diff --git a/graphs/dependency.dot b/graphs/dependency.dot
index eccb285..4708428 100644
--- a/graphs/dependency.dot
+++ b/graphs/dependency.dot
@@ -40,6 +40,15 @@ digraph theorem_deps {
   mfvs_lift_m [label="mfvs_lift_m", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mfvs_lift_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   mfvs_close_m [label="mfvs_close_m", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mfvs_close_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   mfvs_open_m [label="mfvs_open_m", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mfvs_open_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_plug [label="of_trm_plug", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_plug theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_of_red1 [label="mred_of_red1", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_of_red1 theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_context [label="mred_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_context theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_term_context [label="mred_term_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_term_context theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_moccurs [label="of_trm_moccurs", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_moccurs theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_moccurs0 [label="of_trm_moccurs0", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_moccurs0 theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_B_intro [label="mred_B_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_B_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_gc_intro [label="mred_gc_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_gc_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_r_intro [label="mred_r_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_r_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Plus_red1_context [label="Plus_red1_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_red1_context theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   occurs_count_zero [label="occurs_count_zero", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_zero theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   occurs_count_false [label="occurs_count_false", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_false theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
@@ -58,6 +67,12 @@ digraph theorem_deps {
   lemma_2_3_beta_sim [label="lemma_2_3_beta_sim", color="#38761d", fillcolor="#d9ead3", style="rounded,filled", fontcolor="#222222", tooltip="lemma_2_3_beta_sim (section 2.3) theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=box];
   Plus_trans [label="Plus_trans", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Plus_step_trans [label="Plus_step_trans", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_step_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  isubst_meta_in [label="isubst_meta_in", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="isubst_meta_in theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  isubst_meta_notin [label="isubst_meta_notin", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="isubst_meta_notin theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  eqC_in_Es [label="eqC_in_Es", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="eqC_in_Es theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  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];
   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];
@@ -138,6 +153,8 @@ digraph theorem_deps {
   steps_complete -> enumerate_complete;
   of_trm_lift -> of_trm_open;
   mfvs_lift_m -> mfvs_open_m;
+  of_trm_plug -> mred_term_context;
+  of_trm_moccurs -> of_trm_moccurs0;
   occurs_count_zero -> occurs_count_pos;
   occurs_count_false -> occurs_count_zfill_plug;
   occurs_lift_self -> occurs_count_zfill_plug;
diff --git a/graphs/dependency.png b/graphs/dependency.png
index e26f6ea..eace39f 100644
Binary files a/graphs/dependency.png and b/graphs/dependency.png differ
diff --git a/graphs/dependency.svg b/graphs/dependency.svg
index 1de2e31..72ccbcc 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 -->
-<svg width="1294pt" height="2924pt"
- viewBox="0.00 0.00 1293.96 2924.00" 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 2920)">
+<svg width="1294pt" height="3600pt"
+ viewBox="0.00 0.00 1293.96 3600.00" 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 3596)">
 <title>theorem_deps</title>
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-2920 1289.96,-2920 1289.96,4 -4,4"/>
-<text text-anchor="start" x="406.48" y="-2903.4" 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="569.98" y="-2893.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
-<text text-anchor="start" x="599.98" y="-2893.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="606.98" y="-2893.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
-<text text-anchor="start" x="633.98" y="-2893.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="640.98" y="-2893.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
-<text text-anchor="start" x="674.98" y="-2893.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="681.98" y="-2893.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
+<polygon fill="white" stroke="transparent" points="-4,4 -4,-3596 1289.96,-3596 1289.96,4 -4,4"/>
