add(named): the paper measure with machine checked counterexamples to its invariants (M5 metatheory)

README.md

@@ -2,14 +2,36 @@ This was (mostly) me being bored in a Monday evening and in an attempt to formal
 
 ## Graphs
 
-The dependency graph covers the six milestones and every edge runs from a dependency to the result that uses it while the node colour encodes the status proved or stated or planned or blocked:
+The dependency graphs are drawn in greyscale and split by milestone. Every edge runs from a dependency to the result that uses it. A box is a result named after a paper statement, an ellipse is a supporting result, and a dashed box labelled with a milestone is a result defined in another diagram. The overview gives the shape of the whole development:
 
-![Theorem dependency graph](graphs/dependency.svg)
+<p align="center"><img src="graphs/dependency-overview.svg" alt="Theorem dependency by milestone"></p>
+
+The details are then one diagram per milestone:
+
+### M1, binding and reduction
+
+<p align="center"><img src="graphs/dependency-M1.svg" alt="M1 dependencies"></p>
+
+### M2, metatheory and checks
+
+<p align="center"><img src="graphs/dependency-M2.svg" alt="M2 dependencies"></p>
+
+### M3, subsystem
+
+<p align="center"><img src="graphs/dependency-M3.svg" alt="M3 dependencies"></p>
+
+### M4, parallel reduction
+
+<p align="center"><img src="graphs/dependency-M4.svg" alt="M4 dependencies"></p>
+
+### M5, metaterms and the named calculus
+
+<p align="center"><img src="graphs/dependency-M5.svg" alt="M5 dependencies"></p>
 
 The rule sketch gives the transition system generated by the rules B and Gc and R:
 
-![Reduction rules](graphs/rules.svg)
+<p align="center"><img src="graphs/rules.svg" alt="Reduction rules"></p>
 
 The reduction graph gives the bounded reduct set of the term that applies the duplicating identity to a variable with every edge labelled by its rule and with the normal form marked:
 
-![Reduction graph of the duplicating identity](graphs/reduction.svg)
+<p align="center"><img src="graphs/reduction.svg" alt="Reduction graph of the duplicating identity"></p>

graphs/dependency-M1.dot

@@ -0,0 +1,71 @@
+digraph deps {
+  rankdir=LR;
+  bgcolor="transparent";
+  splines=polyline;
+  concentrate=true;
+  nodesep=0.2;
+  ranksep=0.9;
+  node [fontname=Inter, fontsize=9];
+  edge [fontname=Inter, fontsize=8, color="#8f9780", fontcolor="#c9d1bb", arrowsize=0.6, penwidth=0.8];
+  graph [fontname=Inter, fontsize=12, labelloc=t, fontcolor="#c9d1bb", label=<<B>Theorem dependencies M1, 33 results</B>>];
+  close_rec_fvar [label="close_rec_fvar", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="close_rec_fvar theory/Binding.v proved"];
+  close_rec_fvs [label="close_rec_fvs", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="close_rec_fvs theory/Binding.v proved"];
+  close_rec_notin [label="close_rec_notin", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="close_rec_notin theory/Binding.v proved"];
+  fvs_lift [label="fvs_lift", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="fvs_lift theory/Binding.v proved"];
+  lift_0_lc [label="lift_0_lc", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="lift_0_lc theory/Binding.v proved"];
+  lift_lc [label="lift_lc", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="lift_lc theory/Binding.v proved"];
+  open_close [label="open_close", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_close theory/Binding.v proved"];
+  open_rec_bvar [label="open_rec_bvar", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_bvar theory/Binding.v proved"];
+  open_rec_bvar_neq [label="open_rec_bvar_neq", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_bvar_neq theory/Binding.v proved"];
+  open_rec_fvs [label="open_rec_fvs", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_fvs theory/Binding.v proved"];
+  open_rec_occurs_false [label="open_rec_occurs_false", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_occurs_false theory/Binding.v proved"];
+  subst_fvar_other [label="subst_fvar_other", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_fvar_other theory/Binding.v proved"];
+  subst_fvar_self [label="subst_fvar_self", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_fvar_self theory/Binding.v proved"];
+  subst_fvs [label="subst_fvs", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_fvs theory/Binding.v proved"];
+  at_ctx_comp [label="at_ctx_comp", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="at_ctx_comp theory/Reduction.v proved"];
+  has_red_f [label="has_red_f", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="has_red_f theory/Reduction.v proved"];
+  has_red_spec [label="has_red_spec", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="has_red_spec theory/Reduction.v proved"];
+  nf_dec [label="nf_dec", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nf_dec theory/Reduction.v proved"];
+  nf_iff_steps_nil [label="nf_iff_steps_nil", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nf_iff_steps_nil theory/Reduction.v proved"];
+  normal_form_iff_nf [label="normal_form_iff_nf", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="normal_form_iff_nf theory/Reduction.v proved"];
+  plug_comp [label="plug_comp", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="plug_comp theory/Reduction.v proved"];
+  positions_at [label="positions_at", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="positions_at theory/Reduction.v proved"];
+  positions_comp [label="positions_comp", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="positions_comp theory/Reduction.v proved"];
+  positions_hole [label="positions_hole", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="positions_hole theory/Reduction.v proved"];
+  red1_dec [label="red1_dec", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red1_dec theory/Reduction.v proved"];
+  root_steps_complete [label="root_steps_complete", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="root_steps_complete theory/Reduction.v proved"];
+  root_steps_in_steps [label="root_steps_in_steps", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="root_steps_in_steps theory/Reduction.v proved"];
+  root_steps_sound [label="root_steps_sound", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="root_steps_sound theory/Reduction.v proved"];
+  steps_complete [label="steps_complete", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="steps_complete theory/Reduction.v proved"];
+  steps_sound [label="steps_sound", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="steps_sound theory/Reduction.v proved"];
+  steps_sound_red1 [label="steps_sound_red1", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="steps_sound_red1 theory/Reduction.v proved"];
+  zdecs_complete [label="zdecs_complete", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="zdecs_complete theory/Reduction.v proved"];
+  zdecs_sound [label="zdecs_sound", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="zdecs_sound theory/Reduction.v proved"];
+  lift_lc -> lift_0_lc;
+  fvs_lift -> open_rec_fvs;
+  open_rec_bvar -> open_rec_fvs;
+  open_rec_bvar_neq -> open_rec_fvs;
+  open_rec_bvar -> open_close;
+  lift_0_lc -> subst_fvar_self;
+  close_rec_fvs -> subst_fvs;
+  open_rec_fvs -> subst_fvs;
+  zdecs_sound -> root_steps_sound;
+  zdecs_complete -> root_steps_complete;
+  positions_at -> steps_sound;
+  root_steps_sound -> steps_sound;
+  steps_sound -> steps_sound_red1;
+  positions_hole -> root_steps_in_steps;
+  at_ctx_comp -> steps_complete;
+  plug_comp -> steps_complete;
+  positions_comp -> steps_complete;
+  root_steps_complete -> steps_complete;
+  root_steps_in_steps -> steps_complete;
+  has_red_f -> has_red_spec;
+  steps_complete -> has_red_spec;
+  steps_sound -> has_red_spec;
+  has_red_spec -> red1_dec;
+  steps_complete -> nf_iff_steps_nil;
+  steps_sound_red1 -> nf_iff_steps_nil;
+  nf_iff_steps_nil -> normal_form_iff_nf;
+  nf_iff_steps_nil -> nf_dec;
+}

