diff --git a/graphs/dependency.dot b/graphs/dependency.dot
index f75a20f..80a62a0 100644
--- a/graphs/dependency.dot
+++ b/graphs/dependency.dot
@@ -78,6 +78,11 @@ digraph theorem_deps {
   Es_refl_any [label="Es_refl_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_refl_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Es_sym_any [label="Es_sym_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_sym_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Es_trans_any [label="Es_trans_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_trans_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  in_filter_neq [label="in_filter_neq", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="in_filter_neq theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  filter_neq_in [label="filter_neq_in", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="filter_neq_in theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  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];
   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];
@@ -181,6 +186,14 @@ digraph theorem_deps {
   red1r_fv -> lemma_2_1_fv_preserved;
   Plus_red1_context -> lemma_2_3_beta_sim;
   lemma_2_2_full_comp -> lemma_2_3_beta_sim;
+  filter_neq_in -> nplug_fv_mono;
+  in_filter_neq -> nplug_fv_mono;
+  filter_neq_in -> nplug_fv_upper;
+  in_filter_neq -> nplug_fv_upper;
+  filter_neq_in -> nred_core_fv;
+  in_filter_neq -> nred_core_fv;
+  nplug_fv_mono -> nred_core_fv;
+  nplug_fv_upper -> nred_core_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 e52f959..f66b2a0 100644
Binary files a/graphs/dependency.png and b/graphs/dependency.png differ
diff --git a/graphs/dependency.svg b/graphs/dependency.svg
index aa208d0..bee9854 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 3718.00" 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 3714)">
+<svg width="1296pt" height="3882pt"
+ 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)">
 <title>theorem_deps</title>
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-3714 1292.27,-3714 1292.27,4 -4,4"/>
-<text text-anchor="start" x="407.63" y="-3697.4" 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="-3687.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
-<text text-anchor="start" x="601.13" y="-3687.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="608.13" y="-3687.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
-<text text-anchor="start" x="635.13" y="-3687.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="642.13" y="-3687.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
-<text text-anchor="start" x="676.13" y="-3687.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="683.13" y="-3687.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
+<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>
 <!-- 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="-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>
+<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>
 </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="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"/>
+<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"/>
 </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="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"/>
+<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"/>
 </g>
 <!-- subst_notin -->
-<g id="node112" class="node">
+<g id="node117" class="node">
 <title>subst_notin</title>
-<g id="a_node112"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node117"><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="edge68" class="edge">
+<g id="edge76" class="edge">
 <title>open_rec_occurs_false&#45;&gt;subst_notin</title>
-<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"/>
+<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"/>
 </g>
 <!-- full_comp_aux_sub -->
-<g id="node126" class="node">
+<g id="node131" class="node">
 <title>full_comp_aux_sub</title>
-<g id="a_node126"><a xlink:title="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node131"><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="edge78" class="edge">
+<g id="edge86" class="edge">
 <title>open_rec_occurs_false&#45;&gt;full_comp_aux_sub</title>
-<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"/>
+<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"/>
 </g>
 <!-- close_rec_notin -->
 <g id="node7" class="node">
@@ -213,7 +213,7 @@
 </g>
 </g>
 <!-- close_rec_notin&#45;&gt;subst_notin -->
-<g id="edge67" class="edge">
+<g id="edge75" 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="node113" class="node">
+<g id="node118" class="node">
 <title>subst_lc_self</title>
-<g id="a_node113"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node118"><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="edge69" class="edge">
+<g id="edge77" 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="node114" class="node">
+<g id="node119" class="node">
 <title>subst_other</title>
-<g id="a_node114"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node119"><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="edge70" class="edge">
+<g id="edge78" 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="node128" class="node">
+<g id="node133" class="node">
 <title>star_red_sub_full_comp</title>
-<g id="a_node128"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node133"><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="edge85" class="edge">
+<g id="edge93" 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="edge76" class="edge">
+<g id="edge84" 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="-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>
+<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>
 </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="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"/>
