diff --git a/graphs/dependency.dot b/graphs/dependency.dot
index 3171ae0..eccb285 100644
--- a/graphs/dependency.dot
+++ b/graphs/dependency.dot
@@ -32,6 +32,14 @@ digraph theorem_deps {
   enumerate_check [label="enumerate_check", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="enumerate_check theory/Enumerate.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   enumerate_sound [label="enumerate_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="enumerate_sound theory/Enumerate.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   enumerate_complete [label="enumerate_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="enumerate_complete theory/Enumerate.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_fvs [label="of_trm_fvs", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_fvs theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_lift [label="of_trm_lift", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_lift theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_open [label="of_trm_open", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_open theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_close [label="of_trm_close", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_close theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_injective [label="of_trm_injective", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_injective theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  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];
   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];
@@ -128,6 +136,8 @@ digraph theorem_deps {
   check_all_true -> enumerate_check;
   steps_sound -> enumerate_sound;
   steps_complete -> enumerate_complete;
+  of_trm_lift -> of_trm_open;
+  mfvs_lift_m -> mfvs_open_m;
   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 f78ac40..e26f6ea 100644
Binary files a/graphs/dependency.png and b/graphs/dependency.png differ
diff --git a/graphs/dependency.svg b/graphs/dependency.svg
index aa81263..1de2e31 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="2612pt"
- viewBox="0.00 0.00 1293.96 2612.00" 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 2608)">
+<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)">
 <title>theorem_deps</title>
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-2608 1289.96,-2608 1289.96,4 -4,4"/>
-<text text-anchor="start" x="406.48" y="-2591.4" 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="-2581.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
-<text text-anchor="start" x="599.98" y="-2581.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="606.98" y="-2581.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
-<text text-anchor="start" x="633.98" y="-2581.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="640.98" y="-2581.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
-<text text-anchor="start" x="674.98" y="-2581.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
-<text text-anchor="start" x="681.98" y="-2581.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
+<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>
 <!-- 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="node36" class="node">
+<g id="node44" class="node">
 <title>fvs_zplug_lift</title>
-<g id="a_node36"><a xlink:title="fvs_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node44"><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="edge26" class="edge">
+<g id="edge28" 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="node33" class="node">
+<g id="node41" class="node">
 <title>open_rec_zplug_lift</title>
-<g id="a_node33"><a xlink:title="open_rec_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node41"><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="edge18" class="edge">
+<g id="edge20" 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="node34" class="node">
+<g id="node42" class="node">
 <title>full_comp_aux</title>
-<g id="a_node34"><a xlink:title="full_comp_aux theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node42"><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="edge21" class="edge">
+<g id="edge23" 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"/>
 </g>
 <!-- subst_notin -->
-<g id="node84" class="node">
+<g id="node92" class="node">
 <title>subst_notin</title>
-<g id="a_node84"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node92"><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="edge62" class="edge">
+<g id="edge64" 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="node98" class="node">
+<g id="node106" class="node">
 <title>full_comp_aux_sub</title>
-<g id="a_node98"><a xlink:title="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node106"><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="edge72" class="edge">
+<g id="edge74" 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="edge61" class="edge">
+<g id="edge63" 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="node85" class="node">
+<g id="node93" class="node">
 <title>subst_lc_self</title>
-<g id="a_node85"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node93"><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="edge63" class="edge">
+<g id="edge65" 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="node86" class="node">
+<g id="node94" class="node">
 <title>subst_other</title>
-<g id="a_node86"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node94"><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="edge64" class="edge">
+<g id="edge66" 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="node100" class="node">
+<g id="node108" class="node">
 <title>star_red_sub_full_comp</title>
-<g id="a_node100"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node108"><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="edge79" class="edge">
+<g id="edge81" 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"/>
@@ -428,380 +428,464 @@
 </a>
 </g>
 </g>
-<!-- Plus_red1_context -->
+<!-- of_trm_fvs -->
 <g id="node26" class="node">
+<title>of_trm_fvs</title>