+<text text-anchor="start" x="406.48" y="-3579.4" 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="569.98" y="-3569.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
+<text text-anchor="start" x="599.98" y="-3569.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="606.98" y="-3569.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
+<text text-anchor="start" x="633.98" y="-3569.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="640.98" y="-3569.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
+<text text-anchor="start" x="674.98" y="-3569.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="681.98" y="-3569.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
 <!-- fvs_lift -->
 <g id="node1" class="node">
 <title>fvs_lift</title>
@@ -42,16 +42,16 @@
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="198.95,-935.17 205.3,-934.89 200.16,-931.15 198.95,-935.17"/>
 </g>
 <!-- fvs_zplug_lift -->
-<g id="node44" class="node">
+<g id="node53" class="node">
 <title>fvs_zplug_lift</title>
-<g id="a_node44"><a xlink:title="fvs_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node53"><a xlink:title="fvs_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-893" rx="40.98" ry="18"/>
 <text text-anchor="middle" x="239.88" y="-890.8" font-family="Inter" font-size="9.00" fill="#222222">fvs_zplug_lift</text>
 </a>
 </g>
 </g>
 <!-- fvs_lift&#45;&gt;fvs_zplug_lift -->
-<g id="edge28" class="edge">
+<g id="edge30" class="edge">
 <title>fvs_lift&#45;&gt;fvs_zplug_lift</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M92.46,-893C118.9,-893 160.28,-893 192.31,-893"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="192.58,-895.1 198.58,-893 192.58,-890.9 192.58,-895.1"/>
@@ -144,61 +144,61 @@
 </g>
 </g>
 <!-- open_rec_zplug_lift -->
-<g id="node41" class="node">
+<g id="node50" class="node">
 <title>open_rec_zplug_lift</title>
-<g id="a_node41"><a xlink:title="open_rec_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node50"><a xlink:title="open_rec_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-260" rx="56.66" ry="18"/>
 <text text-anchor="middle" x="239.88" y="-257.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_zplug_lift</text>
 </a>
 </g>
 </g>
 <!-- open_rec_occurs_false&#45;&gt;open_rec_zplug_lift -->
-<g id="edge20" class="edge">
+<g id="edge22" class="edge">
 <title>open_rec_occurs_false&#45;&gt;open_rec_zplug_lift</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M100.4,-202.33C121.4,-211.67 148.91,-223.73 173.48,-234 181.21,-237.23 189.48,-240.59 197.46,-243.78"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="197.14,-245.92 203.49,-246.19 198.69,-242.02 197.14,-245.92"/>
 </g>
 <!-- full_comp_aux -->
-<g id="node42" class="node">
+<g id="node51" class="node">
 <title>full_comp_aux</title>
-<g id="a_node42"><a xlink:title="full_comp_aux theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node51"><a xlink:title="full_comp_aux theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-355" rx="44.41" ry="18"/>
 <text text-anchor="middle" x="408.74" y="-352.8" font-family="Inter" font-size="9.00" fill="#222222">full_comp_aux</text>
 </a>
 </g>
 </g>
 <!-- open_rec_occurs_false&#45;&gt;full_comp_aux -->
-<g id="edge23" class="edge">
+<g id="edge25" class="edge">
 <title>open_rec_occurs_false&#45;&gt;full_comp_aux</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M79.07,-204.91C92.06,-223.05 112.71,-251.95 130.48,-277 149.67,-304.05 144.61,-321.67 173.48,-338 231.69,-370.91 311.45,-368.63 361.26,-362.75"/>
-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="361.79,-364.8 367.49,-361.97 361.27,-360.63 361.79,-364.8"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M72.91,-204.91C86.26,-237.92 119.69,-307.71 173.48,-338 231.75,-370.81 311.49,-368.56 361.29,-362.71"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="361.81,-364.76 367.51,-361.94 361.29,-360.59 361.81,-364.76"/>
 </g>
 <!-- subst_notin -->
-<g id="node92" class="node">
+<g id="node107" class="node">
 <title>subst_notin</title>
-<g id="a_node92"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node107"><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="edge64" class="edge">
+<g id="edge66" class="edge">
 <title>open_rec_occurs_false&#45;&gt;subst_notin</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M75.37,-169.12C91.41,-139.31 127.39,-79.25 173.48,-44 181.6,-37.79 191.39,-32.87 200.9,-29.05"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="201.91,-30.91 206.77,-26.81 200.41,-26.98 201.91,-30.91"/>
 </g>
 <!-- full_comp_aux_sub -->
-<g id="node106" class="node">
+<g id="node121" class="node">
 <title>full_comp_aux_sub</title>
-<g id="a_node106"><a xlink:title="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node121"><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="-257" rx="54.7" ry="18"/>
 <text text-anchor="middle" x="408.74" y="-254.8" 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="edge74" class="edge">
+<g id="edge76" class="edge">
 <title>open_rec_occurs_false&#45;&gt;full_comp_aux_sub</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M89.81,-170.29C135.38,-140.25 237.14,-84.29 306.27,-130 346.97,-156.91 315.05,-196.23 349.27,-231 353.28,-235.07 358.03,-238.55 363.07,-241.52"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="362.36,-243.53 368.63,-244.55 364.37,-239.84 362.36,-243.53"/>
@@ -213,7 +213,7 @@
 </g>
 </g>
 <!-- close_rec_notin&#45;&gt;subst_notin -->
-<g id="edge63" class="edge">
