Qed.
 Print Assumptions nplug_fv_mono.
 Print Assumptions nplug_fv_upper.
 Print Assumptions nred_core_fv.
+
+Lemma nplug_esub_fv : forall C X d x u y,
+  y <> x ->
+  In y (nfv (nplug C (NESub (NMVar X d) x u))) ->
+  In y (nfv (nplug C (NMVar X d))) \/ In y (nfv u).
+Proof.
+  induction C; intros X d x u y Hne Hy; simpl in *.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + left. apply in_filter_neq in Hy as [Hy _]. simpl. exact Hy.
+    + right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. left. exact Hy.
+      * right. exact Hy.
+    + left. apply in_app_iff. right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + left. apply in_app_iff. left. exact Hy.
+    + apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. right. exact Hy.
+      * right. exact Hy.
+  - apply in_filter_neq in Hy as [Hy Hz].
+    apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+    + left. apply filter_neq_in; [exact Hy | exact Hz].
+    + right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hz].
+      apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hz].
+      * right. exact Hy.
+    + left. apply in_app_iff. right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + left. apply in_app_iff. left. exact Hy.
+    + apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. right. exact Hy.
+      * right. exact Hy.
+Qed.
+
+Lemma nred_fv : forall t t', nred t t' -> incl (nfv t') (nfv t).
+Proof.
+  intros t t' H. induction H; simpl; unfold incl in *.
+  - intros y Hy. exact Hy.
+  - intros y Hy. apply in_app_iff. left.
+    apply filter_neq_in; [exact Hy | intro E; apply H; subst; exact Hy].
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (nplug_esub_fv C X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (nplug_fv_upper C u (NVar x)) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - apply nplug_fv_mono. exact IHnred.
+Qed.
+
+Print Assumptions nplug_esub_fv.
+Print Assumptions nred_fv.