+<g id="a_node26"><a xlink:title="of_trm_fvs theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1517" rx="34.78" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1514.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_fvs</text>
+</a>
+</g>
+</g>
+<!-- of_trm_lift -->
+<g id="node27" class="node">
+<title>of_trm_lift</title>
+<g id="a_node27"><a xlink:title="of_trm_lift theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1569" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1566.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_lift</text>
+</a>
+</g>
+</g>
+<!-- of_trm_open -->
+<g id="node28" class="node">
+<title>of_trm_open</title>
+<g id="a_node28"><a xlink:title="of_trm_open theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1569" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1566.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_open</text>
+</a>
+</g>
+</g>
+<!-- of_trm_lift&#45;&gt;of_trm_open -->
+<g id="edge14" class="edge">
+<title>of_trm_lift&#45;&gt;of_trm_open</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M99.53,-1569C126.51,-1569 164.79,-1569 194.4,-1569"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="194.58,-1571.1 200.58,-1569 194.58,-1566.9 194.58,-1571.1"/>
+</g>
+<!-- of_trm_close -->
+<g id="node29" class="node">
+<title>of_trm_close</title>
+<g id="a_node29"><a xlink:title="of_trm_close theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1621" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1618.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_close</text>
+</a>
+</g>
+</g>
+<!-- of_trm_injective -->
+<g id="node30" class="node">
+<title>of_trm_injective</title>
+<g id="a_node30"><a xlink:title="of_trm_injective theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1673" rx="48.49" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1670.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_injective</text>
+</a>
+</g>
+</g>
+<!-- mfvs_lift_m -->
+<g id="node31" class="node">
+<title>mfvs_lift_m</title>
+<g id="a_node31"><a xlink:title="mfvs_lift_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1725" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1722.8" font-family="Inter" font-size="9.00" fill="#222222">mfvs_lift_m</text>
+</a>
+</g>
+</g>
+<!-- mfvs_open_m -->
+<g id="node33" class="node">
+<title>mfvs_open_m</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1725" rx="43.1" ry="18"/>
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+</a>
+</g>
+</g>
+<!-- mfvs_lift_m&#45;&gt;mfvs_open_m -->
+<g id="edge15" class="edge">
+<title>mfvs_lift_m&#45;&gt;mfvs_open_m</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M102.85,-1725C128.34,-1725 162.6,-1725 190.36,-1725"/>
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+<!-- mfvs_close_m -->
+<g id="node32" class="node">
+<title>mfvs_close_m</title>
+<g id="a_node32"><a xlink:title="mfvs_close_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1777" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1774.8" font-family="Inter" font-size="9.00" fill="#222222">mfvs_close_m</text>
+</a>
+</g>
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+<!-- Plus_red1_context -->
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 <title>Plus_red1_context</title>
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+<g id="a_node34"><a xlink:title="Plus_red1_context theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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 </a>
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+<g id="node49" class="node">
 <title>lemma_2_3_beta_sim</title>
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+<g id="node37" class="node">
 <title>occurs_count_pos</title>
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+<g id="edge16" class="edge">
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 <title>occurs_lift_self</title>
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+<g id="edge18" class="edge">
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+<g id="edge25" class="edge">
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+<g id="edge29" class="edge">
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 <title>gen_fv_preserved</title>
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 </a>
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 </a>
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 <title>par_red1_iff</title>
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 </a>
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+<g id="node55" class="node">
 <title>par_to_red1_or_eq</title>
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 </a>
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 </g>
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+<g id="edge35" class="edge">
 <title>par_red1_iff&#45;&gt;par_to_red1_or_eq</title>
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+<g id="node56" class="node">
 <title>red1_or_eq_to_par</title>
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+<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>
 </a>
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 <!-- par_red1_iff&#45;&gt;red1_or_eq_to_par -->
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+<g id="edge36" class="edge">
 <title>par_red1_iff&#45;&gt;red1_or_eq_to_par</title>
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 <!-- red1_in_par -->
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+<g id="node53" class="node">
 <title>red1_in_par</title>
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-<text text-anchor="middle" x="65.24" y="-1670.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
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+<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>
 </a>
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 <!-- par_context -->
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+<g id="node54" class="node">
 <title>par_context</title>
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+<g id="a_node54"><a xlink:title="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1">