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+<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"/>
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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="M100.76,-306.4C127.7,-297.35 165.31,-284.71 194.49,-274.91"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="195.19,-276.89 200.2,-272.99 193.85,-272.91 195.19,-276.89"/>
+<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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 <!-- zdecs_nonempty&#45;&gt;full_comp_aux_sub -->
-<g id="edge82" class="edge">
+<g id="edge90" 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="edge77" class="edge">
+<g id="edge85" class="edge">
 <title>occurs_count_zfill_plug&#45;&gt;full_comp_aux_sub</title>
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+<g id="edge87" class="edge">
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-<g id="node83" class="node">
+<g id="node88" class="node">
 <title>gen_fv_preserved</title>
-<g id="a_node83"><a xlink:title="gen_fv_preserved theory/Random.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node88"><a xlink:title="gen_fv_preserved theory/Random.v https://arxiv.org/html/2312.13270v1">
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-<g id="edge46" class="edge">
+<g id="edge54" class="edge">
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-<g id="node136" class="node">
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 <title>corpus_fv_preserved</title>
-<g id="a_node136"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node141"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
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+<g id="edge101" class="edge">
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+<!-- in_filter_neq -->
 <g id="node72" class="node">
+<title>in_filter_neq</title>
+<g id="a_node72"><a xlink:title="in_filter_neq theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2779" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2776.8" font-family="Inter" font-size="9.00" fill="#222222">in_filter_neq</text>
+</a>
+</g>
+</g>
+<!-- nplug_fv_mono -->
+<g id="node74" class="node">
+<title>nplug_fv_mono</title>
+<g id="a_node74"><a xlink:title="nplug_fv_mono theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2779" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2776.8" font-family="Inter" font-size="9.00" fill="#222222">nplug_fv_mono</text>
+</a>
+</g>
+</g>
+<!-- in_filter_neq&#45;&gt;nplug_fv_mono -->
+<g id="edge40" class="edge">
+<title>in_filter_neq&#45;&gt;nplug_fv_mono</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M104.55,-2779C128.92,-2779 160.78,-2779 187.35,-2779"/>
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+</g>
+<!-- nplug_fv_upper -->
+<g id="node75" class="node">
+<title>nplug_fv_upper</title>
+<g id="a_node75"><a xlink:title="nplug_fv_upper theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2727" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2724.8" font-family="Inter" font-size="9.00" fill="#222222">nplug_fv_upper</text>
+</a>
+</g>
+</g>
+<!-- in_filter_neq&#45;&gt;nplug_fv_upper -->
+<g id="edge42" class="edge">
+<title>in_filter_neq&#45;&gt;nplug_fv_upper</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M98.71,-2769.22C126.46,-2760.86 166.56,-2748.78 196.77,-2739.68"/>
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+</g>
+<!-- nred_core_fv -->
+<g id="node76" class="node">
+<title>nred_core_fv</title>
+<g id="a_node76"><a xlink:title="nred_core_fv theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-2753" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-2750.8" font-family="Inter" font-size="9.00" fill="#222222">nred_core_fv</text>
+</a>
+</g>
+</g>
+<!-- in_filter_neq&#45;&gt;nred_core_fv -->
+<g id="edge44" class="edge">
+<title>in_filter_neq&#45;&gt;nred_core_fv</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M97.75,-2789.19C144.06,-2802.65 232.93,-2822.7 306.27,-2805 333.3,-2798.48 361.21,-2783.46 380.9,-2771.23"/>
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+<!-- filter_neq_in -->
+<g id="node73" class="node">
+<title>filter_neq_in</title>
+<g id="a_node73"><a xlink:title="filter_neq_in theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2727" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2724.8" font-family="Inter" font-size="9.00" fill="#222222">filter_neq_in</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in&#45;&gt;nplug_fv_mono -->
+<g id="edge39" class="edge">
+<title>filter_neq_in&#45;&gt;nplug_fv_mono</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M98.71,-2736.78C126.46,-2745.14 166.56,-2757.22 196.77,-2766.32"/>
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+</g>
+<!-- filter_neq_in&#45;&gt;nplug_fv_upper -->
+<g id="edge41" class="edge">
+<title>filter_neq_in&#45;&gt;nplug_fv_upper</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M104.55,-2727C128.92,-2727 160.78,-2727 187.35,-2727"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="187.63,-2729.1 193.63,-2727 187.63,-2724.9 187.63,-2729.1"/>
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+<!-- filter_neq_in&#45;&gt;nred_core_fv -->
+<g id="edge43" class="edge">
+<title>filter_neq_in&#45;&gt;nred_core_fv</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M97.75,-2716.81C144.06,-2703.35 232.93,-2683.3 306.27,-2701 333.3,-2707.52 361.21,-2722.54 380.9,-2734.77"/>
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+<!-- nplug_fv_mono&#45;&gt;nred_core_fv -->
+<g id="edge45" class="edge">
+<title>nplug_fv_mono&#45;&gt;nred_core_fv</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M282.98,-2772.44C307.81,-2768.57 339.18,-2763.68 364.25,-2759.78"/>
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+<!-- nplug_fv_upper&#45;&gt;nred_core_fv -->