graphs/dependency-M1.svg

@@ -0,0 +1,472 @@
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+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.42.4 (0)
+ -->
+<!-- Title: deps Pages: 1 -->
+<svg width="1019pt" height="715pt"
+ viewBox="0.00 0.00 1018.89 715.00" 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(4 711)">
+<title>deps</title>
+<text text-anchor="start" x="375.95" y="-694.4" font-family="Inter" font-weight="bold" font-size="12.00" fill="#c9d1bb">Theorem dependencies M1, 33 results</text>
+<!-- close_rec_fvar -->
+<g id="node1" class="node">
+<title>close_rec_fvar</title>
+<g id="a_node1"><a xlink:title="close_rec_fvar theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-18" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_fvar</text>
+</a>
+</g>
+</g>
+<!-- close_rec_fvs -->
+<g id="node2" class="node">
+<title>close_rec_fvs</title>
+<g id="a_node2"><a xlink:title="close_rec_fvs theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-168" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_fvs</text>
+</a>
+</g>
+</g>
+<!-- subst_fvs -->
+<g id="node14" class="node">
+<title>subst_fvs</title>
+<g id="a_node14"><a xlink:title="subst_fvs theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="442.35" cy="-143" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="442.35" y="-140.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_fvs</text>
+</a>
+</g>
+</g>
+<!-- close_rec_fvs&#45;&gt;subst_fvs -->
+<g id="edge7" class="edge">
+<title>close_rec_fvs&#45;&gt;subst_fvs</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M304.71,-162.33C334.28,-158.14 374.26,-152.49 403.32,-148.38"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="403.71,-150.45 409.35,-147.53 403.12,-146.29 403.71,-150.45"/>
+</g>
+<!-- close_rec_notin -->
+<g id="node3" class="node">
+<title>close_rec_notin</title>
+<g id="a_node3"><a xlink:title="close_rec_notin theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-218" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-215.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_notin</text>
+</a>
+</g>
+</g>
+<!-- fvs_lift -->
+<g id="node4" class="node">
+<title>fvs_lift</title>
+<g id="a_node4"><a xlink:title="fvs_lift theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-168" rx="27" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">fvs_lift</text>
+</a>
+</g>
+</g>
+<!-- open_rec_fvs -->
+<g id="node10" class="node">
+<title>open_rec_fvs</title>
+<g id="a_node10"><a xlink:title="open_rec_fvs theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-118" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_fvs</text>
+</a>
+</g>
+</g>
+<!-- fvs_lift&#45;&gt;open_rec_fvs -->
+<g id="edge2" class="edge">
+<title>fvs_lift&#45;&gt;open_rec_fvs</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M93.13,-161.67C125.07,-153.44 181.37,-138.93 220.26,-128.91"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="221.06,-130.87 226.35,-127.34 220.01,-126.81 221.06,-130.87"/>
+</g>
+<!-- lift_0_lc -->
+<g id="node5" class="node">
+<title>lift_0_lc</title>
+<g id="a_node5"><a xlink:title="lift_0_lc theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-268" rx="28.74" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-265.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">lift_0_lc</text>
+</a>
+</g>
+</g>
+<!-- subst_fvar_self -->
+<g id="node13" class="node">
+<title>subst_fvar_self</title>
+<g id="a_node13"><a xlink:title="subst_fvar_self theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="442.35" cy="-268" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="442.35" y="-265.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_fvar_self</text>
+</a>
+</g>
+</g>
+<!-- lift_0_lc&#45;&gt;subst_fvar_self -->
+<g id="edge6" class="edge">
+<title>lift_0_lc&#45;&gt;subst_fvar_self</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M292.61,-268C318.16,-268 356.61,-268 388.01,-268"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="388.19,-270.1 394.19,-268 388.19,-265.9 388.19,-270.1"/>
+</g>
+<!-- lift_lc -->
+<g id="node6" class="node">
+<title>lift_lc</title>
+<g id="a_node6"><a xlink:title="lift_lc theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-268" rx="27" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-265.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">lift_lc</text>
+</a>
+</g>
+</g>
+<!-- lift_lc&#45;&gt;lift_0_lc -->
+<g id="edge1" class="edge">
+<title>lift_lc&#45;&gt;lift_0_lc</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M94.72,-268C129.24,-268 190.08,-268 228.27,-268"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="228.66,-270.1 234.66,-268 228.66,-265.9 228.66,-270.1"/>
+</g>
+<!-- open_close -->
+<g id="node7" class="node">
+<title>open_close</title>
+<g id="a_node7"><a xlink:title="open_close theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-68" rx="38.21" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-65.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_close</text>
+</a>
+</g>
+</g>
+<!-- open_rec_bvar -->
+<g id="node8" class="node">
+<title>open_rec_bvar</title>
+<g id="a_node8"><a xlink:title="open_rec_bvar theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-68" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-65.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_bvar</text>
+</a>
+</g>
+</g>
+<!-- open_rec_bvar&#45;&gt;open_close -->
+<g id="edge5" class="edge">
+<title>open_rec_bvar&#45;&gt;open_close</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M115.53,-68C147.18,-68 188.62,-68 219.44,-68"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="219.46,-70.1 225.46,-68 219.46,-65.9 219.46,-70.1"/>
+</g>
+<!-- open_rec_bvar&#45;&gt;open_rec_fvs -->
+<g id="edge3" class="edge">
+<title>open_rec_bvar&#45;&gt;open_rec_fvs</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M107.82,-78.12C140.45,-86.53 186.79,-98.47 220.25,-107.09"/>
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+</g>
+<!-- open_rec_bvar_neq -->
+<g id="node9" class="node">
+<title>open_rec_bvar_neq</title>
+<g id="a_node9"><a xlink:title="open_rec_bvar_neq theory/Binding.v proved">
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+<text text-anchor="middle" x="67.55" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_bvar_neq</text>
+</a>
+</g>
+</g>
+<!-- open_rec_bvar_neq&#45;&gt;open_rec_fvs -->
+<g id="edge4" class="edge">
+<title>open_rec_bvar_neq&#45;&gt;open_rec_fvs</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M127.83,-118C155.48,-118 187.99,-118 214.14,-118"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="214.31,-120.1 220.31,-118 214.31,-115.9 214.31,-120.1"/>
+</g>
+<!-- open_rec_fvs&#45;&gt;subst_fvs -->
+<g id="edge8" class="edge">
+<title>open_rec_fvs&#45;&gt;subst_fvs</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M304.71,-123.67C334.28,-127.86 374.26,-133.51 403.32,-137.62"/>
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+</g>
+<!-- open_rec_occurs_false -->
+<g id="node11" class="node">
+<title>open_rec_occurs_false</title>
+<g id="a_node11"><a xlink:title="open_rec_occurs_false theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-318" rx="67.6" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-315.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_occurs_false</text>
+</a>
+</g>
+</g>
+<!-- subst_fvar_other -->
+<g id="node12" class="node">
+<title>subst_fvar_other</title>
+<g id="a_node12"><a xlink:title="subst_fvar_other theory/Binding.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-368" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-365.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_fvar_other</text>
+</a>
+</g>
+</g>
+<!-- at_ctx_comp -->
+<g id="node15" class="node">
+<title>at_ctx_comp</title>
+<g id="a_node15"><a xlink:title="at_ctx_comp theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-618" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-615.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">at_ctx_comp</text>
+</a>
+</g>
+</g>
+<!-- steps_complete -->
+<g id="node29" class="node">
+<title>steps_complete</title>
+<g id="a_node29"><a xlink:title="steps_complete theory/Reduction.v proved">
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+</a>
+</g>
+</g>
+<!-- at_ctx_comp&#45;&gt;steps_complete -->
+<g id="edge15" class="edge">
+<title>at_ctx_comp&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M295.13,-605.79C311,-599.45 327.12,-593 327.12,-593 327.12,-593 392.12,-543 392.12,-543 392.12,-543 398.81,-539.6 407.26,-535.31"/>
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+<!-- has_red_f -->
+<g id="node16" class="node">
+<title>has_red_f</title>
+<g id="a_node16"><a xlink:title="has_red_f theory/Reduction.v proved">
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+<text text-anchor="middle" x="442.35" y="-565.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">has_red_f</text>
+</a>
+</g>
+</g>
+<!-- has_red_spec -->
+<g id="node17" class="node">
+<title>has_red_spec</title>
+<g id="a_node17"><a xlink:title="has_red_spec theory/Reduction.v proved">
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+<text text-anchor="middle" x="614.16" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">has_red_spec</text>
+</a>
+</g>
+</g>
+<!-- has_red_f&#45;&gt;has_red_spec -->
+<g id="edge20" class="edge">
+<title>has_red_f&#45;&gt;has_red_spec</title>
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+<!-- red1_dec -->
+<g id="node25" class="node">
+<title>red1_dec</title>
+<g id="a_node25"><a xlink:title="red1_dec theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="783.08" cy="-518" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="783.08" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red1_dec</text>
+</a>
+</g>
+</g>
+<!-- has_red_spec&#45;&gt;red1_dec -->
+<g id="edge23" class="edge">
+<title>has_red_spec&#45;&gt;red1_dec</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M659.01,-518C684.93,-518 717.47,-518 742.58,-518"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="742.8,-520.1 748.8,-518 742.8,-515.9 742.8,-520.1"/>
+</g>
+<!-- nf_dec -->
+<g id="node18" class="node">
+<title>nf_dec</title>
+<g id="a_node18"><a xlink:title="nf_dec theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="953.16" cy="-475" rx="27.27" ry="18"/>
+<text text-anchor="middle" x="953.16" y="-472.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nf_dec</text>