+<g id="edge65" 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,-988.65 380.23,-988.43 375.12,-984.64 373.87,-988.65"/>
 </g>
 <!-- subst_lc_self -->
-<g id="node93" class="node">
+<g id="node108" class="node">
 <title>subst_lc_self</title>
-<g id="a_node93"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node108"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="577.6" cy="-1049" rx="40.33" ry="18"/>
 <text text-anchor="middle" x="577.6" y="-1046.8" 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="edge65" class="edge">
+<g id="edge67" class="edge">
 <title>subst_fvar_self&#45;&gt;subst_lc_self</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M454.45,-1049C478.12,-1049 507.11,-1049 530.92,-1049"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="531.17,-1051.1 537.17,-1049 531.17,-1046.9 531.17,-1051.1"/>
@@ -288,16 +288,16 @@
 </g>
 </g>
 <!-- subst_other -->
-<g id="node94" class="node">
+<g id="node109" class="node">
 <title>subst_other</title>
-<g id="a_node94"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node109"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1153" rx="37.56" ry="18"/>
 <text text-anchor="middle" x="239.88" y="-1150.8" font-family="Inter" font-size="9.00" fill="#222222">subst_other</text>
 </a>
 </g>
 </g>
 <!-- subst_fvar_other&#45;&gt;subst_other -->
-<g id="edge66" class="edge">
+<g id="edge68" class="edge">
 <title>subst_fvar_other&#45;&gt;subst_other</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M115.25,-1153C140.7,-1153 171.52,-1153 196.02,-1153"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="196.1,-1155.1 202.1,-1153 196.1,-1150.9 196.1,-1155.1"/>
@@ -330,16 +330,16 @@
 </g>
 </g>
 <!-- star_red_sub_full_comp -->
-<g id="node108" class="node">
+<g id="node123" class="node">
 <title>star_red_sub_full_comp</title>
-<g id="a_node108"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node123"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="752.81" cy="-281" rx="65.64" ry="18"/>
 <text text-anchor="middle" x="752.81" y="-278.8" 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="edge81" class="edge">
+<g id="edge83" class="edge">
 <title>Plus_to_Star&#45;&gt;star_red_sub_full_comp</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M615.34,-301.49C636.57,-298.3 663.88,-294.2 688.49,-290.51"/>
 <polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="689.06,-292.54 694.68,-289.58 688.43,-288.39 689.06,-292.54"/>
@@ -512,380 +512,527 @@
 </a>
 </g>
 </g>
-<!-- Plus_red1_context -->
+<!-- of_trm_plug -->
 <g id="node34" class="node">
+<title>of_trm_plug</title>
+<g id="a_node34"><a xlink:title="of_trm_plug theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1829" rx="38.21" ry="18"/>
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+<!-- mred_term_context -->
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+</g>
+</g>
+<!-- of_trm_plug&#45;&gt;mred_term_context -->
+<g id="edge16" class="edge">
+<title>of_trm_plug&#45;&gt;mred_term_context</title>
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+<!-- mred_of_red1 -->
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+<title>mred_of_red1</title>
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+</g>
+<!-- mred_context -->
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+<title>mred_context</title>
+<g id="a_node36"><a xlink:title="mred_context theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
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+<!-- of_trm_moccurs&#45;&gt;of_trm_moccurs0 -->
+<g id="edge17" class="edge">
+<title>of_trm_moccurs&#45;&gt;of_trm_moccurs0</title>
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+<!-- mred_gc_intro -->
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+<!-- mred_r_intro -->
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-<g id="a_node46"><a xlink:title="red1_root_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node55"><a xlink:title="red1_root_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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 </a>
 </g>
 </g>
 <!-- fvs_zplug_lift&#45;&gt;red1_root_fv -->
-<g id="edge29" class="edge">
+<g id="edge31" class="edge">
 <title>fvs_zplug_lift&#45;&gt;red1_root_fv</title>
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 <!-- plug_fv_mono -->
-<g id="node45" class="node">
+<g id="node54" class="node">
 <title>plug_fv_mono</title>
-<g id="a_node45"><a xlink:title="plug_fv_mono theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node54"><a xlink:title="plug_fv_mono theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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 <text text-anchor="middle" x="408.74" y="-942.8" font-family="Inter" font-size="9.00" fill="#222222">plug_fv_mono</text>
 </a>
 </g>
 </g>
 <!-- red1r_fv -->
-<g id="node47" class="node">
+<g id="node56" class="node">
 <title>red1r_fv</title>
-<g id="a_node47"><a xlink:title="red1r_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node56"><a xlink:title="red1r_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="577.6" cy="-919" rx="28.74" ry="18"/>
 <text text-anchor="middle" x="577.6" y="-916.8" font-family="Inter" font-size="9.00" fill="#222222">red1r_fv</text>
 </a>
 </g>
 </g>
 <!-- plug_fv_mono&#45;&gt;red1r_fv -->
-<g id="edge30" class="edge">
+<g id="edge32" class="edge">
 <title>plug_fv_mono&#45;&gt;red1r_fv</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M449.28,-938.84C477.95,-934.37 516.29,-928.4 543.38,-924.18"/>