+<g id="edge46" class="edge">
+<title>nplug_fv_upper&#45;&gt;nred_core_fv</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M282.98,-2733.56C307.81,-2737.43 339.18,-2742.32 364.25,-2746.22"/>
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+<!-- par_red1_iff -->
+<g id="node77" class="node">
 <title>par_red1_iff</title>
-<g id="a_node72"><a xlink:title="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2727" rx="37.56" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2724.8" font-family="Inter" font-size="9.00" fill="#222222">par_red1_iff</text>
+<g id="a_node77"><a xlink:title="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2891" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2888.8" font-family="Inter" font-size="9.00" fill="#222222">par_red1_iff</text>
 </a>
 </g>
 </g>
 <!-- par_to_red1_or_eq -->
-<g id="node75" class="node">
+<g id="node80" class="node">
 <title>par_to_red1_or_eq</title>
-<g id="a_node75"><a xlink:title="par_to_red1_or_eq theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2753" rx="53.23" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-2750.8" font-family="Inter" font-size="9.00" fill="#222222">par_to_red1_or_eq</text>
+<g id="a_node80"><a xlink:title="par_to_red1_or_eq theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2917" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2914.8" font-family="Inter" font-size="9.00" fill="#222222">par_to_red1_or_eq</text>
 </a>
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 </g>
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-<g id="edge39" class="edge">
+<g id="edge47" class="edge">
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+<g id="node81" class="node">
 <title>red1_or_eq_to_par</title>
-<g id="a_node76"><a xlink:title="red1_or_eq_to_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="239.88" y="-2698.8" font-family="Inter" font-size="9.00" fill="#222222">red1_or_eq_to_par</text>
+<g id="a_node81"><a xlink:title="red1_or_eq_to_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2865" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2862.8" font-family="Inter" font-size="9.00" fill="#222222">red1_or_eq_to_par</text>
 </a>
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+<g id="edge48" class="edge">
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+<g id="node78" class="node">
 <title>red1_in_par</title>
-<g id="a_node73"><a xlink:title="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2779" rx="37.56" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2776.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
+<g id="a_node78"><a xlink:title="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2943" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2940.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
 </a>
 </g>
 </g>
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-<g id="node74" class="node">
+<g id="node79" class="node">
 <title>par_context</title>
-<g id="a_node74"><a xlink:title="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2831" rx="38.87" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2828.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
+<g id="a_node79"><a xlink:title="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2995" rx="38.87" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2992.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
 </a>
 </g>
 </g>
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+<g id="node82" class="node">
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2883" rx="41.64" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2880.8" font-family="Inter" font-size="9.00" fill="#222222">par_reflexive</text>
+<g id="a_node82"><a xlink:title="par_reflexive theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3047" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3044.8" font-family="Inter" font-size="9.00" fill="#222222">par_reflexive</text>
 </a>
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-<g id="node78" class="node">
+<g id="node83" class="node">
 <title>gen_list_length</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2935" rx="46.38" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2932.8" font-family="Inter" font-size="9.00" fill="#222222">gen_list_length</text>
+<g id="a_node83"><a xlink:title="gen_list_length theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3099" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3096.8" font-family="Inter" font-size="9.00" fill="#222222">gen_list_length</text>
 </a>
 </g>
 </g>
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+<g id="node84" class="node">
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+<g id="a_node84"><a xlink:title="check_term_true theory/Random.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="923.99" cy="-604" rx="50.61" ry="18"/>
 <text text-anchor="middle" x="923.99" y="-601.8" font-family="Inter" font-size="9.00" fill="#222222">check_term_true</text>
 </a>
 </g>
 </g>
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-<g id="node80" class="node">
+<g id="node85" class="node">
 <title>check_all_true</title>
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+<g id="a_node85"><a xlink:title="check_all_true theory/Random.v https://arxiv.org/html/2312.13270v1">
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 <text text-anchor="middle" x="1082.46" y="-601.8" font-family="Inter" font-size="9.00" fill="#222222">check_all_true</text>
 </a>
 </g>
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-<g id="edge43" class="edge">
+<g id="edge51" class="edge">
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-<g id="node81" class="node">
+<g id="node86" class="node">
 <title>gen_sound</title>