+</a>
+</g>
+</g>
+<!-- nf_iff_steps_nil -->
+<g id="node19" class="node">
+<title>nf_iff_steps_nil</title>
+<g id="a_node19"><a xlink:title="nf_iff_steps_nil theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="783.08" cy="-450" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="783.08" y="-447.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nf_iff_steps_nil</text>
+</a>
+</g>
+</g>
+<!-- nf_iff_steps_nil&#45;&gt;nf_dec -->
+<g id="edge27" class="edge">
+<title>nf_iff_steps_nil&#45;&gt;nf_dec</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M827.36,-456.44C856.36,-460.75 893.89,-466.33 920.2,-470.25"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="920.05,-472.35 926.29,-471.15 920.67,-468.19 920.05,-472.35"/>
+</g>
+<!-- normal_form_iff_nf -->
+<g id="node20" class="node">
+<title>normal_form_iff_nf</title>
+<g id="a_node20"><a xlink:title="normal_form_iff_nf theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="953.16" cy="-425" rx="57.97" ry="18"/>
+<text text-anchor="middle" x="953.16" y="-422.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">normal_form_iff_nf</text>
+</a>
+</g>
+</g>
+<!-- nf_iff_steps_nil&#45;&gt;normal_form_iff_nf -->
+<g id="edge26" class="edge">
+<title>nf_iff_steps_nil&#45;&gt;normal_form_iff_nf</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M827.36,-443.56C847.81,-440.52 872.51,-436.85 894.52,-433.57"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="895.02,-435.62 900.65,-432.66 894.4,-431.47 895.02,-435.62"/>
+</g>
+<!-- plug_comp -->
+<g id="node21" class="node">
+<title>plug_comp</title>
+<g id="a_node21"><a xlink:title="plug_comp theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-568" rx="38.87" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-565.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">plug_comp</text>
+</a>
+</g>
+</g>
+<!-- plug_comp&#45;&gt;steps_complete -->
+<g id="edge16" class="edge">
+<title>plug_comp&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M297.02,-558.83C324.87,-550.95 365.29,-539.52 396.28,-530.75"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="397.15,-532.69 402.35,-529.03 396,-528.65 397.15,-532.69"/>
+</g>
+<!-- positions_at -->
+<g id="node22" class="node">
+<title>positions_at</title>
+<g id="a_node22"><a xlink:title="positions_at theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-368" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-365.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">positions_at</text>
+</a>
+</g>
+</g>
+<!-- steps_sound -->
+<g id="node30" class="node">
+<title>steps_sound</title>
+<g id="a_node30"><a xlink:title="steps_sound theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="442.35" cy="-421" rx="42.29" ry="18"/>
+<text text-anchor="middle" x="442.35" y="-418.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">steps_sound</text>
+</a>
+</g>
+</g>
+<!-- positions_at&#45;&gt;steps_sound -->
+<g id="edge11" class="edge">
+<title>positions_at&#45;&gt;steps_sound</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M297.85,-377.97C327.19,-386.77 370.04,-399.62 401.27,-408.98"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="401,-411.09 407.35,-410.81 402.21,-407.07 401,-411.09"/>
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+<!-- positions_comp -->
+<g id="node23" class="node">
+<title>positions_comp</title>
+<g id="a_node23"><a xlink:title="positions_comp theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-518" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">positions_comp</text>
+</a>
+</g>
+</g>
+<!-- positions_comp&#45;&gt;steps_complete -->
+<g id="edge17" class="edge">
+<title>positions_comp&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M313.37,-518C335.79,-518 362.49,-518 385.69,-518"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="385.82,-520.1 391.82,-518 385.82,-515.9 385.82,-520.1"/>
+</g>
+<!-- positions_hole -->
+<g id="node24" class="node">
+<title>positions_hole</title>
+<g id="a_node24"><a xlink:title="positions_hole theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-668" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-665.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">positions_hole</text>
+</a>
+</g>
+</g>
+<!-- root_steps_in_steps -->
+<g id="node27" class="node">
+<title>root_steps_in_steps</title>
+<g id="a_node27"><a xlink:title="root_steps_in_steps theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-668" rx="60.09" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-665.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">root_steps_in_steps</text>
+</a>
+</g>
+</g>
+<!-- positions_hole&#45;&gt;root_steps_in_steps -->
+<g id="edge14" class="edge">
+<title>positions_hole&#45;&gt;root_steps_in_steps</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M114.05,-668C138.84,-668 169.94,-668 197.28,-668"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="197.41,-670.1 203.41,-668 197.41,-665.9 197.41,-670.1"/>
+</g>
+<!-- root_steps_complete -->
+<g id="node26" class="node">
+<title>root_steps_complete</title>
+<g id="a_node26"><a xlink:title="root_steps_complete theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-468" rx="63.52" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-465.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">root_steps_complete</text>
+</a>
+</g>
+</g>
+<!-- root_steps_complete&#45;&gt;steps_complete -->
+<g id="edge18" class="edge">
+<title>root_steps_complete&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M309.21,-480.62C335.92,-488.17 369.78,-497.75 396.53,-505.32"/>
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+<!-- root_steps_in_steps&#45;&gt;steps_complete -->
+<g id="edge19" class="edge">
+<title>root_steps_in_steps&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M300.61,-653.6C314.5,-648.04 327.12,-643 327.12,-643 327.12,-643 392.12,-543 392.12,-543 392.12,-543 398.81,-539.6 407.26,-535.31"/>
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+<!-- root_steps_sound -->
+<g id="node28" class="node">
+<title>root_steps_sound</title>
+<g id="a_node28"><a xlink:title="root_steps_sound theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="263.61" cy="-418" rx="54.7" ry="18"/>
+<text text-anchor="middle" x="263.61" y="-415.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">root_steps_sound</text>
+</a>
+</g>
+</g>
+<!-- root_steps_sound&#45;&gt;steps_sound -->
+<g id="edge12" class="edge">
+<title>root_steps_sound&#45;&gt;steps_sound</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M318.6,-418.92C342.64,-419.32 370.7,-419.8 393.92,-420.2"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="393.99,-422.3 400.03,-420.3 394.06,-418.1 393.99,-422.3"/>
+</g>
+<!-- steps_complete&#45;&gt;has_red_spec -->
+<g id="edge21" class="edge">
+<title>steps_complete&#45;&gt;has_red_spec</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M492.92,-518C515.11,-518 541.22,-518 563.43,-518"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="563.58,-520.1 569.58,-518 563.58,-515.9 563.58,-520.1"/>
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+<!-- steps_complete&#45;&gt;nf_iff_steps_nil -->
+<g id="edge24" class="edge">
+<title>steps_complete&#45;&gt;nf_iff_steps_nil</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M485.56,-508.76C518.1,-501.64 557.58,-493 557.58,-493 557.58,-493 668.2,-471.81 734.14,-459.18"/>
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+</g>
+<!-- steps_sound&#45;&gt;has_red_spec -->
+<g id="edge22" class="edge">
+<title>steps_sound&#45;&gt;has_red_spec</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M468.4,-435.31C498.66,-452.6 549.57,-481.68 582.24,-500.34"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="581.47,-502.32 587.72,-503.47 583.55,-498.67 581.47,-502.32"/>
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+<!-- steps_sound_red1 -->
+<g id="node31" class="node">
+<title>steps_sound_red1</title>
+<g id="a_node31"><a xlink:title="steps_sound_red1 theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="614.16" cy="-431" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="614.16" y="-428.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">steps_sound_red1</text>
+</a>
+</g>
+</g>
+<!-- steps_sound&#45;&gt;steps_sound_red1 -->
+<g id="edge13" class="edge">
+<title>steps_sound&#45;&gt;steps_sound_red1</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M484.45,-423.42C504.73,-424.61 529.6,-426.08 552.05,-427.4"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="552.2,-429.51 558.31,-427.77 552.44,-425.32 552.2,-429.51"/>
+</g>
+<!-- steps_sound_red1&#45;&gt;nf_iff_steps_nil -->
+<g id="edge25" class="edge">
+<title>steps_sound_red1&#45;&gt;nf_iff_steps_nil</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M667.97,-437.01C688.23,-439.32 711.28,-441.94 731.33,-444.22"/>
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+<!-- zdecs_complete -->
+<g id="node32" class="node">
+<title>zdecs_complete</title>
+<g id="a_node32"><a xlink:title="zdecs_complete theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-468" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-465.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zdecs_complete</text>
+</a>
+</g>
+</g>
+<!-- zdecs_complete&#45;&gt;root_steps_complete -->
+<g id="edge10" class="edge">
+<title>zdecs_complete&#45;&gt;root_steps_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M119.03,-468C141.81,-468 169.07,-468 193.69,-468"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="193.88,-470.1 199.88,-468 193.88,-465.9 193.88,-470.1"/>
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+<!-- zdecs_sound -->
+<g id="node33" class="node">
+<title>zdecs_sound</title>
+<g id="a_node33"><a xlink:title="zdecs_sound theory/Reduction.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="67.55" cy="-418" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="67.55" y="-415.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zdecs_sound</text>
+</a>
+</g>
+</g>
+<!-- zdecs_sound&#45;&gt;root_steps_sound -->
+<g id="edge9" class="edge">
+<title>zdecs_sound&#45;&gt;root_steps_sound</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M110.66,-418C137.51,-418 172.72,-418 202.49,-418"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="202.76,-420.1 208.76,-418 202.76,-415.9 202.76,-420.1"/>
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+</g>
+</svg>

