From 0e1a6659b4bb0a4bb0f5394afb52cc952f093874 Mon Sep 17 00:00:00 2001
From: milner <m@cpl0.zip>
Date: Tue, 22 Sep 2026 23:04:00 +0200
Subject: [PATCH] add(named): congruence of the C equivalence and the R rule
 with capture avoidance (M5 named rules)

---
 graphs/dependency.dot |    7 +
 graphs/dependency.png |  Bin 611667 -> 622615 bytes
 graphs/dependency.svg | 1431 +++++++++++++++++++++--------------------
 graphs/theorems.json  |   53 ++
 theory/NamedMeta.v    |   37 +-
 5 files changed, 840 insertions(+), 688 deletions(-)

diff --git a/graphs/dependency.dot b/graphs/dependency.dot
index 4708428..f75a20f 100644
--- a/graphs/dependency.dot
+++ b/graphs/dependency.dot
@@ -49,6 +49,9 @@ digraph theorem_deps {
   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_mred_of_Plus_red1 [label="Plus_mred_of_Plus_red1", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_mred_of_Plus_red1 theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_full_comp_pure [label="mred_full_comp_pure", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_full_comp_pure theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_plus_one [label="mred_plus_one", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_plus_one 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];
@@ -70,6 +73,8 @@ digraph theorem_deps {
   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_ctx_any [label="Es_ctx_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_ctx_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  nred_R_intro [label="nred_R_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nred_R_intro 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];
@@ -155,6 +160,8 @@ digraph theorem_deps {
   mfvs_lift_m -> mfvs_open_m;
   of_trm_plug -> mred_term_context;
   of_trm_moccurs -> of_trm_moccurs0;
+  Plus_mred_of_Plus_red1 -> mred_full_comp_pure;
+  lemma_2_2_full_comp -> mred_full_comp_pure;
   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 eace39f51214a28e8416386d2c851d3111ccd0dd..e52f9592fb89f5ebbde95a44425f014fb8a6ca0c 100644
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-353" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-350.8" font-family="Inter" font-size="9.00" fill="#222222">full_comp_aux</text>
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+<g id="edge27" class="edge">
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+<g id="node112" class="node">
 <title>subst_notin</title>
-<g id="a_node107"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node112"><a xlink:title="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1">
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+<g id="edge68" class="edge">
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+<g id="node126" class="node">
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-<text text-anchor="middle" x="408.74" y="-254.8" font-family="Inter" font-size="9.00" fill="#222222">full_comp_aux_sub</text>
+<g id="a_node126"><a xlink:title="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-217" rx="54.7" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-214.8" font-family="Inter" font-size="9.00" fill="#222222">full_comp_aux_sub</text>
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+<g id="edge67" class="edge">
 <title>close_rec_notin&#45;&gt;subst_notin</title>
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-<text text-anchor="middle" x="239.88" y="-994.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_fvs</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-959" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-956.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_fvs</text>
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 <g id="node14" class="node">
 <title>subst_fvs</title>
 <g id="a_node14"><a xlink:title="subst_fvs theory/Binding.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="408.74" y="-994.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvs</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-959" rx="32.16" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-956.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvs</text>
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 <title>close_rec_fvs&#45;&gt;subst_fvs</title>
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-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="370.18,-999.1 376.18,-997 370.18,-994.9 370.18,-999.1"/>
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 <title>open_rec_bvar_neq</title>
 <g id="a_node9"><a xlink:title="open_rec_bvar_neq theory/Binding.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="65.24" y="-942.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_bvar_neq</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-907" rx="57.97" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-904.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_bvar_neq</text>
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 <!-- open_rec_bvar_neq&#45;&gt;open_rec_fvs -->
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 <title>open_rec_bvar_neq&#45;&gt;open_rec_fvs</title>
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 <!-- open_rec_fvs&#45;&gt;subst_fvs -->
 <g id="edge8" class="edge">
 <title>open_rec_fvs&#45;&gt;subst_fvs</title>
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 <!-- subst_lc_self -->
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+<g id="node113" class="node">
 <title>subst_lc_self</title>
-<g id="a_node108"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="577.6" y="-1046.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lc_self</text>
+<g id="a_node113"><a xlink:title="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="578.76" cy="-1011" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="578.76" y="-1008.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lc_self</text>
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 <!-- subst_fvar_self&#45;&gt;subst_lc_self -->
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+<g id="edge69" class="edge">
 <title>subst_fvar_self&#45;&gt;subst_lc_self</title>
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 <title>subst_fvar_other</title>
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-<text text-anchor="middle" x="65.24" y="-1150.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvar_other</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1115" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1112.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvar_other</text>
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+<g id="node114" class="node">
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-<text text-anchor="middle" x="239.88" y="-1150.8" font-family="Inter" font-size="9.00" fill="#222222">subst_other</text>
+<g id="a_node114"><a xlink:title="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1115" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1112.8" font-family="Inter" font-size="9.00" fill="#222222">subst_other</text>
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 <!-- subst_fvar_other&#45;&gt;subst_other -->
-<g id="edge68" class="edge">
+<g id="edge70" class="edge">
 <title>subst_fvar_other&#45;&gt;subst_other</title>
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 <title>Star_trans</title>
 <g id="a_node15"><a xlink:title="Star_trans theory/Closure.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="65.24" y="-1202.8" font-family="Inter" font-size="9.00" fill="#222222">Star_trans</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1167" rx="34.78" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1164.8" font-family="Inter" font-size="9.00" fill="#222222">Star_trans</text>