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+<text text-anchor="middle" x="65.24" y="-2034.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
 </a>
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 </a>
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 <!-- gen_list_length -->
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-<text text-anchor="middle" x="65.24" y="-1826.8" font-family="Inter" font-size="9.00" fill="#222222">gen_list_length</text>
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+<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>
 </a>
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 <text text-anchor="middle" x="921.68" y="-738.8" font-family="Inter" font-size="9.00" fill="#222222">check_term_true</text>
 </a>
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 <text text-anchor="middle" x="1080.15" y="-738.8" font-family="Inter" font-size="9.00" fill="#222222">check_all_true</text>
 </a>
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 <!-- check_term_true&#45;&gt;check_all_true -->
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+<g id="edge39" class="edge">
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 </a>
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+<g id="node62" class="node">
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 <text text-anchor="middle" x="752.81" y="-466.8" font-family="Inter" font-size="9.00" fill="#222222">gen_complete</text>
 </a>
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 <text text-anchor="middle" x="239.88" y="-447.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_sound</text>
 </a>
 </g>
 </g>
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+<g id="edge26" class="edge">
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 <title>corpus_decides</title>
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+<g id="edge61" 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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+<g id="edge62" class="edge">
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 <!-- open_rec_app -->
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 </a>
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 <!-- open_rec_lam -->
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 <title>open_rec_lam</title>
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 </a>
 </g>
 </g>
 <!-- open_rec_esub -->
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+<g id="node85" class="node">
 <title>open_rec_esub</title>
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-<text text-anchor="middle" x="65.24" y="-1982.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_esub</text>
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 </a>
 </g>
 </g>
 <!-- close_rec_app -->
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+<g id="node86" class="node">
 <title>close_rec_app</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2037" rx="44.41" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2034.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_app</text>
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 </a>
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 <!-- close_rec_lam -->
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+<g id="node87" class="node">
 <title>close_rec_lam</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2089" rx="44.41" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2086.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_lam</text>
+<g id="a_node87"><a xlink:title="close_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2401" rx="44.41" ry="18"/>
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 </a>
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 </g>
 <!-- close_rec_esub -->
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+<g id="node88" class="node">
 <title>close_rec_esub</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2141" rx="46.53" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2138.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
+<g id="a_node88"><a xlink:title="close_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2453" rx="46.53" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2450.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
 </a>
 </g>
 </g>
 <!-- subst_app -->
-<g id="node81" class="node">
+<g id="node89" class="node">
 <title>subst_app</title>
-<g id="a_node81"><a xlink:title="subst_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2193" rx="34.13" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2190.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
+<g id="a_node89"><a xlink:title="subst_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2505" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2502.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
 </a>
 </g>
 </g>
 <!-- subst_lam -->
-<g id="node82" class="node">
+<g id="node90" class="node">
 <title>subst_lam</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2245" rx="34.13" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2242.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
+<g id="a_node90"><a xlink:title="subst_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2557" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2554.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
 </a>
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 </g>
 <!-- subst_esub -->
-<g id="node83" class="node">
+<g id="node91" class="node">
 <title>subst_esub</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2297" rx="36.25" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2294.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2609" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2606.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
 </a>
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 <!-- open_rec_lc_atom -->
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+<g id="node95" class="node">
 <title>open_rec_lc_atom</title>