-<g id="a_node81"><a xlink:title="gen_sound theory/Random.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node86"><a xlink:title="gen_sound theory/Random.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-457" rx="35.59" ry="18"/>
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 </a>
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 <title>normal_form_iff_nf</title>
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 <title>nf_dec</title>
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 <title>open_rec_app</title>
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-<text text-anchor="middle" x="65.24" y="-2984.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_app</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3151" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3148.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_app</text>
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 <title>open_rec_lam</title>
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-<text text-anchor="middle" x="65.24" y="-3036.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lam</text>
+<g id="a_node109"><a xlink:title="open_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3203" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3200.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lam</text>
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 <title>open_rec_esub</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3255" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3252.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_esub</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3359" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3356.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_lam</text>
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 <!-- close_rec_esub -->
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-<text text-anchor="middle" x="65.24" y="-3244.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
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+<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>
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-<text text-anchor="middle" x="65.24" y="-3296.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
+<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>
 </a>
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 <title>subst_lam</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3515" rx="34.13" ry="18"/>
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 <title>subst_esub</title>
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-<text text-anchor="middle" x="65.24" y="-3400.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
+<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>
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 <title>open_rec_lc_atom</title>
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-<text text-anchor="middle" x="65.24" y="-3452.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
+<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>
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 <title>open_rec_lc</title>
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-<text text-anchor="middle" x="239.88" y="-3452.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
+<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>
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+<g id="node122" class="node">
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-<text text-anchor="middle" x="65.24" y="-3504.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_root_to_red1</text>
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+<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>
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+<g id="node128" class="node">
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 <!-- occurs_zfill&#45;&gt;zfill_occurs0 -->
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+<g id="edge81" class="edge">
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 <!-- R_Gc_disjoint -->
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+<g id="node129" class="node">
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 <!-- zfill_occurs0&#45;&gt;R_Gc_disjoint -->
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+<g id="edge82" class="edge">
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 </a>
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 <!-- red_gc_measure -->
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+<g id="node135" class="node">
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 </g>
 <!-- es_count_plug_mono&#45;&gt;red_gc_measure -->
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 <!-- red_gc_terminates -->
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diff --git a/graphs/theorems.json b/graphs/theorems.json
index 672b9e8..7536fca 100644
--- a/graphs/theorems.json
+++ b/graphs/theorems.json
@@ -789,6 +789,67 @@
       "file": "theory/NamedMeta.v",
       "depends": []
     },
+    {
+      "id": "in_filter_neq",
+      "label": "in_filter_neq",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "filter_neq_in",
+      "label": "filter_neq_in",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "nplug_fv_mono",
+      "label": "nplug_fv_mono",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq"
+      ]
+    },
+    {
+      "id": "nplug_fv_upper",
+      "label": "nplug_fv_upper",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq"
+      ]
+    },
+    {
+      "id": "nred_core_fv",
+      "label": "nred_core_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq",
+        "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 a267974..c9ada7e 100644
--- a/theory/NamedMeta.v
+++ b/theory/NamedMeta.v
@@ -139,3 +139,104 @@ Print Assumptions eqC_in_Es.