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 <title>red1_star</title>
 <g id="a_node16"><a xlink:title="red1_star theory/Closure.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="65.24" y="-1254.8" font-family="Inter" font-size="9.00" fill="#222222">red1_star</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1219" rx="31.35" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1216.8" font-family="Inter" font-size="9.00" fill="#222222">red1_star</text>
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@@ -324,32 +324,32 @@
 <g id="node17" class="node">
 <title>Plus_to_Star</title>
 <g id="a_node17"><a xlink:title="Plus_to_Star theory/Closure.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="577.6" y="-304.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_to_Star</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="578.76" cy="-249" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="578.76" y="-246.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_to_Star</text>
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+<g id="node128" class="node">
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-<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>
+<g id="a_node128"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-223" rx="65.64" ry="18"/>
+<text text-anchor="middle" x="755.12" y="-220.8" font-family="Inter" font-size="9.00" fill="#222222">star_red_sub_full_comp</text>
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+<g id="edge85" class="edge">
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-<text text-anchor="middle" x="65.24" y="-1306.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red1_context</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1271" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1268.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red1_context</text>
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-<text text-anchor="middle" x="65.24" y="-1358.8" font-family="Inter" font-size="9.00" fill="#222222">star_one_trans</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1323" rx="45.07" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1320.8" font-family="Inter" font-size="9.00" fill="#222222">star_one_trans</text>
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-<text text-anchor="middle" x="65.24" y="-1462.8" font-family="Inter" font-size="9.00" fill="#222222">in_comb</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1427" rx="30.04" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1424.8" font-family="Inter" font-size="9.00" fill="#222222">in_comb</text>
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-<text text-anchor="middle" x="239.88" y="-1436.8" font-family="Inter" font-size="9.00" fill="#222222">size_all_terms</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1401" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1398.8" font-family="Inter" font-size="9.00" fill="#222222">size_all_terms</text>
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 <title>in_comb&#45;&gt;size_all_terms</title>
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 <!-- pow2_pos -->
 <g id="node21" class="node">
 <title>pow2_pos</title>
 <g id="a_node21"><a xlink:title="pow2_pos theory/Enumerate.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="65.24" y="-1410.8" font-family="Inter" font-size="9.00" fill="#222222">pow2_pos</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1375" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1372.8" font-family="Inter" font-size="9.00" fill="#222222">pow2_pos</text>
 </a>
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 <!-- pow2_pos&#45;&gt;size_all_terms -->
 <g id="edge10" class="edge">
 <title>pow2_pos&#45;&gt;size_all_terms</title>
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 <!-- enumerate_check -->
 <g id="node23" class="node">
 <title>enumerate_check</title>
 <g id="a_node23"><a xlink:title="enumerate_check theory/Enumerate.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="1232.27" y="-738.8" font-family="Inter" font-size="9.00" fill="#222222">enumerate_check</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="1234.58" cy="-604" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="1234.58" y="-601.8" font-family="Inter" font-size="9.00" fill="#222222">enumerate_check</text>
 </a>
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@@ -414,8 +414,8 @@
 <g id="node24" class="node">
 <title>enumerate_sound</title>
 <g id="a_node24"><a xlink:title="enumerate_sound theory/Enumerate.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="752.81" y="-708.8" font-family="Inter" font-size="9.00" fill="#222222">enumerate_sound</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-509" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="755.12" y="-506.8" font-family="Inter" font-size="9.00" fill="#222222">enumerate_sound</text>
 </a>
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@@ -423,8 +423,8 @@
 <g id="node25" class="node">
 <title>enumerate_complete</title>
 <g id="a_node25"><a xlink:title="enumerate_complete theory/Enumerate.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="752.81" cy="-521" rx="61.55" ry="18"/>
-<text text-anchor="middle" x="752.81" y="-518.8" font-family="Inter" font-size="9.00" fill="#222222">enumerate_complete</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-803" rx="61.55" ry="18"/>
+<text text-anchor="middle" x="755.12" y="-800.8" font-family="Inter" font-size="9.00" fill="#222222">enumerate_complete</text>
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@@ -432,8 +432,8 @@
 <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>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1479" rx="34.78" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1476.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_fvs</text>
 </a>
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@@ -441,8 +441,8 @@
 <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>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1531" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1528.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_lift</text>
 </a>
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@@ -450,23 +450,23 @@
 <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>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1531" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1528.8" 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>
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 </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">
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-<text text-anchor="middle" x="65.24" y="-1618.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_close</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1583" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1580.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_close</text>
 </a>
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@@ -474,8 +474,8 @@
 <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>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1635" rx="48.49" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1632.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_injective</text>
 </a>
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@@ -483,8 +483,8 @@
 <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>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1687" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1684.8" font-family="Inter" font-size="9.00" fill="#222222">mfvs_lift_m</text>
 </a>
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@@ -492,23 +492,23 @@
 <g id="node33" class="node">