graphs/dependency-M2.dot

@@ -0,0 +1,115 @@
+digraph deps {
+  rankdir=LR;
+  bgcolor="transparent";
+  splines=polyline;
+  concentrate=true;
+  nodesep=0.2;
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+  node [fontname=Inter, fontsize=9];
+  edge [fontname=Inter, fontsize=8, color="#8f9780", fontcolor="#c9d1bb", arrowsize=0.6, penwidth=0.8];
+  graph [fontname=Inter, fontsize=12, labelloc=t, fontcolor="#c9d1bb", label=<<B>Theorem dependencies M2, 53 results</B>>];
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+  subst_fvar_self [label="subst_fvar_self\\n(M1)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="subst_fvar_self theory/Binding.v proved"];
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+  Plus_to_Star [label="Plus_to_Star", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Plus_to_Star theory/Closure.v proved"];
+  Star_red1_context [label="Star_red1_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Star_red1_context theory/Closure.v proved"];
+  Star_trans [label="Star_trans", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Star_trans theory/Closure.v proved"];
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+  star_one_trans [label="star_one_trans", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="star_one_trans theory/Closure.v proved"];
+  enumerate_check [label="enumerate_check", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="enumerate_check theory/Enumerate.v proved"];
+  enumerate_complete [label="enumerate_complete", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="enumerate_complete theory/Enumerate.v proved"];
+  enumerate_sound [label="enumerate_sound", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="enumerate_sound theory/Enumerate.v proved"];
+  in_comb [label="in_comb", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="in_comb theory/Enumerate.v proved"];
+  pow2_pos [label="pow2_pos", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="pow2_pos theory/Enumerate.v proved"];
+  size_all_terms [label="size_all_terms", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="size_all_terms theory/Enumerate.v proved"];
+  Plus_red1_context [label="Plus_red1_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Plus_red1_context theory/Metatheory.v proved"];
+  Plus_step_trans [label="Plus_step_trans", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Plus_step_trans theory/Metatheory.v proved"];
+  Plus_trans [label="Plus_trans", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Plus_trans theory/Metatheory.v proved"];
+  full_comp_aux [label="full_comp_aux", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="full_comp_aux theory/Metatheory.v proved"];
+  fvs_zplug_lift [label="fvs_zplug_lift", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="fvs_zplug_lift theory/Metatheory.v proved"];
+  lemma_2_1_fv_preserved [label="lemma_2_1_fv_preserved", shape=box, style="rounded,filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="lemma_2_1_fv_preserved (section 2.1) theory/Metatheory.v proved"];
+  lemma_2_2_full_comp [label="lemma_2_2_full_comp", shape=box, style="rounded,filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="lemma_2_2_full_comp (section 2.2) theory/Metatheory.v proved"];
+  lemma_2_3_beta_sim [label="lemma_2_3_beta_sim", shape=box, style="rounded,filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="lemma_2_3_beta_sim (section 2.3) theory/Metatheory.v proved"];
+  occurs_count_false [label="occurs_count_false", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="occurs_count_false theory/Metatheory.v proved"];
+  occurs_count_pos [label="occurs_count_pos", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="occurs_count_pos theory/Metatheory.v proved"];
+  occurs_count_zero [label="occurs_count_zero", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="occurs_count_zero theory/Metatheory.v proved"];
+  occurs_count_zfill_plug [label="occurs_count_zfill_plug", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="occurs_count_zfill_plug theory/Metatheory.v proved"];
+  occurs_lift_self [label="occurs_lift_self", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="occurs_lift_self theory/Metatheory.v proved"];
+  open_rec_zplug_lift [label="open_rec_zplug_lift", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_zplug_lift theory/Metatheory.v proved"];
+  plug_fv_mono [label="plug_fv_mono", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="plug_fv_mono theory/Metatheory.v proved"];
+  red1_root_fv [label="red1_root_fv", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red1_root_fv theory/Metatheory.v proved"];
+  red1r_fv [label="red1r_fv", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red1r_fv theory/Metatheory.v proved"];
+  zdecs_nonempty [label="zdecs_nonempty", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="zdecs_nonempty theory/Metatheory.v proved"];
+  check_all_true [label="check_all_true", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="check_all_true theory/Random.v proved"];
+  check_term_true [label="check_term_true", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="check_term_true theory/Random.v proved"];
+  gen_complete [label="gen_complete", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="gen_complete theory/Random.v proved"];
+  gen_fv_preserved [label="gen_fv_preserved", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="gen_fv_preserved theory/Random.v proved"];
+  gen_list_length [label="gen_list_length", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="gen_list_length theory/Random.v proved"];
+  gen_sound [label="gen_sound", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="gen_sound theory/Random.v proved"];
+  close_rec_app [label="close_rec_app", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="close_rec_app theory/Substitution.v proved"];
+  close_rec_esub [label="close_rec_esub", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="close_rec_esub theory/Substitution.v proved"];
+  close_rec_lam [label="close_rec_lam", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="close_rec_lam theory/Substitution.v proved"];
+  open_rec_app [label="open_rec_app", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_app theory/Substitution.v proved"];
+  open_rec_esub [label="open_rec_esub", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_esub theory/Substitution.v proved"];
+  open_rec_lam [label="open_rec_lam", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_lam theory/Substitution.v proved"];
+  open_rec_lc [label="open_rec_lc", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_lc theory/Substitution.v proved"];
+  open_rec_lc_atom [label="open_rec_lc_atom", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="open_rec_lc_atom theory/Substitution.v proved"];
+  subst_app [label="subst_app", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_app theory/Substitution.v proved"];
+  subst_esub [label="subst_esub", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_esub theory/Substitution.v proved"];
+  subst_lam [label="subst_lam", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_lam theory/Substitution.v proved"];
+  subst_lc_self [label="subst_lc_self", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_lc_self theory/Substitution.v proved"];
+  subst_notin [label="subst_notin", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_notin theory/Substitution.v proved"];
+  subst_other [label="subst_other", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subst_other theory/Substitution.v proved"];
+  corpus_complete [label="corpus_complete", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="corpus_complete theory/Tests.v proved"];
+  corpus_decides [label="corpus_decides", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="corpus_decides theory/Tests.v proved"];
+  corpus_fv_preserved [label="corpus_fv_preserved", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="corpus_fv_preserved theory/Tests.v proved"];
+  corpus_sound [label="corpus_sound", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="corpus_sound theory/Tests.v proved"];
+  in_comb -> size_all_terms;
+  pow2_pos -> size_all_terms;
+  check_all_true -> enumerate_check;
+  steps_sound -> enumerate_sound;
+  steps_complete -> enumerate_complete;
+  occurs_count_zero -> occurs_count_pos;
+  occurs_count_false -> occurs_count_zfill_plug;
+  occurs_lift_self -> occurs_count_zfill_plug;
+  occurs_lift_self -> open_rec_zplug_lift;
+  open_rec_occurs_false -> open_rec_zplug_lift;
+  occurs_count_zero -> full_comp_aux;
+  occurs_count_zfill_plug -> full_comp_aux;
+  open_rec_occurs_false -> full_comp_aux;
+  open_rec_zplug_lift -> full_comp_aux;
+  zdecs_nonempty -> full_comp_aux;
+  zdecs_sound -> full_comp_aux;
+  full_comp_aux -> lemma_2_2_full_comp;
+  fvs_lift -> fvs_zplug_lift;
+  fvs_zplug_lift -> red1_root_fv;
+  plug_fv_mono -> red1r_fv;
+  red1_root_fv -> red1r_fv;
+  red1r_fv -> lemma_2_1_fv_preserved;
+  Plus_red1_context -> lemma_2_3_beta_sim;
+  lemma_2_2_full_comp -> lemma_2_3_beta_sim;
+  has_red_spec -> check_term_true;
+  steps_sound_red1 -> check_term_true;
+  check_term_true -> check_all_true;
+  steps_sound -> gen_sound;
+  steps_complete -> gen_complete;
+  lemma_2_1_fv_preserved -> gen_fv_preserved;
+  steps_sound_red1 -> gen_fv_preserved;
+  close_rec_notin -> subst_notin;
+  open_rec_occurs_false -> subst_notin;
+  subst_fvar_self -> subst_lc_self;
+  subst_fvar_other -> subst_other;
+  open_rec_lc_atom -> open_rec_lc;
+  steps_sound -> corpus_sound;
+  steps_complete -> corpus_complete;
+  has_red_spec -> corpus_decides;
+  lemma_2_1_fv_preserved -> corpus_fv_preserved;
+  steps_sound_red1 -> corpus_fv_preserved;
+}