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 </g>
 <!-- red1_root_fv&#45;&gt;red1r_fv -->
-<g id="edge31" class="edge">
+<g id="edge33" class="edge">
 <title>red1_root_fv&#45;&gt;red1r_fv</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M445.53,-898.58C474.47,-903.09 514.9,-909.39 543.18,-913.79"/>
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 </g>
 <!-- lemma_2_1_fv_preserved -->
-<g id="node48" class="node">
+<g id="node57" class="node">
 <title>lemma_2_1_fv_preserved</title>
-<g id="a_node48"><a xlink:title="lemma_2_1_fv_preserved (section 2.1) theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node57"><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="752.81" y="-916.8" font-family="Inter" font-size="9.00" fill="#222222">lemma_2_1_fv_preserved</text>
 </a>
 </g>
 </g>
 <!-- red1r_fv&#45;&gt;lemma_2_1_fv_preserved -->
-<g id="edge32" class="edge">
+<g id="edge34" class="edge">
 <title>red1r_fv&#45;&gt;lemma_2_1_fv_preserved</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M606.82,-919C628.38,-919 659,-919 686.54,-919"/>
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 </g>
 <!-- gen_fv_preserved -->
-<g id="node63" class="node">
+<g id="node78" class="node">
 <title>gen_fv_preserved</title>
-<g id="a_node63"><a xlink:title="gen_fv_preserved theory/Random.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node78"><a xlink:title="gen_fv_preserved theory/Random.v https://arxiv.org/html/2312.13270v1">
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 <text text-anchor="middle" x="921.68" y="-827.8" font-family="Inter" font-size="9.00" fill="#222222">gen_fv_preserved</text>
 </a>
 </g>
 </g>
 <!-- lemma_2_1_fv_preserved&#45;&gt;gen_fv_preserved -->
-<g id="edge42" class="edge">
+<g id="edge44" class="edge">
 <title>lemma_2_1_fv_preserved&#45;&gt;gen_fv_preserved</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M801.67,-900.89C807.45,-898.39 813.22,-895.73 818.63,-893 844.23,-880.09 871.81,-862.84 891.84,-849.65"/>
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 <!-- corpus_fv_preserved -->
-<g id="node116" class="node">
+<g id="node131" class="node">
 <title>corpus_fv_preserved</title>
-<g id="a_node116"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node131"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
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 <text text-anchor="middle" x="921.68" y="-897.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_fv_preserved</text>
 </a>
 </g>
 </g>
 <!-- lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved -->
-<g id="edge89" class="edge">
+<g id="edge91" class="edge">
 <title>lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved</title>
 <path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M813.11,-912.25C827.95,-910.56 843.9,-908.74 858.85,-907.04"/>
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 <!-- Plus_trans -->
-<g id="node50" class="node">
+<g id="node59" class="node">
 <title>Plus_trans</title>
-<g id="a_node50"><a xlink:title="Plus_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1829" rx="34.78" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1826.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_trans</text>
+<g id="a_node59"><a xlink:title="Plus_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2193" rx="34.78" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2190.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_trans</text>
 </a>
 </g>
 </g>
 <!-- Plus_step_trans -->
-<g id="node51" class="node">
+<g id="node60" class="node">
 <title>Plus_step_trans</title>
-<g id="a_node51"><a xlink:title="Plus_step_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1881" rx="47.19" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1878.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_step_trans</text>
+<g id="a_node60"><a xlink:title="Plus_step_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2245" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2242.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_step_trans</text>
+</a>
+</g>
+</g>
+<!-- isubst_meta_in -->
+<g id="node61" class="node">
+<title>isubst_meta_in</title>
+<g id="a_node61"><a xlink:title="isubst_meta_in theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2297" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2294.8" font-family="Inter" font-size="9.00" fill="#222222">isubst_meta_in</text>
+</a>
+</g>
+</g>
+<!-- isubst_meta_notin -->
+<g id="node62" class="node">
+<title>isubst_meta_notin</title>
+<g id="a_node62"><a xlink:title="isubst_meta_notin theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2349" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2346.8" font-family="Inter" font-size="9.00" fill="#222222">isubst_meta_notin</text>
+</a>
+</g>
+</g>
+<!-- eqC_in_Es -->
+<g id="node63" class="node">
+<title>eqC_in_Es</title>
+<g id="a_node63"><a xlink:title="eqC_in_Es theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2401" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2398.8" font-family="Inter" font-size="9.00" fill="#222222">eqC_in_Es</text>
+</a>
+</g>
+</g>
+<!-- Es_refl_any -->
+<g id="node64" class="node">
+<title>Es_refl_any</title>
+<g id="a_node64"><a xlink:title="Es_refl_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2453" rx="36.9" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2450.8" font-family="Inter" font-size="9.00" fill="#222222">Es_refl_any</text>