 <title>mfvs_open_m</title>
 <g id="a_node33"><a xlink:title="mfvs_open_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1725" rx="43.1" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-1722.8" font-family="Inter" font-size="9.00" fill="#222222">mfvs_open_m</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1687" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1684.8" font-family="Inter" font-size="9.00" fill="#222222">mfvs_open_m</text>
 </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>
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+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M102.85,-1687C128.34,-1687 162.6,-1687 190.36,-1687"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="190.54,-1689.1 196.54,-1687 190.54,-1684.9 190.54,-1689.1"/>
 </g>
 <!-- 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>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1739" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1736.8" font-family="Inter" font-size="9.00" fill="#222222">mfvs_close_m</text>
 </a>
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@@ -516,8 +516,8 @@
 <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"/>
-<text text-anchor="middle" x="65.24" y="-1826.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_plug</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1791" rx="38.21" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1788.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_plug</text>
 </a>
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@@ -525,23 +525,23 @@
 <g id="node37" class="node">
 <title>mred_term_context</title>
 <g id="a_node37"><a xlink:title="mred_term_context theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1829" rx="56.66" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-1826.8" font-family="Inter" font-size="9.00" fill="#222222">mred_term_context</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1791" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1788.8" font-family="Inter" font-size="9.00" fill="#222222">mred_term_context</text>
 </a>
 </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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+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M103.7,-1791C125.09,-1791 152.45,-1791 176.96,-1791"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="177.12,-1793.1 183.12,-1791 177.12,-1788.9 177.12,-1793.1"/>
 </g>
 <!-- mred_of_red1 -->
 <g id="node35" class="node">
 <title>mred_of_red1</title>
 <g id="a_node35"><a xlink:title="mred_of_red1 theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1881" rx="41.64" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1878.8" font-family="Inter" font-size="9.00" fill="#222222">mred_of_red1</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1843" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1840.8" font-family="Inter" font-size="9.00" fill="#222222">mred_of_red1</text>
 </a>
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@@ -549,8 +549,8 @@
 <g id="node36" class="node">
 <title>mred_context</title>
 <g id="a_node36"><a xlink:title="mred_context theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1933" rx="42.95" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1930.8" font-family="Inter" font-size="9.00" fill="#222222">mred_context</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1895" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1892.8" font-family="Inter" font-size="9.00" fill="#222222">mred_context</text>
 </a>
 </g>
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@@ -558,8 +558,8 @@
 <g id="node38" class="node">
 <title>of_trm_moccurs</title>
 <g id="a_node38"><a xlink:title="of_trm_moccurs theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1985" rx="49.8" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-1982.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_moccurs</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1947" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1944.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_moccurs</text>
 </a>
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@@ -567,23 +567,23 @@
 <g id="node39" class="node">
 <title>of_trm_moccurs0</title>
 <g id="a_node39"><a xlink:title="of_trm_moccurs0 theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1985" rx="53.23" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-1982.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_moccurs0</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-1947" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-1944.8" font-family="Inter" font-size="9.00" fill="#222222">of_trm_moccurs0</text>
 </a>
 </g>
 </g>
 <!-- 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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-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="180.73,-1987.1 186.73,-1985 180.73,-1982.9 180.73,-1987.1"/>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M115.25,-1947C135.57,-1947 159.31,-1947 180.53,-1947"/>
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 </g>
 <!-- mred_B_intro -->
 <g id="node40" class="node">
 <title>mred_B_intro</title>
 <g id="a_node40"><a xlink:title="mred_B_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2037" rx="41.64" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2034.8" font-family="Inter" font-size="9.00" fill="#222222">mred_B_intro</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-1999" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-1996.8" font-family="Inter" font-size="9.00" fill="#222222">mred_B_intro</text>
 </a>
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@@ -591,8 +591,8 @@
 <g id="node41" class="node">
 <title>mred_gc_intro</title>
 <g id="a_node41"><a xlink:title="mred_gc_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2089" rx="43.76" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2086.8" font-family="Inter" font-size="9.00" fill="#222222">mred_gc_intro</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2051" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2048.8" font-family="Inter" font-size="9.00" fill="#222222">mred_gc_intro</text>
 </a>
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@@ -600,1153 +600,1210 @@
 <g id="node42" class="node">
 <title>mred_r_intro</title>
 <g id="a_node42"><a xlink:title="mred_r_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2141" rx="40.33" ry="18"/>
-<text text-anchor="middle" x="65.24" y="-2138.8" font-family="Inter" font-size="9.00" fill="#222222">mred_r_intro</text>
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2103" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2100.8" font-family="Inter" font-size="9.00" fill="#222222">mred_r_intro</text>
+</a>
+</g>
+</g>
+<!-- Plus_mred_of_Plus_red1 -->
+<g id="node43" class="node">
+<title>Plus_mred_of_Plus_red1</title>
+<g id="a_node43"><a xlink:title="Plus_mred_of_Plus_red1 theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="578.76" cy="-405" rx="67.6" ry="18"/>
+<text text-anchor="middle" x="578.76" y="-402.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_mred_of_Plus_red1</text>
+</a>
+</g>
+</g>
+<!-- mred_full_comp_pure -->
+<g id="node44" class="node">
+<title>mred_full_comp_pure</title>
+<g id="a_node44"><a xlink:title="mred_full_comp_pure theory/Metaterm.v https://arxiv.org/html/2312.13270v1">
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+</g>
+<!-- Plus_mred_of_Plus_red1&#45;&gt;mred_full_comp_pure -->
+<g id="edge18" class="edge">
+<title>Plus_mred_of_Plus_red1&#45;&gt;mred_full_comp_pure</title>
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+<!-- mred_plus_one -->
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+<g id="edge19" class="edge">