graphs/dependency-M2.svg

@@ -0,0 +1,826 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.42.4 (0)
+ -->
+<!-- Title: deps Pages: 1 -->
+<svg width="1445pt" height="1800pt"
+ viewBox="0.00 0.00 1444.74 1800.00" 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 1796)">
+<title>deps</title>
+<text text-anchor="start" x="588.87" y="-1779.4" font-family="Inter" font-weight="bold" font-size="12.00" fill="#c9d1bb">Theorem dependencies M2, 53 results</text>
+<!-- close_rec_notin -->
+<g id="node1" class="node">
+<title>close_rec_notin</title>
+<g id="a_node1"><a xlink:title="close_rec_notin theory/Binding.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M119.5,-321C119.5,-321 27.5,-321 27.5,-321 21.5,-321 15.5,-315 15.5,-309 15.5,-309 15.5,-297 15.5,-297 15.5,-291 21.5,-285 27.5,-285 27.5,-285 119.5,-285 119.5,-285 125.5,-285 131.5,-291 131.5,-297 131.5,-297 131.5,-309 131.5,-309 131.5,-315 125.5,-321 119.5,-321"/>
+<text text-anchor="middle" x="73.5" y="-300.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_notin\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- subst_notin -->
+<g id="node58" class="node">
+<title>subst_notin</title>
+<g id="a_node58"><a xlink:title="subst_notin theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-303" rx="39.67" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-300.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_notin</text>
+</a>
+</g>
+</g>
+<!-- close_rec_notin&#45;&gt;subst_notin -->
+<g id="edge32" class="edge">
+<title>close_rec_notin&#45;&gt;subst_notin</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M131.59,-303C164.68,-303 205.74,-303 236.51,-303"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="236.52,-305.1 242.52,-303 236.52,-300.9 236.52,-305.1"/>
+</g>
+<!-- fvs_lift -->
+<g id="node2" class="node">
+<title>fvs_lift</title>
+<g id="a_node2"><a xlink:title="fvs_lift theory/Binding.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M99,-371C99,-371 48,-371 48,-371 42,-371 36,-365 36,-359 36,-359 36,-347 36,-347 36,-341 42,-335 48,-335 48,-335 99,-335 99,-335 105,-335 111,-341 111,-347 111,-347 111,-359 111,-359 111,-365 105,-371 99,-371"/>
+<text text-anchor="middle" x="73.5" y="-350.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">fvs_lift\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- fvs_zplug_lift -->
+<g id="node26" class="node">
+<title>fvs_zplug_lift</title>
+<g id="a_node26"><a xlink:title="fvs_zplug_lift theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-353" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-350.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">fvs_zplug_lift</text>
+</a>
+</g>
+</g>
+<!-- fvs_lift&#45;&gt;fvs_zplug_lift -->
+<g id="edge18" class="edge">
+<title>fvs_lift&#45;&gt;fvs_zplug_lift</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M111.02,-353C145.02,-353 195.93,-353 233.25,-353"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="233.59,-355.1 239.59,-353 233.59,-350.9 233.59,-355.1"/>
+</g>
+<!-- has_red_spec -->
+<g id="node3" class="node">
+<title>has_red_spec</title>
+<g id="a_node3"><a xlink:title="has_red_spec theory/Reduction.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M884.03,-271C884.03,-271 801.03,-271 801.03,-271 795.03,-271 789.03,-265 789.03,-259 789.03,-259 789.03,-247 789.03,-247 789.03,-241 795.03,-235 801.03,-235 801.03,-235 884.03,-235 884.03,-235 890.03,-235 896.03,-241 896.03,-247 896.03,-247 896.03,-259 896.03,-259 896.03,-265 890.03,-271 884.03,-271"/>
+<text text-anchor="middle" x="842.53" y="-250.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">has_red_spec\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- check_term_true -->
+<g id="node41" class="node">
+<title>check_term_true</title>
+<g id="a_node41"><a xlink:title="check_term_true theory/Random.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="1036.54" cy="-278" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="1036.54" y="-275.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">check_term_true</text>
+</a>
+</g>
+</g>
+<!-- has_red_spec&#45;&gt;check_term_true -->
+<g id="edge25" class="edge">
+<title>has_red_spec&#45;&gt;check_term_true</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M896.26,-259.86C922.26,-263.25 953.62,-267.33 980,-270.77"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="980.02,-272.89 986.24,-271.58 980.56,-268.72 980.02,-272.89"/>
+</g>
+<!-- corpus_decides -->
+<g id="node61" class="node">
+<title>corpus_decides</title>
+<g id="a_node61"><a xlink:title="corpus_decides theory/Tests.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="1036.54" cy="-228" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="1036.54" y="-225.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">corpus_decides</text>
+</a>
+</g>
+</g>
+<!-- has_red_spec&#45;&gt;corpus_decides -->
+<g id="edge39" class="edge">
+<title>has_red_spec&#45;&gt;corpus_decides</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M896.26,-246.14C923.4,-242.6 956.38,-238.31 983.44,-234.78"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="983.79,-236.86 989.47,-234 983.25,-232.69 983.79,-236.86"/>
+</g>
+<!-- open_rec_occurs_false -->
+<g id="node4" class="node">
+<title>open_rec_occurs_false</title>
+<g id="a_node4"><a xlink:title="open_rec_occurs_false theory/Binding.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M135,-271C135,-271 12,-271 12,-271 6,-271 0,-265 0,-259 0,-259 0,-247 0,-247 0,-241 6,-235 12,-235 12,-235 135,-235 135,-235 141,-235 147,-241 147,-247 147,-247 147,-259 147,-259 147,-265 141,-271 135,-271"/>
+<text text-anchor="middle" x="73.5" y="-250.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_occurs_false\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- full_comp_aux -->
+<g id="node25" class="node">
+<title>full_comp_aux</title>
+<g id="a_node25"><a xlink:title="full_comp_aux theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="474.45" cy="-143" rx="46.53" ry="18"/>
+<text text-anchor="middle" x="474.45" y="-140.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">full_comp_aux</text>
+</a>
+</g>
+</g>
+<!-- open_rec_occurs_false&#45;&gt;full_comp_aux -->
+<g id="edge13" class="edge">
+<title>open_rec_occurs_false&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M147.28,-250.39C229.76,-247.42 352.87,-243 352.87,-243 352.87,-243 413.53,-192.7 448.8,-163.44"/>
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+</g>
+<!-- open_rec_zplug_lift -->
+<g id="node35" class="node">
+<title>open_rec_zplug_lift</title>
+<g id="a_node35"><a xlink:title="open_rec_zplug_lift theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-218" rx="59.43" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-215.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_zplug_lift</text>
+</a>
+</g>
+</g>
+<!-- open_rec_occurs_false&#45;&gt;open_rec_zplug_lift -->
+<g id="edge10" class="edge">
+<title>open_rec_occurs_false&#45;&gt;open_rec_zplug_lift</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M147.17,-240.71C172.4,-236.44 200.31,-231.72 224.13,-227.69"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="224.53,-229.76 230.09,-226.68 223.82,-225.61 224.53,-229.76"/>
+</g>
+<!-- open_rec_occurs_false&#45;&gt;subst_notin -->
+<g id="edge33" class="edge">
+<title>open_rec_occurs_false&#45;&gt;subst_notin</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M147.17,-270.56C178.67,-278.17 214.37,-286.8 241.04,-293.24"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="240.6,-295.29 246.93,-294.66 241.59,-291.21 240.6,-295.29"/>
+</g>
+<!-- steps_complete -->
+<g id="node5" class="node">
+<title>steps_complete</title>
+<g id="a_node5"><a xlink:title="steps_complete theory/Reduction.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M120,-471C120,-471 27,-471 27,-471 21,-471 15,-465 15,-459 15,-459 15,-447 15,-447 15,-441 21,-435 27,-435 27,-435 120,-435 120,-435 126,-435 132,-441 132,-447 132,-447 132,-459 132,-459 132,-465 126,-471 120,-471"/>
+<text text-anchor="middle" x="73.5" y="-450.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">steps_complete\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- enumerate_complete -->
+<g id="node17" class="node">
+<title>enumerate_complete</title>
+<g id="a_node17"><a xlink:title="enumerate_complete theory/Enumerate.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-503" rx="65.48" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-500.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">enumerate_complete</text>
+</a>
+</g>
+</g>
+<!-- steps_complete&#45;&gt;enumerate_complete -->
+<g id="edge5" class="edge">
+<title>steps_complete&#45;&gt;enumerate_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M132.14,-466.93C161.73,-474.08 197.58,-482.74 226.74,-489.78"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="226.53,-491.89 232.85,-491.26 227.51,-487.81 226.53,-491.89"/>
+</g>
+<!-- gen_complete -->
+<g id="node42" class="node">
+<title>gen_complete</title>
+<g id="a_node42"><a xlink:title="gen_complete theory/Random.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-453" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-450.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">gen_complete</text>
+</a>
+</g>
+</g>
+<!-- steps_complete&#45;&gt;gen_complete -->
+<g id="edge29" class="edge">
+<title>steps_complete&#45;&gt;gen_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M132.14,-453C162.85,-453 200.3,-453 230.01,-453"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="230.23,-455.1 236.23,-453 230.23,-450.9 230.23,-455.1"/>
+</g>
+<!-- corpus_complete -->
+<g id="node60" class="node">
+<title>corpus_complete</title>
+<g id="a_node60"><a xlink:title="corpus_complete theory/Tests.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-403" rx="54.04" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-400.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">corpus_complete</text>
+</a>
+</g>
+</g>
+<!-- steps_complete&#45;&gt;corpus_complete -->
+<g id="edge38" class="edge">
+<title>steps_complete&#45;&gt;corpus_complete</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M132.14,-439.07C163.65,-431.46 202.26,-422.13 232.32,-414.87"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="232.87,-416.89 238.21,-413.44 231.89,-412.81 232.87,-416.89"/>
+</g>
+<!-- steps_sound -->
+<g id="node6" class="node">
+<title>steps_sound</title>
+<g id="a_node6"><a xlink:title="steps_sound theory/Reduction.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M113,-621C113,-621 34,-621 34,-621 28,-621 22,-615 22,-609 22,-609 22,-597 22,-597 22,-591 28,-585 34,-585 34,-585 113,-585 113,-585 119,-585 125,-591 125,-597 125,-597 125,-609 125,-609 125,-615 119,-621 113,-621"/>
+<text text-anchor="middle" x="73.5" y="-600.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">steps_sound\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- enumerate_sound -->
+<g id="node18" class="node">
+<title>enumerate_sound</title>
+<g id="a_node18"><a xlink:title="enumerate_sound theory/Enumerate.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-653" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-650.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">enumerate_sound</text>
+</a>
+</g>
+</g>
+<!-- steps_sound&#45;&gt;enumerate_sound -->
+<g id="edge4" class="edge">
+<title>steps_sound&#45;&gt;enumerate_sound</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M125.12,-615.23C157.34,-623.02 198.9,-633.06 231.07,-640.83"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="230.63,-642.89 236.96,-642.25 231.62,-638.8 230.63,-642.89"/>
+</g>
+<!-- gen_sound -->
+<g id="node45" class="node">
+<title>gen_sound</title>
+<g id="a_node45"><a xlink:title="gen_sound theory/Random.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-603" rx="38.21" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-600.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">gen_sound</text>
+</a>
+</g>
+</g>
+<!-- steps_sound&#45;&gt;gen_sound -->
+<g id="edge28" class="edge">
+<title>steps_sound&#45;&gt;gen_sound</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M125.12,-603C159.82,-603 205.35,-603 238.31,-603"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="238.32,-605.1 244.32,-603 238.32,-600.9 238.32,-605.1"/>
+</g>
+<!-- corpus_sound -->
+<g id="node63" class="node">
+<title>corpus_sound</title>
+<g id="a_node63"><a xlink:title="corpus_sound theory/Tests.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-553" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-550.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">corpus_sound</text>
+</a>
+</g>
+</g>
+<!-- steps_sound&#45;&gt;corpus_sound -->
+<g id="edge37" class="edge">
+<title>steps_sound&#45;&gt;corpus_sound</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M125.12,-590.77C159.25,-582.52 203.87,-571.74 236.7,-563.81"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="237.35,-565.81 242.69,-562.36 236.36,-561.73 237.35,-565.81"/>
+</g>
+<!-- steps_sound_red1 -->
+<g id="node7" class="node">
+<title>steps_sound_red1</title>
+<g id="a_node7"><a xlink:title="steps_sound_red1 theory/Reduction.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M894.53,-346C894.53,-346 790.53,-346 790.53,-346 784.53,-346 778.53,-340 778.53,-334 778.53,-334 778.53,-322 778.53,-322 778.53,-316 784.53,-310 790.53,-310 790.53,-310 894.53,-310 894.53,-310 900.53,-310 906.53,-316 906.53,-322 906.53,-322 906.53,-334 906.53,-334 906.53,-340 900.53,-346 894.53,-346"/>