 Print Assumptions Es_refl_any.
 Print Assumptions Es_sym_any.
 Print Assumptions Es_trans_any.
+
+Lemma in_filter_neq : forall y x l,
+  In y (filter (fun z => negb (Nat.eqb z x)) l) -> In y l /\ y <> x.
+Proof.
+  intros y x l H. apply filter_In in H. destruct H as [Hl Hp]. split; [exact Hl |].
+  apply Nat.eqb_neq. destruct (Nat.eqb y x) eqn:E; [| reflexivity].
+  simpl in Hp. discriminate Hp.
+Qed.
+
+Lemma filter_neq_in : forall y x l,
+  In y l -> y <> x -> In y (filter (fun z => negb (Nat.eqb z x)) l).
+Proof.
+  intros y x l Hl Hne. apply filter_In. split; [exact Hl |].
+  apply Nat.eqb_neq in Hne. rewrite Hne. reflexivity.
+Qed.
+
+
+Lemma nplug_fv_mono : forall C t s,
+  incl (nfv t) (nfv s) -> incl (nfv (nplug C t)) (nfv (nplug C s)).
+Proof.
+  induction C; intros t s Hincl; unfold incl in *; simpl in *.
+  - apply Hincl.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply (IHC t s Hincl). exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. exact Hy.
+    + apply in_app_iff. right. apply (IHC t s Hincl). exact Hy.
+  - intros y Hy. apply in_filter_neq in Hy as [Hy Hne].
+    apply filter_neq_in.
+    + apply (IHC t s Hincl). exact Hy.
+    + exact Hne.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply in_filter_neq in Hy as [Hy Hne].
+      apply filter_neq_in.
+      * apply (IHC t s Hincl). exact Hy.
+      * exact Hne.
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. exact Hy.
+    + apply in_app_iff. right. apply (IHC t s Hincl). exact Hy.
+Qed.
+
+Lemma nplug_fv_upper : forall C t s,
+  incl (nfv (nplug C t)) (nfv (nplug C s) ++ nfv t).
+Proof.
+  induction C; intros t s; unfold incl in *; simpl in *.
+  - intros y Hy. apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. left. exact Hy.
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply in_app_iff. left. exact Hy.
+    + apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+      * apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_filter_neq in Hy as [Hy Hne].
+    apply (IHC t s) 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.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply in_app_iff. left. exact Hy.
+    + apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+      * apply in_app_iff. right. exact Hy.
+Qed.
+
+Inductive nred_core : ntrm -> ntrm -> Prop :=
+| ncore_B : forall t x u, nred_core (NApp (NLam x t) u) (NESub t x u)
+| ncore_Gc : forall t x u, ~ In x (nfv t) -> nred_core (NESub t x u) t
+| ncore_R : forall C x u phi,
+    In x phi -> (forall y, In y (nfv u) -> In y phi) -> cavoid C phi = true ->
+    nred_core (NESub (nplug C (NVar x)) x u) (NESub (nplug C u) x u)
+| ncore_ctx : forall C t t', nred_core t t' -> nred_core (nplug C t) (nplug C t').
+
+Lemma nred_core_fv : forall t t', nred_core 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_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_core.
+Qed.
+
+Print Assumptions nplug_fv_mono.
+Print Assumptions nplug_fv_upper.
+Print Assumptions nred_core_fv.