+<title>lemma_2_2_full_comp&#45;&gt;mred_full_comp_pure</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M633.02,-353C650.3,-353 669.68,-353 687.79,-353"/>
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-<g id="node55" class="node">
+<g id="node58" class="node">
 <title>red1_root_fv</title>
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+<g id="edge33" class="edge">
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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>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-907" rx="42.95" ry="18"/>
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+<g id="node59" class="node">
 <title>red1r_fv</title>
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+<g id="edge34" class="edge">
 <title>plug_fv_mono&#45;&gt;red1r_fv</title>
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+<g id="edge35" class="edge">
 <title>red1_root_fv&#45;&gt;red1r_fv</title>
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 <title>lemma_2_1_fv_preserved</title>
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+<g id="edge36" class="edge">
 <title>red1r_fv&#45;&gt;lemma_2_1_fv_preserved</title>
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 <title>gen_fv_preserved</title>
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+<g id="a_node83"><a xlink:title="gen_fv_preserved theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="923.99" cy="-755" rx="52.58" ry="18"/>
+<text text-anchor="middle" x="923.99" y="-752.8" font-family="Inter" font-size="9.00" fill="#222222">gen_fv_preserved</text>
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 </g>
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+<g id="edge46" class="edge">
 <title>lemma_2_1_fv_preserved&#45;&gt;gen_fv_preserved</title>
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 <title>corpus_fv_preserved</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="923.99" cy="-831" rx="60.09" ry="18"/>
+<text text-anchor="middle" x="923.99" y="-828.8" 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="edge91" class="edge">
+<g id="edge93" class="edge">
 <title>lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved</title>
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+<g id="node62" class="node">
 <title>Plus_trans</title>
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-<text text-anchor="middle" x="65.24" y="-2190.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_trans</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2207" rx="34.78" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2204.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_trans</text>
 </a>
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 <title>Plus_step_trans</title>
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-<text text-anchor="middle" x="65.24" y="-2242.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_step_trans</text>
+<g id="a_node63"><a xlink:title="Plus_step_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2259" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2256.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_step_trans</text>
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+<g id="node64" class="node">
 <title>isubst_meta_in</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2311" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2308.8" font-family="Inter" font-size="9.00" fill="#222222">isubst_meta_in</text>
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 <title>isubst_meta_notin</title>
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-<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>
+<g id="a_node65"><a xlink:title="isubst_meta_notin theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2363" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2360.8" font-family="Inter" font-size="9.00" fill="#222222">isubst_meta_notin</text>
 </a>
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 <title>eqC_in_Es</title>
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-<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>
+<g id="a_node66"><a xlink:title="eqC_in_Es theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2415" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2412.8" font-family="Inter" font-size="9.00" fill="#222222">eqC_in_Es</text>
+</a>
+</g>
+</g>
+<!-- Es_ctx_any -->
+<g id="node67" class="node">
+<title>Es_ctx_any</title>
+<g id="a_node67"><a xlink:title="Es_ctx_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2467" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2464.8" font-family="Inter" font-size="9.00" fill="#222222">Es_ctx_any</text>
+</a>
+</g>
+</g>
+<!-- nred_R_intro -->
+<g id="node68" class="node">
+<title>nred_R_intro</title>
+<g id="a_node68"><a xlink:title="nred_R_intro theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2519" rx="39.67" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2516.8" font-family="Inter" font-size="9.00" fill="#222222">nred_R_intro</text>
 </a>
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 </g>
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+<g id="node69" class="node">
 <title>Es_refl_any</title>
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-<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>
+<g id="a_node69"><a xlink:title="Es_refl_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2571" rx="36.9" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2568.8" font-family="Inter" font-size="9.00" fill="#222222">Es_refl_any</text>
 </a>
 </g>
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 <!-- Es_sym_any -->
-<g id="node65" class="node">
+<g id="node70" class="node">
 <title>Es_sym_any</title>
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-<text text-anchor="middle" x="65.24" y="-2502.8" font-family="Inter" font-size="9.00" fill="#222222">Es_sym_any</text>
+<g id="a_node70"><a xlink:title="Es_sym_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2623" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2620.8" font-family="Inter" font-size="9.00" fill="#222222">Es_sym_any</text>
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 <!-- Es_trans_any -->
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 <title>Es_trans_any</title>
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-<text text-anchor="middle" x="65.24" y="-2554.8" font-family="Inter" font-size="9.00" fill="#222222">Es_trans_any</text>
+<g id="a_node71"><a xlink:title="Es_trans_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1">
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+<text text-anchor="middle" x="65.24" y="-2672.8" font-family="Inter" font-size="9.00" fill="#222222">Es_trans_any</text>
 </a>
 </g>
 </g>
 <!-- par_red1_iff -->
-<g id="node67" class="node">
+<g id="node72" class="node">
 <title>par_red1_iff</title>
-<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>
+<g id="a_node72"><a xlink:title="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2727" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2724.8" font-family="Inter" font-size="9.00" fill="#222222">par_red1_iff</text>
 </a>
 </g>
 </g>
 <!-- par_to_red1_or_eq -->
-<g id="node70" class="node">
+<g id="node75" class="node">
 <title>par_to_red1_or_eq</title>
-<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>
+<g id="a_node75"><a xlink:title="par_to_red1_or_eq theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-2753" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2750.8" font-family="Inter" font-size="9.00" fill="#222222">par_to_red1_or_eq</text>
 </a>
 </g>
 </g>
 <!-- par_red1_iff&#45;&gt;par_to_red1_or_eq -->
-<g id="edge37" class="edge">
+<g id="edge39" class="edge">