+<text text-anchor="middle" x="842.53" y="-325.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">steps_sound_red1\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- steps_sound_red1&#45;&gt;check_term_true -->
+<g id="edge26" class="edge">
+<title>steps_sound_red1&#45;&gt;check_term_true</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M906.67,-311.56C932.97,-304.71 963.01,-296.89 987.47,-290.52"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="988.29,-292.48 993.56,-288.93 987.23,-288.41 988.29,-292.48"/>
+</g>
+<!-- gen_fv_preserved -->
+<g id="node43" class="node">
+<title>gen_fv_preserved</title>
+<g id="a_node43"><a xlink:title="gen_fv_preserved theory/Random.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="1036.54" cy="-378" rx="55.35" ry="18"/>
+<text text-anchor="middle" x="1036.54" y="-375.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">gen_fv_preserved</text>
+</a>
+</g>
+</g>
+<!-- steps_sound_red1&#45;&gt;gen_fv_preserved -->
+<g id="edge31" class="edge">
+<title>steps_sound_red1&#45;&gt;gen_fv_preserved</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M906.67,-344.44C932.67,-351.21 962.32,-358.93 986.62,-365.26"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="986.35,-367.36 992.68,-366.84 987.41,-363.29 986.35,-367.36"/>
+</g>
+<!-- corpus_fv_preserved -->
+<g id="node62" class="node">
+<title>corpus_fv_preserved</title>
+<g id="a_node62"><a xlink:title="corpus_fv_preserved theory/Tests.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="1036.54" cy="-328" rx="63.52" ry="18"/>
+<text text-anchor="middle" x="1036.54" y="-325.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">corpus_fv_preserved</text>
+</a>
+</g>
+</g>
+<!-- steps_sound_red1&#45;&gt;corpus_fv_preserved -->
+<g id="edge41" class="edge">
+<title>steps_sound_red1&#45;&gt;corpus_fv_preserved</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M906.67,-328C925.99,-328 947.32,-328 966.98,-328"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="967.02,-330.1 973.02,-328 967.02,-325.9 967.02,-330.1"/>
+</g>
+<!-- subst_fvar_other -->
+<g id="node8" class="node">
+<title>subst_fvar_other</title>
+<g id="a_node8"><a xlink:title="subst_fvar_other theory/Binding.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M122.5,-721C122.5,-721 24.5,-721 24.5,-721 18.5,-721 12.5,-715 12.5,-709 12.5,-709 12.5,-697 12.5,-697 12.5,-691 18.5,-685 24.5,-685 24.5,-685 122.5,-685 122.5,-685 128.5,-685 134.5,-691 134.5,-697 134.5,-697 134.5,-709 134.5,-709 134.5,-715 128.5,-721 122.5,-721"/>
+<text text-anchor="middle" x="73.5" y="-700.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_fvar_other\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- subst_other -->
+<g id="node59" class="node">
+<title>subst_other</title>
+<g id="a_node59"><a xlink:title="subst_other theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-703" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-700.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_other</text>
+</a>
+</g>
+</g>
+<!-- subst_fvar_other&#45;&gt;subst_other -->
+<g id="edge35" class="edge">
+<title>subst_fvar_other&#45;&gt;subst_other</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M134.63,-703C166.83,-703 205.86,-703 235.58,-703"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="235.78,-705.1 241.78,-703 235.78,-700.9 235.78,-705.1"/>
+</g>
+<!-- subst_fvar_self -->
+<g id="node9" class="node">
+<title>subst_fvar_self</title>
+<g id="a_node9"><a xlink:title="subst_fvar_self theory/Binding.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M118,-771C118,-771 29,-771 29,-771 23,-771 17,-765 17,-759 17,-759 17,-747 17,-747 17,-741 23,-735 29,-735 29,-735 118,-735 118,-735 124,-735 130,-741 130,-747 130,-747 130,-759 130,-759 130,-765 124,-771 118,-771"/>
+<text text-anchor="middle" x="73.5" y="-750.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_fvar_self\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- subst_lc_self -->
+<g id="node57" class="node">
+<title>subst_lc_self</title>
+<g id="a_node57"><a xlink:title="subst_lc_self theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-753" rx="42.29" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-750.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_lc_self</text>
+</a>
+</g>
+</g>
+<!-- subst_fvar_self&#45;&gt;subst_lc_self -->
+<g id="edge34" class="edge">
+<title>subst_fvar_self&#45;&gt;subst_lc_self</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M130.23,-753C162.66,-753 203.11,-753 234.01,-753"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="234.06,-755.1 240.06,-753 234.06,-750.9 234.06,-755.1"/>
+</g>
+<!-- zdecs_sound -->
+<g id="node10" class="node">
+<title>zdecs_sound</title>
+<g id="a_node10"><a xlink:title="zdecs_sound theory/Reduction.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M322.44,-186C322.44,-186 242.44,-186 242.44,-186 236.44,-186 230.44,-180 230.44,-174 230.44,-174 230.44,-162 230.44,-162 230.44,-156 236.44,-150 242.44,-150 242.44,-150 322.44,-150 322.44,-150 328.44,-150 334.44,-156 334.44,-162 334.44,-162 334.44,-174 334.44,-174 334.44,-180 328.44,-186 322.44,-186"/>
+<text text-anchor="middle" x="282.44" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zdecs_sound\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- zdecs_sound&#45;&gt;full_comp_aux -->
+<g id="edge16" class="edge">
+<title>zdecs_sound&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M334.61,-161.27C362.3,-157.62 396.36,-153.14 423.82,-149.53"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="424.25,-151.59 429.93,-148.73 423.7,-147.43 424.25,-151.59"/>
+</g>
+<!-- Plus_to_Star -->
+<g id="node11" class="node">
+<title>Plus_to_Star</title>
+<g id="a_node11"><a xlink:title="Plus_to_Star theory/Closure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-803" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-800.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Plus_to_Star</text>
+</a>
+</g>
+</g>
+<!-- Star_red1_context -->
+<g id="node12" class="node">
+<title>Star_red1_context</title>
+<g id="a_node12"><a xlink:title="Star_red1_context theory/Closure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-853" rx="57.97" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-850.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Star_red1_context</text>
+</a>
+</g>
+</g>
+<!-- Star_trans -->
+<g id="node13" class="node">
+<title>Star_trans</title>
+<g id="a_node13"><a xlink:title="Star_trans theory/Closure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-903" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-900.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Star_trans</text>
+</a>
+</g>
+</g>
+<!-- red1_star -->
+<g id="node14" class="node">
+<title>red1_star</title>
+<g id="a_node14"><a xlink:title="red1_star theory/Closure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-953" rx="34.13" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-950.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red1_star</text>
+</a>
+</g>
+</g>
+<!-- star_one_trans -->
+<g id="node15" class="node">
+<title>star_one_trans</title>
+<g id="a_node15"><a xlink:title="star_one_trans theory/Closure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1003" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1000.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">star_one_trans</text>
+</a>
+</g>
+</g>
+<!-- enumerate_check -->
+<g id="node16" class="node">
+<title>enumerate_check</title>
+<g id="a_node16"><a xlink:title="enumerate_check theory/Enumerate.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="1380.16" cy="-278" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="1380.16" y="-275.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">enumerate_check</text>
+</a>
+</g>
+</g>
+<!-- in_comb -->
+<g id="node19" class="node">
+<title>in_comb</title>
+<g id="a_node19"><a xlink:title="in_comb theory/Enumerate.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1103" rx="31.51" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1100.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">in_comb</text>
+</a>
+</g>
+</g>
+<!-- size_all_terms -->
+<g id="node21" class="node">
+<title>size_all_terms</title>
+<g id="a_node21"><a xlink:title="size_all_terms theory/Enumerate.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-1078" rx="45.07" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-1075.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">size_all_terms</text>
+</a>
+</g>
+</g>
+<!-- in_comb&#45;&gt;size_all_terms -->
+<g id="edge1" class="edge">
+<title>in_comb&#45;&gt;size_all_terms</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M105.03,-1099.31C138.65,-1095.25 193.17,-1088.66 232.7,-1083.89"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="233.21,-1085.94 238.92,-1083.14 232.71,-1081.77 233.21,-1085.94"/>
+</g>
+<!-- pow2_pos -->
+<g id="node20" class="node">
+<title>pow2_pos</title>
+<g id="a_node20"><a xlink:title="pow2_pos theory/Enumerate.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1053" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1050.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">pow2_pos</text>
+</a>
+</g>
+</g>
+<!-- pow2_pos&#45;&gt;size_all_terms -->
+<g id="edge2" class="edge">
+<title>pow2_pos&#45;&gt;size_all_terms</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M108.67,-1057.13C142.56,-1061.22 194.8,-1067.53 232.97,-1072.14"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="232.76,-1074.23 238.97,-1072.87 233.27,-1070.07 232.76,-1074.23"/>
+</g>
+<!-- Plus_red1_context -->
+<g id="node22" class="node">
+<title>Plus_red1_context</title>
+<g id="a_node22"><a xlink:title="Plus_red1_context theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="654.03" cy="-93" rx="57.32" ry="18"/>
+<text text-anchor="middle" x="654.03" y="-90.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Plus_red1_context</text>
+</a>
+</g>
+</g>
+<!-- lemma_2_3_beta_sim -->
+<g id="node29" class="node">
+<title>lemma_2_3_beta_sim</title>
+<g id="a_node29"><a xlink:title="lemma_2_3_beta_sim (section 2.3) theory/Metatheory.v proved">
+<path fill="transparent" stroke="#a9b492" d="M887.03,-136C887.03,-136 798.03,-136 798.03,-136 792.03,-136 786.03,-130 786.03,-124 786.03,-124 786.03,-112 786.03,-112 786.03,-106 792.03,-100 798.03,-100 798.03,-100 887.03,-100 887.03,-100 893.03,-100 899.03,-106 899.03,-112 899.03,-112 899.03,-124 899.03,-124 899.03,-130 893.03,-136 887.03,-136"/>
+<text text-anchor="middle" x="842.53" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">lemma_2_3_beta_sim</text>
+</a>
+</g>
+</g>
+<!-- Plus_red1_context&#45;&gt;lemma_2_3_beta_sim -->
+<g id="edge23" class="edge">
+<title>Plus_red1_context&#45;&gt;lemma_2_3_beta_sim</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M706.98,-99.96C729.54,-102.99 756.14,-106.55 779.71,-109.71"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="779.72,-111.83 785.95,-110.55 780.28,-107.67 779.72,-111.83"/>
+</g>
+<!-- Plus_step_trans -->
+<g id="node23" class="node">
+<title>Plus_step_trans</title>
+<g id="a_node23"><a xlink:title="Plus_step_trans theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1153" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1150.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Plus_step_trans</text>
+</a>
+</g>
+</g>
+<!-- Plus_trans -->
+<g id="node24" class="node">
+<title>Plus_trans</title>
+<g id="a_node24"><a xlink:title="Plus_trans theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1203" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1200.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Plus_trans</text>
+</a>
+</g>
+</g>
+<!-- lemma_2_2_full_comp -->
+<g id="node28" class="node">
+<title>lemma_2_2_full_comp</title>
+<g id="a_node28"><a xlink:title="lemma_2_2_full_comp (section 2.2) theory/Metatheory.v proved">
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+</a>
+</g>
+</g>
+<!-- full_comp_aux&#45;&gt;lemma_2_2_full_comp -->
+<g id="edge17" class="edge">
+<title>full_comp_aux&#45;&gt;lemma_2_2_full_comp</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M521.65,-143C542.41,-143 567.24,-143 589.7,-143"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="589.96,-145.1 595.96,-143 589.96,-140.9 589.96,-145.1"/>
+</g>
+<!-- red1_root_fv -->
+<g id="node37" class="node">
+<title>red1_root_fv</title>
+<g id="a_node37"><a xlink:title="red1_root_fv theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="474.45" cy="-353" rx="42.29" ry="18"/>
+<text text-anchor="middle" x="474.45" y="-350.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red1_root_fv</text>
+</a>
+</g>
+</g>
+<!-- fvs_zplug_lift&#45;&gt;red1_root_fv -->
+<g id="edge19" class="edge">
+<title>fvs_zplug_lift&#45;&gt;red1_root_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M325.61,-353C355.33,-353 395.18,-353 426.01,-353"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="426.05,-355.1 432.05,-353 426.05,-350.9 426.05,-355.1"/>
+</g>
+<!-- lemma_2_1_fv_preserved -->
+<g id="node27" class="node">
+<title>lemma_2_1_fv_preserved</title>
+<g id="a_node27"><a xlink:title="lemma_2_1_fv_preserved (section 2.1) theory/Metatheory.v proved">
+<path fill="transparent" stroke="#a9b492" d="M896.03,-396C896.03,-396 789.03,-396 789.03,-396 783.03,-396 777.03,-390 777.03,-384 777.03,-384 777.03,-372 777.03,-372 777.03,-366 783.03,-360 789.03,-360 789.03,-360 896.03,-360 896.03,-360 902.03,-360 908.03,-366 908.03,-372 908.03,-372 908.03,-384 908.03,-384 908.03,-390 902.03,-396 896.03,-396"/>