-<g id="a_node87"><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="-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">open_rec_lc_atom</text>
+<g id="a_node95"><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="-2661" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2658.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
 </a>
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 <!-- open_rec_lc -->
-<g id="node88" class="node">
+<g id="node96" class="node">
 <title>open_rec_lc</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2349" rx="39.02" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-2346.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
+<g id="a_node96"><a xlink:title="open_rec_lc theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2661" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2658.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
 </a>
 </g>
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 <!-- open_rec_lc_atom&#45;&gt;open_rec_lc -->
-<g id="edge65" class="edge">
+<g id="edge67" class="edge">
 <title>open_rec_lc_atom&#45;&gt;open_rec_lc</title>
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 <!-- red_sub_root_to_red1 -->
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+<g id="node97" class="node">
 <title>red_sub_root_to_red1</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2401" rx="60.09" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2398.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_root_to_red1</text>
+<g id="a_node97"><a xlink:title="red_sub_root_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2713" rx="60.09" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2710.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_root_to_red1</text>
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 <!-- red_sub_to_red1 -->
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+<g id="node98" class="node">
 <title>red_sub_to_red1</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2401" rx="47.84" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-2398.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_to_red1</text>
+<g id="a_node98"><a xlink:title="red_sub_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2713" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2710.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_to_red1</text>
 </a>
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 </g>
 <!-- red_sub_root_to_red1&#45;&gt;red_sub_to_red1 -->
-<g id="edge66" class="edge">
+<g id="edge68" class="edge">
 <title>red_sub_root_to_red1&#45;&gt;red_sub_to_red1</title>
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 <!-- red_sub_context -->
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 <title>red_sub_context</title>
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-<text text-anchor="middle" x="65.24" y="-2450.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_context</text>
+<g id="a_node99"><a xlink:title="red_sub_context theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2765" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2762.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_context</text>
 </a>
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 <!-- Star_red_sub_context -->
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 <title>Star_red_sub_context</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2453" rx="61.55" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-2450.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red_sub_context</text>
+<g id="a_node105"><a xlink:title="Star_red_sub_context theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2765" rx="61.55" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2762.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red_sub_context</text>
 </a>
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 <!-- red_sub_context&#45;&gt;Star_red_sub_context -->
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+<g id="edge71" class="edge">
 <title>red_sub_context&#45;&gt;Star_red_sub_context</title>
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 <!-- red_sub_gc -->
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 <title>red_sub_gc</title>
-<g id="a_node92"><a xlink:title="red_sub_gc theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node100"><a xlink:title="red_sub_gc theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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 <!-- red_sub_gc&#45;&gt;full_comp_aux_sub -->
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+<g id="edge76" class="edge">
 <title>red_sub_gc&#45;&gt;full_comp_aux_sub</title>
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 <title>red_gc_to_red_sub</title>
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 <text text-anchor="middle" x="408.74" y="-67.8" font-family="Inter" font-size="9.00" fill="#222222">red_gc_to_red_sub</text>
 </a>
 </g>
 </g>
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 <title>red_sub_r</title>
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 </a>
 </g>
 </g>
 <!-- red_sub_r&#45;&gt;full_comp_aux_sub -->
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+<g id="edge77" class="edge">
 <title>red_sub_r&#45;&gt;full_comp_aux_sub</title>
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+<g id="node102" class="node">
 <title>occurs_zfill</title>
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 </a>
 </g>
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 <!-- zfill_occurs0 -->
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+<g id="node103" class="node">
 <title>zfill_occurs0</title>
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 </a>
 </g>
 </g>
 <!-- occurs_zfill&#45;&gt;zfill_occurs0 -->
-<g id="edge67" class="edge">
+<g id="edge69" class="edge">
 <title>occurs_zfill&#45;&gt;zfill_occurs0</title>
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 <!-- R_Gc_disjoint -->
-<g id="node96" class="node">
+<g id="node104" class="node">
 <title>R_Gc_disjoint</title>