 <title>par_red1_iff&#45;&gt;par_to_red1_or_eq</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M101.18,-2614.26C125.17,-2617.88 157.5,-2622.74 184.81,-2626.86"/>
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 </g>
 <!-- red1_or_eq_to_par -->
-<g id="node71" class="node">
+<g id="node76" class="node">
 <title>red1_or_eq_to_par</title>
-<g id="a_node71"><a xlink:title="red1_or_eq_to_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="239.88" y="-2580.8" font-family="Inter" font-size="9.00" fill="#222222">red1_or_eq_to_par</text>
+<g id="a_node76"><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="-2701" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-2698.8" font-family="Inter" font-size="9.00" fill="#222222">red1_or_eq_to_par</text>
 </a>
 </g>
 </g>
 <!-- par_red1_iff&#45;&gt;red1_or_eq_to_par -->
-<g id="edge38" class="edge">
+<g id="edge40" 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,-2603.74C125.17,-2600.12 157.5,-2595.26 184.81,-2591.14"/>
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 <!-- red1_in_par -->
-<g id="node68" class="node">
+<g id="node73" class="node">
 <title>red1_in_par</title>
-<g id="a_node68"><a xlink:title="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="65.24" y="-2658.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
+<g id="a_node73"><a xlink:title="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2779" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2776.8" font-family="Inter" font-size="9.00" fill="#222222">red1_in_par</text>
 </a>
 </g>
 </g>
 <!-- par_context -->
-<g id="node69" class="node">
+<g id="node74" class="node">
 <title>par_context</title>
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-<text text-anchor="middle" x="65.24" y="-2710.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2831" rx="38.87" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2828.8" font-family="Inter" font-size="9.00" fill="#222222">par_context</text>
 </a>
 </g>
 </g>
 <!-- par_reflexive -->
-<g id="node72" class="node">
+<g id="node77" class="node">
 <title>par_reflexive</title>
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-<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>
+<g id="a_node77"><a xlink:title="par_reflexive theory/Parallel.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2883" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2880.8" font-family="Inter" font-size="9.00" fill="#222222">par_reflexive</text>
 </a>
 </g>
 </g>
 <!-- gen_list_length -->
-<g id="node73" class="node">
+<g id="node78" class="node">
 <title>gen_list_length</title>
-<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>
+<g id="a_node78"><a xlink:title="gen_list_length theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2935" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2932.8" font-family="Inter" font-size="9.00" fill="#222222">gen_list_length</text>
 </a>
 </g>
 </g>
 <!-- check_term_true -->
-<g id="node74" class="node">
+<g id="node79" class="node">
 <title>check_term_true</title>
-<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>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="923.99" cy="-604" rx="50.61" ry="18"/>
+<text text-anchor="middle" x="923.99" y="-601.8" font-family="Inter" font-size="9.00" fill="#222222">check_term_true</text>
 </a>
 </g>
 </g>
 <!-- check_all_true -->
-<g id="node75" class="node">
+<g id="node80" class="node">
 <title>check_all_true</title>
-<g id="a_node75"><a xlink:title="check_all_true theory/Random.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="1080.15" cy="-741" rx="44.41" ry="18"/>
-<text text-anchor="middle" x="1080.15" y="-738.8" font-family="Inter" font-size="9.00" fill="#222222">check_all_true</text>
+<g id="a_node80"><a xlink:title="check_all_true theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="1082.46" cy="-604" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="1082.46" y="-601.8" font-family="Inter" font-size="9.00" fill="#222222">check_all_true</text>
 </a>
 </g>
 </g>
 <!-- check_term_true&#45;&gt;check_all_true -->
-<g id="edge41" class="edge">
+<g id="edge43" class="edge">
 <title>check_term_true&#45;&gt;check_all_true</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M972.62,-741C990.78,-741 1011.29,-741 1029.43,-741"/>
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+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M974.92,-604C993.08,-604 1013.6,-604 1031.74,-604"/>
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 </g>
 <!-- check_all_true&#45;&gt;enumerate_check -->
 <g id="edge11" class="edge">
 <title>check_all_true&#45;&gt;enumerate_check</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M1124.97,-741C1139.72,-741 1156.42,-741 1172.14,-741"/>
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 </g>
 <!-- gen_sound -->
-<g id="node76" class="node">
+<g id="node81" class="node">
 <title>gen_sound</title>
-<g id="a_node76"><a xlink:title="gen_sound theory/Random.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="752.81" cy="-867" rx="35.59" ry="18"/>
-<text text-anchor="middle" x="752.81" y="-864.8" font-family="Inter" font-size="9.00" fill="#222222">gen_sound</text>
+<g id="a_node81"><a xlink:title="gen_sound theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-457" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="755.12" y="-454.8" font-family="Inter" font-size="9.00" fill="#222222">gen_sound</text>
 </a>
 </g>
 </g>
 <!-- gen_complete -->
-<g id="node77" class="node">
+<g id="node82" class="node">
 <title>gen_complete</title>
-<g id="a_node77"><a xlink:title="gen_complete theory/Random.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="752.81" cy="-469" rx="43.76" ry="18"/>
-<text text-anchor="middle" x="752.81" y="-466.8" font-family="Inter" font-size="9.00" fill="#222222">gen_complete</text>
+<g id="a_node82"><a xlink:title="gen_complete theory/Random.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-751" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="755.12" y="-748.8" font-family="Inter" font-size="9.00" fill="#222222">gen_complete</text>
 </a>
 </g>
 </g>
 <!-- zdecs_sound -->
-<g id="node79" class="node">
+<g id="node84" class="node">
 <title>zdecs_sound</title>
-<g id="a_node79"><a xlink:title="zdecs_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<g id="a_node84"><a xlink:title="zdecs_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
 <ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-450" rx="40.98" ry="18"/>
 <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>
 <!-- zdecs_sound&#45;&gt;full_comp_aux -->
-<g id="edge28" class="edge">
+<g id="edge30" class="edge">
 <title>zdecs_sound&#45;&gt;full_comp_aux</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M272.44,-439.05C283.39,-434.84 295.59,-429.67 306.27,-424 333.51,-409.55 362.38,-389.38 382.33,-374.58"/>
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 </g>
 <!-- root_steps_sound -->
-<g id="node82" class="node">
+<g id="node87" class="node">
 <title>root_steps_sound</title>
-<g id="a_node82"><a xlink:title="root_steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-729" rx="51.27" ry="18"/>
-<text text-anchor="middle" x="408.74" y="-726.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_sound</text>
+<g id="a_node87"><a xlink:title="root_steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-451" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-448.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_sound</text>
 </a>
 </g>
 </g>
 <!-- zdecs_sound&#45;&gt;root_steps_sound -->
-<g id="edge46" class="edge">
+<g id="edge48" class="edge">