+<text text-anchor="middle" x="842.53" y="-375.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">lemma_2_1_fv_preserved</text>
+</a>
+</g>
+</g>
+<!-- lemma_2_1_fv_preserved&#45;&gt;gen_fv_preserved -->
+<g id="edge30" class="edge">
+<title>lemma_2_1_fv_preserved&#45;&gt;gen_fv_preserved</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M908.27,-378C929.77,-378 953.63,-378 974.81,-378"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="975.01,-380.1 981.01,-378 975.01,-375.9 975.01,-380.1"/>
+</g>
+<!-- lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved -->
+<g id="edge40" class="edge">
+<title>lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M908.27,-361.14C932.66,-354.79 960.06,-347.66 983.18,-341.64"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="983.99,-343.59 989.27,-340.05 982.94,-339.53 983.99,-343.59"/>
+</g>
+<!-- lemma_2_2_full_comp&#45;&gt;lemma_2_3_beta_sim -->
+<g id="edge24" class="edge">
+<title>lemma_2_2_full_comp&#45;&gt;lemma_2_3_beta_sim</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M712.26,-135.33C733.69,-132.46 758.13,-129.18 779.94,-126.26"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="780.35,-128.32 786.02,-125.44 779.79,-124.16 780.35,-128.32"/>
+</g>
+<!-- occurs_count_false -->
+<g id="node30" class="node">
+<title>occurs_count_false</title>
+<g id="a_node30"><a xlink:title="occurs_count_false theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-118" rx="58.78" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_count_false</text>
+</a>
+</g>
+</g>
+<!-- occurs_count_zfill_plug -->
+<g id="node33" class="node">
+<title>occurs_count_zfill_plug</title>
+<g id="a_node33"><a xlink:title="occurs_count_zfill_plug theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-118" rx="70.37" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_count_zfill_plug</text>
+</a>
+</g>
+</g>
+<!-- occurs_count_false&#45;&gt;occurs_count_zfill_plug -->
+<g id="edge7" class="edge">
+<title>occurs_count_false&#45;&gt;occurs_count_zfill_plug</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M132.42,-118C155.13,-118 181.49,-118 205.67,-118"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="205.77,-120.1 211.77,-118 205.77,-115.9 205.77,-120.1"/>
+</g>
+<!-- occurs_count_pos -->
+<g id="node31" class="node">
+<title>occurs_count_pos</title>
+<g id="a_node31"><a xlink:title="occurs_count_pos theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="474.45" cy="-18" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="474.45" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_count_pos</text>
+</a>
+</g>
+</g>
+<!-- occurs_count_zero -->
+<g id="node32" class="node">
+<title>occurs_count_zero</title>
+<g id="a_node32"><a xlink:title="occurs_count_zero theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-18" rx="58.13" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_count_zero</text>
+</a>
+</g>
+</g>
+<!-- occurs_count_zero&#45;&gt;full_comp_aux -->
+<g id="edge11" class="edge">
+<title>occurs_count_zero&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M321,-31.52C337.36,-37.41 352.87,-43 352.87,-43 352.87,-43 413.53,-93.3 448.8,-122.56"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="447.85,-124.5 453.81,-126.71 450.53,-121.26 447.85,-124.5"/>
+</g>
+<!-- occurs_count_zero&#45;&gt;occurs_count_pos -->
+<g id="edge6" class="edge">
+<title>occurs_count_zero&#45;&gt;occurs_count_pos</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M340.96,-18C363.32,-18 388.98,-18 411.73,-18"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="411.75,-20.1 417.75,-18 411.75,-15.9 411.75,-20.1"/>
+</g>
+<!-- occurs_count_zfill_plug&#45;&gt;full_comp_aux -->
+<g id="edge12" class="edge">
+<title>occurs_count_zfill_plug&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M345.39,-126.15C370.85,-129.5 399.93,-133.33 423.93,-136.48"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="423.7,-138.57 429.92,-137.27 424.25,-134.41 423.7,-138.57"/>
+</g>
+<!-- occurs_lift_self -->
+<g id="node34" class="node">
+<title>occurs_lift_self</title>
+<g id="a_node34"><a xlink:title="occurs_lift_self theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-185" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-182.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_lift_self</text>
+</a>
+</g>
+</g>
+<!-- occurs_lift_self&#45;&gt;occurs_count_zfill_plug -->
+<g id="edge8" class="edge">
+<title>occurs_lift_self&#45;&gt;occurs_count_zfill_plug</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M110.07,-173.48C143.75,-162.58 194.6,-146.11 232.17,-133.95"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="233.02,-135.88 238.08,-132.04 231.73,-131.89 233.02,-135.88"/>
+</g>
+<!-- occurs_lift_self&#45;&gt;open_rec_zplug_lift -->
+<g id="edge9" class="edge">
+<title>occurs_lift_self&#45;&gt;open_rec_zplug_lift</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M117.38,-191.84C148.03,-196.73 189.63,-203.36 223.27,-208.72"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="223.19,-210.84 229.45,-209.71 223.86,-206.69 223.19,-210.84"/>
+</g>
+<!-- open_rec_zplug_lift&#45;&gt;full_comp_aux -->
+<g id="edge14" class="edge">
+<title>open_rec_zplug_lift&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M321.41,-204.33C337.62,-198.49 352.87,-193 352.87,-193 352.87,-193 401.07,-173.02 436.52,-158.32"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="437.35,-160.24 442.09,-156.01 435.74,-156.36 437.35,-160.24"/>
+</g>
+<!-- plug_fv_mono -->
+<g id="node36" class="node">
+<title>plug_fv_mono</title>
+<g id="a_node36"><a xlink:title="plug_fv_mono theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="474.45" cy="-403" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="474.45" y="-400.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">plug_fv_mono</text>
+</a>
+</g>
+</g>
+<!-- red1r_fv -->
+<g id="node38" class="node">
+<title>red1r_fv</title>
+<g id="a_node38"><a xlink:title="red1r_fv theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="654.03" cy="-378" rx="31.35" ry="18"/>
+<text text-anchor="middle" x="654.03" y="-375.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red1r_fv</text>
+</a>
+</g>
+</g>
+<!-- plug_fv_mono&#45;&gt;red1r_fv -->
+<g id="edge20" class="edge">
+<title>plug_fv_mono&#45;&gt;red1r_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M518.44,-396.95C548.69,-392.69 588.71,-387.06 617.23,-383.04"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="617.85,-385.07 623.5,-382.16 617.26,-380.92 617.85,-385.07"/>
+</g>
+<!-- red1_root_fv&#45;&gt;red1r_fv -->
+<g id="edge21" class="edge">
+<title>red1_root_fv&#45;&gt;red1r_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M514.86,-358.55C545.58,-362.87 587.86,-368.82 617.53,-373"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="617.43,-375.11 623.67,-373.87 618.02,-370.95 617.43,-375.11"/>
+</g>
+<!-- red1r_fv&#45;&gt;lemma_2_1_fv_preserved -->
+<g id="edge22" class="edge">
+<title>red1r_fv&#45;&gt;lemma_2_1_fv_preserved</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M685.41,-378C708.44,-378 741.05,-378 770.52,-378"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="770.75,-380.1 776.75,-378 770.75,-375.9 770.75,-380.1"/>
+</g>
+<!-- zdecs_nonempty -->
+<g id="node39" class="node">
+<title>zdecs_nonempty</title>
+<g id="a_node39"><a xlink:title="zdecs_nonempty theory/Metatheory.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-68" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-65.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zdecs_nonempty</text>
+</a>
+</g>
+</g>
+<!-- zdecs_nonempty&#45;&gt;full_comp_aux -->
+<g id="edge15" class="edge">
+<title>zdecs_nonempty&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M319.78,-81.09C336.58,-87.13 352.87,-93 352.87,-93 352.87,-93 401.07,-112.98 436.52,-127.68"/>
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+</g>
+<!-- check_all_true -->
+<g id="node40" class="node">
+<title>check_all_true</title>
+<g id="a_node40"><a xlink:title="check_all_true theory/Random.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="1211.82" cy="-278" rx="46.53" ry="18"/>
+<text text-anchor="middle" x="1211.82" y="-275.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">check_all_true</text>
+</a>
+</g>
+</g>
+<!-- check_all_true&#45;&gt;enumerate_check -->
+<g id="edge3" class="edge">
+<title>check_all_true&#45;&gt;enumerate_check</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M1258.71,-278C1276.76,-278 1297.76,-278 1317.13,-278"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="1317.35,-280.1 1323.35,-278 1317.35,-275.9 1317.35,-280.1"/>
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+<!-- check_term_true&#45;&gt;check_all_true -->
+<g id="edge27" class="edge">
+<title>check_term_true&#45;&gt;check_all_true</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M1090.47,-278C1112.21,-278 1137.31,-278 1158.94,-278"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="1158.95,-280.1 1164.95,-278 1158.95,-275.9 1158.95,-280.1"/>
+</g>
+<!-- gen_list_length -->
+<g id="node44" class="node">
+<title>gen_list_length</title>
+<g id="a_node44"><a xlink:title="gen_list_length theory/Random.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1253" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1250.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">gen_list_length</text>
+</a>
+</g>
+</g>
+<!-- close_rec_app -->
+<g id="node46" class="node">
+<title>close_rec_app</title>
+<g id="a_node46"><a xlink:title="close_rec_app theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1303" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1300.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_app</text>
+</a>
+</g>
+</g>
+<!-- close_rec_esub -->
+<g id="node47" class="node">
+<title>close_rec_esub</title>
+<g id="a_node47"><a xlink:title="close_rec_esub theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1353" rx="48.49" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1350.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_esub</text>
+</a>
+</g>
+</g>
+<!-- close_rec_lam -->
+<g id="node48" class="node">
+<title>close_rec_lam</title>
+<g id="a_node48"><a xlink:title="close_rec_lam theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1403" rx="45.72" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1400.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">close_rec_lam</text>
+</a>
+</g>
+</g>
+<!-- open_rec_app -->
+<g id="node49" class="node">
+<title>open_rec_app</title>
+<g id="a_node49"><a xlink:title="open_rec_app theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1453" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1450.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_app</text>
+</a>
+</g>
+</g>
+<!-- open_rec_esub -->
+<g id="node50" class="node">
+<title>open_rec_esub</title>
+<g id="a_node50"><a xlink:title="open_rec_esub theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1503" rx="48.49" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1500.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_esub</text>
+</a>
+</g>
+</g>
+<!-- open_rec_lam -->
+<g id="node51" class="node">
+<title>open_rec_lam</title>
+<g id="a_node51"><a xlink:title="open_rec_lam theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1553" rx="45.72" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1550.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_lam</text>
+</a>
+</g>
+</g>
+<!-- open_rec_lc -->
+<g id="node52" class="node">
+<title>open_rec_lc</title>
+<g id="a_node52"><a xlink:title="open_rec_lc theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="282.44" cy="-1603" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="282.44" y="-1600.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_lc</text>
+</a>
+</g>
+</g>
+<!-- open_rec_lc_atom -->
+<g id="node53" class="node">
+<title>open_rec_lc_atom</title>
+<g id="a_node53"><a xlink:title="open_rec_lc_atom theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1603" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1600.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_lc_atom</text>
+</a>
+</g>
+</g>
+<!-- open_rec_lc_atom&#45;&gt;open_rec_lc -->
+<g id="edge36" class="edge">
+<title>open_rec_lc_atom&#45;&gt;open_rec_lc</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M130.23,-1603C163.33,-1603 204.79,-1603 235.92,-1603"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="236.01,-1605.1 242.01,-1603 236.01,-1600.9 236.01,-1605.1"/>
+</g>
+<!-- subst_app -->
+<g id="node54" class="node">
+<title>subst_app</title>
+<g id="a_node54"><a xlink:title="subst_app theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1653" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1650.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_app</text>
+</a>
+</g>
+</g>
+<!-- subst_esub -->
+<g id="node55" class="node">
+<title>subst_esub</title>
+<g id="a_node55"><a xlink:title="subst_esub theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1703" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1700.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_esub</text>
+</a>
+</g>
+</g>
+<!-- subst_lam -->
+<g id="node56" class="node">
+<title>subst_lam</title>
+<g id="a_node56"><a xlink:title="subst_lam theory/Substitution.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="73.5" cy="-1753" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="73.5" y="-1750.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subst_lam</text>
+</a>
+</g>
+</g>
+</g>
+</svg>