+</a>
+</g>
+</g>
+<!-- Es_sym_any -->
+<g id="node65" class="node">
+<title>Es_sym_any</title>
+<g id="a_node65"><a xlink:title="Es_sym_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2505" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2502.8" font-family="Inter" font-size="9.00" fill="#222222">Es_sym_any</text>
+</a>
+</g>
+</g>
+<!-- Es_trans_any -->
+<g id="node66" class="node">
+<title>Es_trans_any</title>
+<g id="a_node66"><a xlink:title="Es_trans_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2557" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2554.8" font-family="Inter" font-size="9.00" fill="#222222">Es_trans_any</text>
 </a>
 </g>
 </g>
 <!-- par_red1_iff -->
-<g id="node52" class="node">
+<g id="node67" class="node">
 <title>par_red1_iff</title>
-<g id="a_node52"><a xlink:title="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1933" rx="37.56" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1930.8" font-family="Inter" font-size="9.00" fill="#222222">par_red1_iff</text>
+<g id="a_node67"><a xlink:title="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2609" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2606.8" font-family="Inter" font-size="9.00" fill="#222222">par_red1_iff</text>
 </a>
 </g>
 </g>
 <!-- par_to_red1_or_eq -->
-<g id="node55" class="node">
+<g id="node70" class="node">
 <title>par_to_red1_or_eq</title>
-<g id="a_node55"><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="-1959" rx="53.23" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-1956.8" font-family="Inter" font-size="9.00" fill="#222222">par_to_red1_or_eq</text>
+<g id="a_node70"><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="-2635" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2632.8" font-family="Inter" font-size="9.00" fill="#222222">par_to_red1_or_eq</text>
 </a>
 </g>
 </g>
 <!-- par_red1_iff&#45;&gt;par_to_red1_or_eq -->
-<g id="edge35" class="edge">
+<g id="edge37" class="edge">
 <title>par_red1_iff&#45;&gt;par_to_red1_or_eq</title>
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 </g>
 <!-- red1_or_eq_to_par -->
-<g id="node56" class="node">
+<g id="node71" class="node">
 <title>red1_or_eq_to_par</title>
-<g id="a_node56"><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="-1907" rx="53.23" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-1904.8" font-family="Inter" font-size="9.00" fill="#222222">red1_or_eq_to_par</text>
+<g id="a_node71"><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="-2583" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2580.8" font-family="Inter" font-size="9.00" fill="#222222">red1_or_eq_to_par</text>
 </a>
 </g>
 </g>
 <!-- par_red1_iff&#45;&gt;red1_or_eq_to_par -->
-<g id="edge36" class="edge">
+<g id="edge38" class="edge">
 <title>par_red1_iff&#45;&gt;red1_or_eq_to_par</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M101.18,-1927.74C125.17,-1924.12 157.5,-1919.26 184.81,-1915.14"/>
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 <!-- red1_in_par -->
-<g id="node53" class="node">
+<g id="node68" class="node">
 <title>red1_in_par</title>
-<g id="a_node53"><a xlink:title="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1985" rx="37.56" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1982.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
+<g id="a_node68"><a xlink:title="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2661" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2658.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
 </a>
 </g>
 </g>
 <!-- par_context -->
-<g id="node54" class="node">
+<g id="node69" class="node">
 <title>par_context</title>
-<g id="a_node54"><a xlink:title="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2037" rx="38.87" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2034.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
+<g id="a_node69"><a xlink:title="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2713" rx="38.87" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2710.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
 </a>
 </g>
 </g>
 <!-- par_reflexive -->
-<g id="node57" class="node">
+<g id="node72" class="node">
 <title>par_reflexive</title>
-<g id="a_node57"><a xlink:title="par_reflexive theory/Parallel.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2089" rx="41.64" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2086.8" font-family="Inter" font-size="9.00" fill="#222222">par_reflexive</text>
+<g id="a_node72"><a xlink:title="par_reflexive theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2765" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2762.8" font-family="Inter" font-size="9.00" fill="#222222">par_reflexive</text>
 </a>
 </g>
 </g>
 <!-- gen_list_length -->
-<g id="node58" class="node">
+<g id="node73" class="node">
 <title>gen_list_length</title>
-<g id="a_node58"><a xlink:title="gen_list_length theory/Random.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2141" rx="46.38" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2138.8" font-family="Inter" font-size="9.00" fill="#222222">gen_list_length</text>
+<g id="a_node73"><a xlink:title="gen_list_length theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2817" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2814.8" font-family="Inter" font-size="9.00" fill="#222222">gen_list_length</text>