 <title>zdecs_sound&#45;&gt;root_steps_sound</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M277.52,-457.42C288.2,-461.28 298.96,-467.14 306.27,-476 371.62,-555.21 285.52,-622.5 349.27,-703 353,-707.7 357.75,-711.58 362.93,-714.77"/>
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 </g>
 <!-- zdecs_sound&#45;&gt;full_comp_aux_sub -->
-<g id="edge81" class="edge">
+<g id="edge83" class="edge">
 <title>zdecs_sound&#45;&gt;full_comp_aux_sub</title>
-<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M276.31,-441.47C286.94,-437.54 297.96,-431.93 306.27,-424 339.8,-392 324.91,-368.43 349.27,-329 360.28,-311.19 375.19,-292.9 387.2,-279.21"/>
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 <!-- zdecs_complete -->
-<g id="node80" class="node">
+<g id="node85" class="node">
 <title>zdecs_complete</title>
-<g id="a_node80"><a xlink:title="zdecs_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-677" rx="49.15" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-674.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_complete</text>
+<g id="a_node85"><a xlink:title="zdecs_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-699" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-696.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_complete</text>
 </a>
 </g>
 </g>
 <!-- root_steps_complete -->
-<g id="node83" class="node">
+<g id="node88" class="node">
 <title>root_steps_complete</title>
-<g id="a_node83"><a xlink:title="root_steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
-<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-677" rx="59.43" ry="18"/>
-<text text-anchor="middle" x="408.74" y="-674.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_complete</text>
+<g id="a_node88"><a xlink:title="root_steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-699" rx="59.43" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-696.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_complete</text>
 </a>
 </g>
 </g>
 <!-- zdecs_complete&#45;&gt;root_steps_complete -->
-<g id="edge47" class="edge">
+<g id="edge49" class="edge">
 <title>zdecs_complete&#45;&gt;root_steps_complete</title>
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-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="343.23,-679.1 349.23,-677 343.23,-674.9 343.23,-679.1"/>
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-<g id="node81" class="node">
+<g id="node86" class="node">
 <title>positions_at</title>
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 <title>steps_sound&#45;&gt;enumerate_sound</title>
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+<g id="edge90" class="edge">
 <title>steps_sound&#45;&gt;corpus_sound</title>
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 <!-- at_ctx_comp -->
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 <title>at_ctx_comp</title>
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+<g id="edge54" class="edge">
 <title>at_ctx_comp&#45;&gt;steps_complete</title>
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 <g id="edge13" class="edge">
 <title>steps_complete&#45;&gt;enumerate_complete</title>
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+<g id="edge60" class="edge">
 <title>steps_complete&#45;&gt;has_red_spec</title>
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+<g id="edge63" class="edge">
 <title>steps_complete&#45;&gt;nf_iff_steps_nil</title>
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 <!-- corpus_complete -->
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 <title>corpus_complete</title>
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-<text text-anchor="middle" x="752.81" y="-570.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_complete</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="755.12" cy="-699" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="755.12" y="-696.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_complete</text>
 </a>
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 </g>
 <!-- steps_complete&#45;&gt;corpus_complete -->
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+<g id="edge91" class="edge">
 <title>steps_complete&#45;&gt;corpus_complete</title>
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-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="695.2,-575.1 701.2,-573 695.2,-570.9 695.2,-575.1"/>
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 <!-- has_red_f -->
-<g id="node92" class="node">
+<g id="node97" class="node">
 <title>has_red_f</title>
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-<text text-anchor="middle" x="577.6" y="-656.8" font-family="Inter" font-size="9.00" fill="#222222">has_red_f</text>
+<g id="a_node97"><a xlink:title="has_red_f theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="578.76" cy="-561" rx="32.16" ry="18"/>
+<text text-anchor="middle" x="578.76" y="-558.8" font-family="Inter" font-size="9.00" fill="#222222">has_red_f</text>
 </a>
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 </g>
 <!-- has_red_f&#45;&gt;has_red_spec -->
-<g id="edge57" class="edge">
+<g id="edge59" class="edge">
 <title>has_red_f&#45;&gt;has_red_spec</title>
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-<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="703.92,-661.1 709.92,-659 703.92,-656.9 703.92,-661.1"/>
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 <!-- has_red_spec&#45;&gt;check_term_true -->
-<g id="edge39" class="edge">
+<g id="edge41" class="edge">
 <title>has_red_spec&#45;&gt;check_term_true</title>
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 <!-- red1_dec -->
-<g id="node94" class="node">
+<g id="node99" class="node">
 <title>red1_dec</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="921.68" cy="-689" rx="30.7" ry="18"/>
-<text text-anchor="middle" x="921.68" y="-686.8" font-family="Inter" font-size="9.00" fill="#222222">red1_dec</text>
+<g id="a_node99"><a xlink:title="red1_dec theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="923.99" cy="-500" rx="30.7" ry="18"/>
+<text text-anchor="middle" x="923.99" y="-497.8" font-family="Inter" font-size="9.00" fill="#222222">red1_dec</text>
 </a>
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 </g>
 <!-- has_red_spec&#45;&gt;red1_dec -->
-<g id="edge60" class="edge">
+<g id="edge62" class="edge">
 <title>has_red_spec&#45;&gt;red1_dec</title>
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 <!-- corpus_decides -->
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 <title>corpus_decides</title>
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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>
+<g id="a_node135"><a xlink:title="corpus_decides theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="923.99" cy="-552" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="923.99" y="-549.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_decides</text>
 </a>
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 </g>
 <!-- has_red_spec&#45;&gt;corpus_decides -->
-<g id="edge90" class="edge">
+<g id="edge92" class="edge">
 <title>has_red_spec&#45;&gt;corpus_decides</title>
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 <!-- normal_form_iff_nf -->
-<g id="node96" class="node">
+<g id="node101" class="node">
 <title>normal_form_iff_nf</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="1080.15" cy="-611" rx="55.35" ry="18"/>
-<text text-anchor="middle" x="1080.15" y="-608.8" font-family="Inter" font-size="9.00" fill="#222222">normal_form_iff_nf</text>
+<g id="a_node101"><a xlink:title="normal_form_iff_nf theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="1082.46" cy="-708" rx="55.35" ry="18"/>
+<text text-anchor="middle" x="1082.46" y="-705.8" font-family="Inter" font-size="9.00" fill="#222222">normal_form_iff_nf</text>
 </a>