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-<text text-anchor="middle" x="408.74" y="-2502.8" font-family="Inter" font-size="9.00" fill="#222222">R_Gc_disjoint</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-2817" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-2814.8" font-family="Inter" font-size="9.00" fill="#222222">R_Gc_disjoint</text>
 </a>
 </g>
 </g>
 <!-- zfill_occurs0&#45;&gt;R_Gc_disjoint -->
-<g id="edge68" class="edge">
+<g id="edge70" class="edge">
 <title>zfill_occurs0&#45;&gt;R_Gc_disjoint</title>
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 </g>
 <!-- plus_red_sub_full_comp -->
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+<g id="node107" class="node">
 <title>plus_red_sub_full_comp</title>
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 <text text-anchor="middle" x="577.6" y="-252.8" font-family="Inter" font-size="9.00" fill="#222222">plus_red_sub_full_comp</text>
 </a>
 </g>
 </g>
 <!-- full_comp_aux_sub&#45;&gt;plus_red_sub_full_comp -->
-<g id="edge78" class="edge">
+<g id="edge80" class="edge">
 <title>full_comp_aux_sub&#45;&gt;plus_red_sub_full_comp</title>
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 <!-- plus_red_sub_full_comp&#45;&gt;star_red_sub_full_comp -->
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+<g id="edge82" class="edge">
 <title>plus_red_sub_full_comp&#45;&gt;star_red_sub_full_comp</title>
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 <!-- es_count_plug_mono -->
-<g id="node101" class="node">
+<g id="node109" class="node">
 <title>es_count_plug_mono</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2869" rx="60.74" ry="18"/>
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 </a>
 </g>
 </g>
 <!-- red_gc_measure -->
-<g id="node102" class="node">
+<g id="node110" class="node">
 <title>red_gc_measure</title>
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 </a>
 </g>
 </g>
 <!-- es_count_plug_mono&#45;&gt;red_gc_measure -->
-<g id="edge81" class="edge">
+<g id="edge83" class="edge">
 <title>es_count_plug_mono&#45;&gt;red_gc_measure</title>
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 <!-- red_gc_terminates -->
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+<g id="node111" class="node">
 <title>red_gc_terminates</title>
-<g id="a_node103"><a xlink:title="red_gc_terminates theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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 </g>
 <!-- red_gc_measure&#45;&gt;red_gc_terminates -->
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+<g id="edge84" class="edge">
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diff --git a/graphs/theorems.json b/graphs/theorems.json
index 9a78af6..e2477fd 100644
--- a/graphs/theorems.json
+++ b/graphs/theorems.json
@@ -9,6 +9,7 @@
     "html": "https://arxiv.org/html/2312.13270v1"
   },
   "milestones": [
+    "M0",
     "M1",
     "M2",
     "M3",
@@ -288,6 +289,90 @@
         "steps_complete"
       ]
     },
+    {
+      "id": "of_trm_fvs",
+      "label": "of_trm_fvs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_lift",
+      "label": "of_trm_lift",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_open",
+      "label": "of_trm_open",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_lift"
+      ]
+    },
+    {
+      "id": "of_trm_close",
+      "label": "of_trm_close",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_injective",
+      "label": "of_trm_injective",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_lift_m",
+      "label": "mfvs_lift_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_close_m",
+      "label": "mfvs_close_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_open_m",
+      "label": "mfvs_open_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "mfvs_lift_m"
+      ]
+    },
     {
       "id": "Plus_red1_context",
       "label": "Plus_red1_context",
diff --git a/theory/Metaterm.v b/theory/Metaterm.v
index 2337fe6..629fb77 100644
--- a/theory/Metaterm.v
+++ b/theory/Metaterm.v
@@ -110,3 +110,54 @@ Print Assumptions of_trm_lift.
 Print Assumptions of_trm_open.
 Print Assumptions of_trm_close.
 Print Assumptions of_trm_injective.
+
+Lemma mfvs_lift_m : forall t k, mfvs (lift_m k t) = mfvs t.
+Proof.
+  induction t; intros k; simpl.
+  - destruct (Nat.ltb n k); reflexivity.
+  - reflexivity.
+  - reflexivity.
+  - rewrite IHt1, IHt2. reflexivity.
+  - rewrite IHt. reflexivity.
+  - rewrite IHt1, IHt2. reflexivity.
+Qed.
+
+Lemma mfvs_close_m : forall t x k y, In y (mfvs (close_rec_m x k t)) -> In y (mfvs t).
+Proof.
+  induction t; intros x k y H; simpl in *.
+  - destruct H.
+  - destruct (Nat.eqb a x) eqn:E.
+    + destruct H.
+    + destruct H as [Hy | Hf]. subst y. simpl. left. reflexivity. destruct Hf.
+  - exact H.
+  - apply in_app_iff in H as [H|H]; apply in_app_iff.
+    + left. apply (IHt1 x k y). exact H.
+    + right. apply (IHt2 x k y). exact H.
+  - apply (IHt x (S k) y). exact H.
+  - apply in_app_iff in H as [H|H]; apply in_app_iff.
+    + left. apply (IHt1 x (S k) y). exact H.
+    + right. apply (IHt2 x k y). exact H.
+Qed.
+
+Lemma mfvs_open_m : forall t k u y,
+  In y (mfvs (open_rec_m k u t)) -> In y (mfvs u) \/ In y (mfvs t).
+Proof.
+  induction t; intros k u y H; simpl in *.
+  - destruct (Nat.eqb n k) eqn:E.
+    + apply Nat.eqb_eq in E. subst n. simpl in H.
+      left. rewrite <- (mfvs_lift_m u k). exact H.
+    + simpl in H. destruct H.
+  - right. exact H.
+  - right. exact H.
+  - apply in_app_iff in H as [H|H].
+    + destruct (IHt1 k u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; left; exact Hr.
+    + destruct (IHt2 k u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; right; exact Hr.
+  - destruct (IHt (S k) u y H) as [Hl|Hr]. left; exact Hl. right; exact Hr.
+  - apply in_app_iff in H as [H|H].
+    + destruct (IHt1 (S k) u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; left; exact Hr.
+    + destruct (IHt2 k u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; right; exact Hr.
+Qed.
+
+Print Assumptions mfvs_lift_m.
+Print Assumptions mfvs_close_m.
+Print Assumptions mfvs_open_m.