 </a>
 </g>
 </g>
 <!-- check_term_true -->
-<g id="node59" class="node">
+<g id="node74" class="node">
 <title>check_term_true</title>
-<g id="a_node59"><a xlink:title="check_term_true theory/Random.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node74"><a xlink:title="check_term_true theory/Random.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="921.68" cy="-741" rx="50.61" ry="18"/>
 <text text-anchor="middle" x="921.68" y="-738.8" font-family="Inter" font-size="9.00" fill="#222222">check_term_true</text>
 </a>
 </g>
 </g>
 <!-- check_all_true -->
-<g id="node60" class="node">
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 <title>check_all_true</title>
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 </a>
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 <title>red1_dec</title>
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 </a>
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 <text text-anchor="middle" x="921.68" y="-582.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_decides</text>
 </a>
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 <!-- has_red_spec&#45;&gt;corpus_decides -->
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 <title>normal_form_iff_nf</title>
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 <!-- nf_iff_steps_nil&#45;&gt;normal_form_iff_nf -->
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+<g id="edge63" class="edge">
 <title>nf_iff_steps_nil&#45;&gt;normal_form_iff_nf</title>
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 <title>nf_dec</title>
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 <text text-anchor="middle" x="1080.15" y="-660.8" font-family="Inter" font-size="9.00" fill="#222222">nf_dec</text>
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-<text text-anchor="middle" x="65.24" y="-2190.8" 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="-2869" rx="43.1" ry="18"/>
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 <title>open_rec_lam</title>
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-<text text-anchor="middle" x="65.24" y="-2242.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lam</text>
+<g id="a_node99"><a xlink:title="open_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2921" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2918.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lam</text>
 </a>
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 </g>
 <!-- open_rec_esub -->
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 <title>open_rec_esub</title>
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-<text text-anchor="middle" x="65.24" y="-2294.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_esub</text>
+<g id="a_node100"><a xlink:title="open_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2973" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2970.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_esub</text>
 </a>
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 <!-- close_rec_app -->
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 <title>close_rec_app</title>
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-<text text-anchor="middle" x="65.24" y="-2346.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_app</text>
+<g id="a_node101"><a xlink:title="close_rec_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3025" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3022.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_app</text>
 </a>
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 <!-- close_rec_lam -->
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 <title>close_rec_lam</title>
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-<text text-anchor="middle" x="65.24" y="-2398.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_lam</text>
+<g id="a_node102"><a xlink:title="close_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3077" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3074.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_lam</text>
 </a>
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 <!-- close_rec_esub -->
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 <title>close_rec_esub</title>
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-<text text-anchor="middle" x="65.24" y="-2450.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
+<g id="a_node103"><a xlink:title="close_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3129" rx="46.53" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3126.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
 </a>
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 <title>subst_app</title>
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-<text text-anchor="middle" x="65.24" y="-2502.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
+<g id="a_node104"><a xlink:title="subst_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3181" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3178.8" 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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-<text text-anchor="middle" x="65.24" y="-2554.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
+<g id="a_node105"><a xlink:title="subst_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3233" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3230.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