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 </g>
 <!-- nf_iff_steps_nil&#45;&gt;normal_form_iff_nf -->
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+<g id="edge65" class="edge">
 <title>nf_iff_steps_nil&#45;&gt;normal_form_iff_nf</title>
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 <!-- nf_dec -->
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 <title>nf_dec</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="1080.15" cy="-663" rx="27" ry="18"/>
-<text text-anchor="middle" x="1080.15" y="-660.8" font-family="Inter" font-size="9.00" fill="#222222">nf_dec</text>
+<g id="a_node102"><a xlink:title="nf_dec theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="1082.46" cy="-656" rx="27" ry="18"/>
+<text text-anchor="middle" x="1082.46" y="-653.8" font-family="Inter" font-size="9.00" fill="#222222">nf_dec</text>
 </a>
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 <!-- nf_iff_steps_nil&#45;&gt;nf_dec -->
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+<g id="edge66" class="edge">
 <title>nf_iff_steps_nil&#45;&gt;nf_dec</title>
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 <!-- open_rec_app -->
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 <title>open_rec_app</title>
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-<text text-anchor="middle" x="65.24" y="-2866.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_app</text>
+<g id="a_node103"><a xlink:title="open_rec_app theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-2987" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-2984.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_app</text>
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 <!-- open_rec_lam -->
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-<text text-anchor="middle" x="65.24" y="-2918.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lam</text>
+<g id="a_node104"><a xlink:title="open_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3039" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3036.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lam</text>
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 <!-- open_rec_esub -->
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 <title>open_rec_esub</title>
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-<text text-anchor="middle" x="65.24" y="-2970.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_esub</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3091" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3088.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_esub</text>
 </a>
 </g>
 </g>
 <!-- close_rec_app -->
-<g id="node101" class="node">
+<g id="node106" class="node">
 <title>close_rec_app</title>
-<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>
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+<text text-anchor="middle" x="65.24" y="-3140.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_app</text>
 </a>
 </g>
 </g>
 <!-- close_rec_lam -->
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+<g id="node107" class="node">
 <title>close_rec_lam</title>
-<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>
+<g id="a_node107"><a xlink:title="close_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3195" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3192.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_lam</text>
 </a>
 </g>
 </g>
 <!-- close_rec_esub -->
-<g id="node103" class="node">
+<g id="node108" class="node">
 <title>close_rec_esub</title>
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-<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>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3247" rx="46.53" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3244.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_esub</text>
 </a>
 </g>
 </g>
 <!-- subst_app -->
-<g id="node104" class="node">
+<g id="node109" class="node">
 <title>subst_app</title>
-<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>
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+<text text-anchor="middle" x="65.24" y="-3296.8" font-family="Inter" font-size="9.00" fill="#222222">subst_app</text>
 </a>
 </g>
 </g>
 <!-- subst_lam -->
-<g id="node105" class="node">
+<g id="node110" class="node">
 <title>subst_lam</title>
-<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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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3351" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3348.8" font-family="Inter" font-size="9.00" fill="#222222">subst_lam</text>
 </a>
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 <!-- subst_esub -->
-<g id="node106" class="node">
+<g id="node111" class="node">
 <title>subst_esub</title>
-<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>
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+<text text-anchor="middle" x="65.24" y="-3400.8" font-family="Inter" font-size="9.00" fill="#222222">subst_esub</text>
 </a>
 </g>
 </g>
 <!-- open_rec_lc_atom -->
-<g id="node110" class="node">
+<g id="node115" class="node">
 <title>open_rec_lc_atom</title>
-<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>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3455" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-3452.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc_atom</text>
 </a>
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 <!-- open_rec_lc -->
-<g id="node111" class="node">
+<g id="node116" class="node">
 <title>open_rec_lc</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3337" rx="39.02" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-3334.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
+<g id="a_node116"><a xlink:title="open_rec_lc theory/Substitution.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3455" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-3452.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_lc</text>
 </a>
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 <!-- open_rec_lc_atom&#45;&gt;open_rec_lc -->
-<g id="edge69" class="edge">
+<g id="edge71" class="edge">
 <title>open_rec_lc_atom&#45;&gt;open_rec_lc</title>
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 <!-- red_sub_root_to_red1 -->
-<g id="node112" class="node">
+<g id="node117" class="node">
 <title>red_sub_root_to_red1</title>
-<g id="a_node112"><a xlink:title="red_sub_root_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="65.24" y="-3386.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_root_to_red1</text>
+<g id="a_node117"><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="-3507" rx="60.09" ry="18"/>
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 <!-- red_sub_to_red1 -->
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+<g id="node118" class="node">
 <title>red_sub_to_red1</title>
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-<text text-anchor="middle" x="239.88" y="-3386.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_to_red1</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3507" rx="47.84" ry="18"/>
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 <!-- red_sub_root_to_red1&#45;&gt;red_sub_to_red1 -->
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 <title>red_sub_root_to_red1&#45;&gt;red_sub_to_red1</title>
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 <!-- red_sub_context -->
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+<g id="node119" class="node">
 <title>red_sub_context</title>
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-<text text-anchor="middle" x="65.24" y="-3438.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_context</text>