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 <title>subst_esub</title>
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-<text text-anchor="middle" x="65.24" y="-2606.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
+<g id="a_node106"><a xlink:title="subst_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3285" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3282.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
 </a>
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 <title>open_rec_lc_atom</title>
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-<text text-anchor="middle" x="65.24" y="-2658.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
+<g id="a_node110"><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="-3337" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3334.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
 </a>
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 <title>open_rec_lc</title>
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 <title>red_sub_root_to_red1</title>
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+<g id="edge73" class="edge">
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 <!-- red_sub_r&#45;&gt;full_comp_aux_sub -->
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 <title>zfill_occurs0</title>
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diff --git a/graphs/index.txt b/graphs/index.txt
index 8e8e42a..1bdbd77 100644
--- a/graphs/index.txt
+++ b/graphs/index.txt
@@ -1,4 +1,27 @@
 status	milestone	kind	name	file
+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	eqC_in_Es	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
+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_gc_intro	theory/Metaterm.v
+proved	M0	infra	mred_of_red1	theory/Metaterm.v
+proved	M0	infra	mred_r_intro	theory/Metaterm.v
+proved	M0	infra	mred_term_context	theory/Metaterm.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
+proved	M0	infra	of_trm_lift	theory/Metaterm.v
+proved	M0	infra	of_trm_moccurs	theory/Metaterm.v
+proved	M0	infra	of_trm_moccurs0	theory/Metaterm.v
+proved	M0	infra	of_trm_open	theory/Metaterm.v
+proved	M0	infra	of_trm_plug	theory/Metaterm.v
 proved	M1	infra	at_ctx_comp	theory/Reduction.v
 proved	M1	infra	close_rec_fvar	theory/Binding.v
 proved	M1	infra	close_rec_fvs	theory/Binding.v
diff --git a/graphs/theorems.json b/graphs/theorems.json
index e2477fd..1ed9d7a 100644
--- a/graphs/theorems.json
+++ b/graphs/theorems.json
@@ -373,6 +373,100 @@
         "mfvs_lift_m"
       ]
     },
+    {
+      "id": "of_trm_plug",
+      "label": "of_trm_plug",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_of_red1",
+      "label": "mred_of_red1",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_context",
+      "label": "mred_context",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_term_context",
+      "label": "mred_term_context",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_plug"
+      ]
+    },
+    {
+      "id": "of_trm_moccurs",
+      "label": "of_trm_moccurs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_moccurs0",
+      "label": "of_trm_moccurs0",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_moccurs"
+      ]
+    },
+    {
+      "id": "mred_B_intro",
+      "label": "mred_B_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_gc_intro",
+      "label": "mred_gc_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_r_intro",
+      "label": "mred_r_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
     {
       "id": "Plus_red1_context",
       "label": "Plus_red1_context",
@@ -582,6 +676,66 @@
       "file": "theory/Metatheory.v",
       "depends": []
     },
+    {
+      "id": "isubst_meta_in",
+      "label": "isubst_meta_in",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "isubst_meta_notin",
+      "label": "isubst_meta_notin",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "eqC_in_Es",
+      "label": "eqC_in_Es",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_refl_any",
+      "label": "Es_refl_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_sym_any",
+      "label": "Es_sym_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_trans_any",
+      "label": "Es_trans_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
     {
       "id": "par_red1_iff",
       "label": "par_red1_iff",
diff --git a/scripts/audit.sh b/scripts/audit.sh
index 54a8cac..7ad07f4 100755
--- a/scripts/audit.sh
+++ b/scripts/audit.sh
@@ -35,7 +35,7 @@ done < <(find theory -name '*.v' -print0 2>/dev/null)
 echo "== Print Assumptions =="
 WORK="$(mktemp -d ./.audit-work.XXXXXX)"
 cp theory/*.v "$WORK"/
-for f in ExecReducer Binding Reduction Metatheory Closure Subsystem Substitution Parallel Metaterm Tests Random Enumerate; do
+for f in ExecReducer Binding Reduction Metatheory Closure Subsystem Substitution Parallel Metaterm NamedMeta Tests Random Enumerate; do
   echo "-- $f"
   out="$(rocq compile -Q "$WORK" LambdaSub "$WORK/$f.v" 2>&1 || true)"
   echo "$out"
diff --git a/theory/NamedMeta.v b/theory/NamedMeta.v
new file mode 100644
index 0000000..25640d4
--- /dev/null
+++ b/theory/NamedMeta.v
@@ -0,0 +1,106 @@
+From Stdlib Require Import List Bool Arith Lia PeanoNat.
+Import ListNotations.
+
+Definition atom := nat.
+
+Inductive ntrm : Type :=
+| NVar : atom -> ntrm
+| NApp : ntrm -> ntrm -> ntrm
+| NLam : atom -> ntrm -> ntrm
+| NESub : ntrm -> atom -> ntrm -> ntrm
+| NMVar : atom -> list atom -> ntrm.