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+<text text-anchor="middle" x="65.24" y="-3556.8" 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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+<g id="node125" class="node">
 <title>Star_red_sub_context</title>
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-<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3441" rx="61.55" ry="18"/>
-<text text-anchor="middle" x="239.88" y="-3438.8" font-family="Inter" font-size="9.00" fill="#222222">Star_red_sub_context</text>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-3559" rx="61.55" ry="18"/>
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 <!-- red_sub_context&#45;&gt;Star_red_sub_context -->
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+<g id="edge75" class="edge">
 <title>red_sub_context&#45;&gt;Star_red_sub_context</title>
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 <!-- red_sub_gc -->
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+<g id="node120" class="node">
 <title>red_sub_gc</title>
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+<g id="a_node120"><a xlink:title="red_sub_gc theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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 <text text-anchor="middle" x="239.88" y="-67.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_gc</text>
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 <!-- red_sub_gc&#45;&gt;full_comp_aux_sub -->
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+<g id="edge80" class="edge">
 <title>red_sub_gc&#45;&gt;full_comp_aux_sub</title>
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 <!-- red_gc_to_red_sub -->
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 <title>red_gc_to_red_sub</title>
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+<g id="a_node132"><a xlink:title="red_gc_to_red_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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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>
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 <!-- red_sub_gc&#45;&gt;red_gc_to_red_sub -->
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+<g id="edge89" class="edge">
 <title>red_sub_gc&#45;&gt;red_gc_to_red_sub</title>
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 <!-- red_sub_r -->
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+<g id="node121" class="node">
 <title>red_sub_r</title>
-<g id="a_node116"><a xlink:title="red_sub_r theory/Subsystem.v https://arxiv.org/html/2312.13270v1">
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-<text text-anchor="middle" x="239.88" y="-153.8" font-family="Inter" font-size="9.00" fill="#222222">red_sub_r</text>
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 <!-- red_sub_r&#45;&gt;full_comp_aux_sub -->
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+<g id="edge81" class="edge">
 <title>red_sub_r&#45;&gt;full_comp_aux_sub</title>
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 <title>occurs_zfill</title>
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+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-3611" rx="37.56" ry="18"/>
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 <!-- zfill_occurs0 -->
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+<g id="node123" class="node">
 <title>zfill_occurs0</title>
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 <title>occurs_zfill&#45;&gt;zfill_occurs0</title>
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 <!-- full_comp_aux_sub&#45;&gt;plus_red_sub_full_comp -->
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+<g id="edge84" class="edge">
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 <!-- plus_red_sub_full_comp&#45;&gt;star_red_sub_full_comp -->
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+<g id="edge86" class="edge">
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+<g id="node129" class="node">
 <title>es_count_plug_mono</title>
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 </a>
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 <!-- red_gc_measure -->
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+<g id="node130" class="node">
 <title>red_gc_measure</title>
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 </g>
 <!-- es_count_plug_mono&#45;&gt;red_gc_measure -->
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+<g id="edge87" class="edge">
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diff --git a/graphs/theorems.json b/graphs/theorems.json
index 1ed9d7a..672b9e8 100644
--- a/graphs/theorems.json
+++ b/graphs/theorems.json
@@ -467,6 +467,39 @@
       "file": "theory/Metaterm.v",
       "depends": []
     },
+    {
+      "id": "Plus_mred_of_Plus_red1",
+      "label": "Plus_mred_of_Plus_red1",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_full_comp_pure",
+      "label": "mred_full_comp_pure",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "Plus_mred_of_Plus_red1",
+        "lemma_2_2_full_comp"
+      ]
+    },
+    {
+      "id": "mred_plus_one",
+      "label": "mred_plus_one",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
     {
       "id": "Plus_red1_context",
       "label": "Plus_red1_context",
@@ -706,6 +739,26 @@
       "file": "theory/NamedMeta.v",
       "depends": []
     },
+    {
+      "id": "Es_ctx_any",
+      "label": "Es_ctx_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "nred_R_intro",
+      "label": "nred_R_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
     {
       "id": "Es_refl_any",
       "label": "Es_refl_any",
diff --git a/theory/NamedMeta.v b/theory/NamedMeta.v
index 25640d4..a267974 100644
--- a/theory/NamedMeta.v
+++ b/theory/NamedMeta.v
@@ -70,7 +70,21 @@ Inductive Es : ntrm -> ntrm -> Prop :=
 | 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).
+    Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u)
+| Es_ctx : forall C t t', Es t t' -> Es (nplug C t) (nplug C t').
+
+Fixpoint cbinders (C : nctx) : list atom :=
+  match C with
+  | nHole => []
+  | nAppL C1 _ => cbinders C1
+  | nAppR _ C1 => cbinders C1
+  | nLam x C1 => x :: cbinders C1
+  | nESubL C1 _ _ => cbinders C1
+  | nESubR _ _ C1 => cbinders C1
+  end.
+
+Definition cavoid (C : nctx) (phi : list atom) : bool :=
+  forallb (fun y => negb (existsb (Nat.eqb y) phi)) (cbinders C).
 
 Inductive nred : ntrm -> ntrm -> Prop :=
 | nred_B : forall t x u, nred (NApp (NLam x t) u) (NESub t x u)
@@ -82,6 +96,11 @@ Inductive nred : ntrm -> ntrm -> Prop :=
     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_R : forall C x u phi,
+    In x phi ->
+    (forall y, In y (nfv u) -> In y phi) ->
+    cavoid C phi = true ->
+    nred (NESub (nplug C (NVar x)) x u) (NESub (nplug C 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,
@@ -89,6 +108,20 @@ Lemma eqC_in_Es : forall t x u y 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_ctx_any : forall C t t', Es t t' -> Es (nplug C t) (nplug C t').
+Proof. intros. apply Es_ctx. exact H. Qed.
+
+Lemma nred_R_intro : forall C x u phi,
+  In x phi -> (forall y, In y (nfv u) -> In y phi) -> cavoid C phi = true ->
+  nred (NESub (nplug C (NVar x)) x u) (NESub (nplug C u) x u).
+Proof.
+  intros C x u phi Hx Hu Hc.
+  apply nred_R with (phi := phi).
+  - exact Hx.
+  - exact Hu.
+  - exact Hc.
+Qed.
+
 Lemma Es_refl_any : forall t, Es t t.
 Proof. intros. apply Es_refl. Qed.
 
@@ -100,6 +133,8 @@ Proof. intros. apply Es_trans with (u := u); assumption. Qed.
 
 Print Assumptions isubst_meta_in.
 Print Assumptions isubst_meta_notin.
+Print Assumptions Es_ctx_any.
+Print Assumptions nred_R_intro.
 Print Assumptions eqC_in_Es.
 Print Assumptions Es_refl_any.
 Print Assumptions Es_sym_any.