+
+Fixpoint nfv (t : ntrm) : list atom :=
+  match t with
+  | NVar x => [x]
+  | NApp a b => nfv a ++ nfv b
+  | NLam x a => filter (fun y => negb (Nat.eqb y x)) (nfv a)
+  | NESub a x b => filter (fun y => negb (Nat.eqb y x)) (nfv a) ++ nfv b
+  | NMVar _ d => d
+  end.
+
+Definition isubst_meta (X : atom) (d : list atom) (x : atom) (v : ntrm) : ntrm :=
+  if existsb (Nat.eqb x) d then NESub (NMVar X d) x v else NMVar X d.
+
+Lemma isubst_meta_in : forall X d x v, In x d -> isubst_meta X d x v = NESub (NMVar X d) x v.
+Proof.
+  intros X d x v H. unfold isubst_meta.
+  destruct (existsb (Nat.eqb x) d) eqn:E; [reflexivity |].
+  assert (Hex : existsb (Nat.eqb x) d = true)
+    by (apply existsb_exists; exists x; split; [exact H | apply Nat.eqb_refl]).
+  rewrite Hex in E. discriminate E.
+Qed.
+
+Lemma isubst_meta_notin : forall X d x v, ~ In x d -> isubst_meta X d x v = NMVar X d.
+Proof.
+  intros X d x v H. unfold isubst_meta.
+  destruct (existsb (Nat.eqb x) d) eqn:E; [| reflexivity].
+  exfalso. apply existsb_exists in E. destruct E as [y [Hy HE]].
+  apply Nat.eqb_eq in HE. subst y. apply H. exact Hy.
+Qed.
+
+Inductive nctx : Type :=
+| nHole : nctx
+| nAppL : nctx -> ntrm -> nctx
+| nAppR : ntrm -> nctx -> nctx
+| nLam : atom -> nctx -> nctx
+| nESubL : nctx -> atom -> ntrm -> nctx
+| nESubR : ntrm -> atom -> nctx -> nctx.
+
+Fixpoint nplug (C : nctx) (t : ntrm) : ntrm :=
+  match C with
+  | nHole => t
+  | nAppL C1 u => NApp (nplug C1 t) u
+  | nAppR u C1 => NApp u (nplug C1 t)
+  | nLam x C1 => NLam x (nplug C1 t)
+  | nESubL C1 x u => NESub (nplug C1 t) x u
+  | nESubR t1 x C1 => NESub t1 x (nplug C1 t)
+  end.
+
+Fixpoint is_subst_chain (C : nctx) : bool :=
+  match C with
+  | nHole => true
+  | nESubL C1 _ _ => is_subst_chain C1
+  | _ => false
+  end.
+
+Inductive Es : ntrm -> ntrm -> Prop :=
+| Es_refl : forall t, Es t t
+| Es_sym : forall t u, Es t u -> Es u t
+| Es_trans : forall t u v, Es t u -> Es u v -> Es t v
+| Es_C : forall t x u y v,
+    y <> x -> ~ In y (nfv u) -> ~ In x (nfv v) ->
+    Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u).
+
+Inductive nred : ntrm -> ntrm -> Prop :=
+| nred_B : forall t x u, nred (NApp (NLam x t) u) (NESub t x u)
+| nred_Gc : forall t x u, ~ In x (nfv t) -> nred (NESub t x u) t
+| nred_RX : forall C X d x u phi,
+    In x d ->
+    In x phi ->
+    (forall y, In y (nfv u) -> In y phi) ->
+    is_subst_chain C = false ->
+    nred (NESub (nplug C (NMVar X d)) x u)
+         (NESub (nplug C (NESub (NMVar X d) x u)) x u)
+| nred_ctx : forall C t t', nred t t' -> nred (nplug C t) (nplug C t').
+
+Lemma eqC_in_Es : forall t x u y v,
+  y <> x -> ~ In y (nfv u) -> ~ In x (nfv v) ->
+  Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u).
+Proof. intros. apply Es_C; assumption. Qed.
+
+Lemma Es_refl_any : forall t, Es t t.
+Proof. intros. apply Es_refl. Qed.
+
+Lemma Es_sym_any : forall t u, Es t u -> Es u t.
+Proof. intros. apply Es_sym. exact H. Qed.
+
+Lemma Es_trans_any : forall t u v, Es t u -> Es u v -> Es t v.
+Proof. intros. apply Es_trans with (u := u); assumption. Qed.
+
+Print Assumptions isubst_meta_in.
+Print Assumptions isubst_meta_notin.
+Print Assumptions eqC_in_Es.
+Print Assumptions Es_refl_any.
+Print Assumptions Es_sym_any.
+Print Assumptions Es_trans_any.