graphs/dependency-M3.dot

@@ -0,0 +1,52 @@
+digraph deps {
+  rankdir=LR;
+  bgcolor="transparent";
+  splines=polyline;
+  concentrate=true;
+  nodesep=0.2;
+  ranksep=0.9;
+  node [fontname=Inter, fontsize=9];
+  edge [fontname=Inter, fontsize=8, color="#8f9780", fontcolor="#c9d1bb", arrowsize=0.6, penwidth=0.8];
+  graph [fontname=Inter, fontsize=12, labelloc=t, fontcolor="#c9d1bb", label=<<B>Theorem dependencies M3, 16 results</B>>];
+  Plus_to_Star [label="Plus_to_Star\\n(M2)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="Plus_to_Star theory/Closure.v proved"];
+  occurs_count_zero [label="occurs_count_zero\\n(M2)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="occurs_count_zero theory/Metatheory.v proved"];
+  occurs_count_zfill_plug [label="occurs_count_zfill_plug\\n(M2)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="occurs_count_zfill_plug theory/Metatheory.v proved"];
+  open_rec_occurs_false [label="open_rec_occurs_false\\n(M1)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="open_rec_occurs_false theory/Binding.v proved"];
+  open_rec_zplug_lift [label="open_rec_zplug_lift\\n(M2)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="open_rec_zplug_lift theory/Metatheory.v proved"];
+  zdecs_nonempty [label="zdecs_nonempty\\n(M2)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="zdecs_nonempty theory/Metatheory.v proved"];
+  zdecs_sound [label="zdecs_sound\\n(M1)", shape=box, style="rounded,dashed,filled", fillcolor="transparent", color="#8f9780", penwidth=0.9, fontcolor="#c9d1bb", tooltip="zdecs_sound theory/Reduction.v proved"];
+  R_Gc_disjoint [label="R_Gc_disjoint", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="R_Gc_disjoint theory/Subsystem.v proved"];
+  Star_red_sub_context [label="Star_red_sub_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Star_red_sub_context theory/Subsystem.v proved"];
+  es_count_plug_mono [label="es_count_plug_mono", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="es_count_plug_mono theory/Subsystem.v proved"];
+  full_comp_aux_sub [label="full_comp_aux_sub", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="full_comp_aux_sub theory/Subsystem.v proved"];
+  occurs_zfill [label="occurs_zfill", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="occurs_zfill theory/Subsystem.v proved"];
+  plus_red_sub_full_comp [label="plus_red_sub_full_comp", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="plus_red_sub_full_comp theory/Subsystem.v proved"];
+  red_gc_measure [label="red_gc_measure", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_gc_measure theory/Subsystem.v proved"];
+  red_gc_terminates [label="red_gc_terminates", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_gc_terminates theory/Subsystem.v proved"];
+  red_gc_to_red_sub [label="red_gc_to_red_sub", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_gc_to_red_sub theory/Subsystem.v proved"];
+  red_sub_context [label="red_sub_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_sub_context theory/Subsystem.v proved"];
+  red_sub_gc [label="red_sub_gc", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_sub_gc theory/Subsystem.v proved"];
+  red_sub_r [label="red_sub_r", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_sub_r theory/Subsystem.v proved"];
+  red_sub_root_to_red1 [label="red_sub_root_to_red1", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_sub_root_to_red1 theory/Subsystem.v proved"];
+  red_sub_to_red1 [label="red_sub_to_red1", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="red_sub_to_red1 theory/Subsystem.v proved"];
+  star_red_sub_full_comp [label="star_red_sub_full_comp", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="star_red_sub_full_comp theory/Subsystem.v proved"];
+  zfill_occurs0 [label="zfill_occurs0", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="zfill_occurs0 theory/Subsystem.v proved"];
+  red_sub_root_to_red1 -> red_sub_to_red1;
+  occurs_zfill -> zfill_occurs0;
+  zfill_occurs0 -> R_Gc_disjoint;
+  red_sub_context -> Star_red_sub_context;
+  occurs_count_zero -> full_comp_aux_sub;
+  occurs_count_zfill_plug -> full_comp_aux_sub;
+  open_rec_occurs_false -> full_comp_aux_sub;
+  open_rec_zplug_lift -> full_comp_aux_sub;
+  red_sub_gc -> full_comp_aux_sub;
+  red_sub_r -> full_comp_aux_sub;
+  zdecs_nonempty -> full_comp_aux_sub;
+  zdecs_sound -> full_comp_aux_sub;
+  full_comp_aux_sub -> plus_red_sub_full_comp;
+  Plus_to_Star -> star_red_sub_full_comp;
+  plus_red_sub_full_comp -> star_red_sub_full_comp;
+  es_count_plug_mono -> red_gc_measure;
+  red_gc_measure -> red_gc_terminates;
+  red_sub_gc -> red_gc_to_red_sub;
+}

graphs/dependency-M3.svg

@@ -0,0 +1,328 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.42.4 (0)
+ -->
+<!-- Title: deps Pages: 1 -->
+<svg width="772pt" height="615pt"
+ viewBox="0.00 0.00 771.85 615.00" 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(4 611)">
+<title>deps</title>
+<text text-anchor="start" x="252.42" y="-594.4" font-family="Inter" font-weight="bold" font-size="12.00" fill="#c9d1bb">Theorem dependencies M3, 16 results</text>
+<!-- Plus_to_Star -->
+<g id="node1" class="node">
+<title>Plus_to_Star</title>
+<g id="a_node1"><a xlink:title="Plus_to_Star theory/Closure.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M524.88,-261C524.88,-261 447.88,-261 447.88,-261 441.88,-261 435.88,-255 435.88,-249 435.88,-249 435.88,-237 435.88,-237 435.88,-231 441.88,-225 447.88,-225 447.88,-225 524.88,-225 524.88,-225 530.88,-225 536.88,-231 536.88,-237 536.88,-237 536.88,-249 536.88,-249 536.88,-255 530.88,-261 524.88,-261"/>
+<text text-anchor="middle" x="486.38" y="-240.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Plus_to_Star\n(M2)</text>
+</a>
+</g>
+</g>
+<!-- star_red_sub_full_comp -->
+<g id="node22" class="node">
+<title>star_red_sub_full_comp</title>
+<g id="a_node22"><a xlink:title="star_red_sub_full_comp theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="693.41" cy="-218" rx="70.37" ry="18"/>
+<text text-anchor="middle" x="693.41" y="-215.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">star_red_sub_full_comp</text>
+</a>
+</g>
+</g>
+<!-- Plus_to_Star&#45;&gt;star_red_sub_full_comp -->
+<g id="edge14" class="edge">
+<title>Plus_to_Star&#45;&gt;star_red_sub_full_comp</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M537.01,-236.95C563.01,-233.78 595.27,-229.85 623.64,-226.39"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="623.93,-228.47 629.63,-225.66 623.42,-224.3 623.93,-228.47"/>
+</g>
+<!-- occurs_count_zero -->
+<g id="node2" class="node">
+<title>occurs_count_zero</title>
+<g id="a_node2"><a xlink:title="occurs_count_zero theory/Metatheory.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M129.5,-386C129.5,-386 22.5,-386 22.5,-386 16.5,-386 10.5,-380 10.5,-374 10.5,-374 10.5,-362 10.5,-362 10.5,-356 16.5,-350 22.5,-350 22.5,-350 129.5,-350 129.5,-350 135.5,-350 141.5,-356 141.5,-362 141.5,-362 141.5,-374 141.5,-374 141.5,-380 135.5,-386 129.5,-386"/>
+<text text-anchor="middle" x="76" y="-365.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_count_zero\n(M2)</text>
+</a>
+</g>
+</g>
+<!-- full_comp_aux_sub -->
+<g id="node11" class="node">
+<title>full_comp_aux_sub</title>
+<g id="a_node11"><a xlink:title="full_comp_aux_sub theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="283.4" cy="-193" rx="58.78" ry="18"/>
+<text text-anchor="middle" x="283.4" y="-190.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">full_comp_aux_sub</text>
+</a>
+</g>
+</g>
+<!-- occurs_count_zero&#45;&gt;full_comp_aux_sub -->
+<g id="edge5" class="edge">
+<title>occurs_count_zero&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M131.11,-349.96C142.94,-346.02 152,-343 152,-343 152,-343 226.9,-256.84 263.17,-215.12"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="264.83,-216.41 267.18,-210.5 261.66,-213.65 264.83,-216.41"/>
+</g>
+<!-- occurs_count_zfill_plug -->
+<g id="node3" class="node">
+<title>occurs_count_zfill_plug</title>
+<g id="a_node3"><a xlink:title="occurs_count_zfill_plug theory/Metatheory.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M140,-336C140,-336 12,-336 12,-336 6,-336 0,-330 0,-324 0,-324 0,-312 0,-312 0,-306 6,-300 12,-300 12,-300 140,-300 140,-300 146,-300 152,-306 152,-312 152,-312 152,-324 152,-324 152,-330 146,-336 140,-336"/>
+<text text-anchor="middle" x="76" y="-315.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_count_zfill_plug\n(M2)</text>
+</a>
+</g>
+</g>
+<!-- occurs_count_zfill_plug&#45;&gt;full_comp_aux_sub -->
+<g id="edge6" class="edge">
+<title>occurs_count_zfill_plug&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M131.11,-299.96C142.94,-296.02 152,-293 152,-293 152,-293 217.44,-242.81 255.6,-213.55"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="257.09,-215.06 260.57,-209.74 254.53,-211.72 257.09,-215.06"/>
+</g>
+<!-- open_rec_occurs_false -->
+<g id="node4" class="node">
+<title>open_rec_occurs_false</title>
+<g id="a_node4"><a xlink:title="open_rec_occurs_false theory/Binding.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M137.5,-286C137.5,-286 14.5,-286 14.5,-286 8.5,-286 2.5,-280 2.5,-274 2.5,-274 2.5,-262 2.5,-262 2.5,-256 8.5,-250 14.5,-250 14.5,-250 137.5,-250 137.5,-250 143.5,-250 149.5,-256 149.5,-262 149.5,-262 149.5,-274 149.5,-274 149.5,-280 143.5,-286 137.5,-286"/>
+<text text-anchor="middle" x="76" y="-265.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_occurs_false\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- open_rec_occurs_false&#45;&gt;full_comp_aux_sub -->
+<g id="edge7" class="edge">
+<title>open_rec_occurs_false&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M131.11,-249.96C142.94,-246.02 152,-243 152,-243 152,-243 202,-223.83 240.07,-209.23"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="241.22,-211.04 246.07,-206.93 239.72,-207.12 241.22,-211.04"/>
+</g>
+<!-- open_rec_zplug_lift -->
+<g id="node5" class="node">
+<title>open_rec_zplug_lift</title>
+<g id="a_node5"><a xlink:title="open_rec_zplug_lift theory/Metatheory.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M130.5,-236C130.5,-236 21.5,-236 21.5,-236 15.5,-236 9.5,-230 9.5,-224 9.5,-224 9.5,-212 9.5,-212 9.5,-206 15.5,-200 21.5,-200 21.5,-200 130.5,-200 130.5,-200 136.5,-200 142.5,-206 142.5,-212 142.5,-212 142.5,-224 142.5,-224 142.5,-230 136.5,-236 130.5,-236"/>
+<text text-anchor="middle" x="76" y="-215.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">open_rec_zplug_lift\n(M2)</text>
+</a>
+</g>
+</g>
+<!-- open_rec_zplug_lift&#45;&gt;full_comp_aux_sub -->
+<g id="edge8" class="edge">
+<title>open_rec_zplug_lift&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M142.55,-210.02C168.1,-206.91 197.18,-203.37 222.19,-200.33"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="222.75,-202.38 228.45,-199.57 222.24,-198.21 222.75,-202.38"/>
+</g>
+<!-- zdecs_nonempty -->
+<g id="node6" class="node">
+<title>zdecs_nonempty</title>
+<g id="a_node6"><a xlink:title="zdecs_nonempty theory/Metatheory.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M125.5,-186C125.5,-186 26.5,-186 26.5,-186 20.5,-186 14.5,-180 14.5,-174 14.5,-174 14.5,-162 14.5,-162 14.5,-156 20.5,-150 26.5,-150 26.5,-150 125.5,-150 125.5,-150 131.5,-150 137.5,-156 137.5,-162 137.5,-162 137.5,-174 137.5,-174 137.5,-180 131.5,-186 125.5,-186"/>
+<text text-anchor="middle" x="76" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zdecs_nonempty\n(M2)</text>
+</a>
+</g>
+</g>
+<!-- zdecs_nonempty&#45;&gt;full_comp_aux_sub -->
+<g id="edge11" class="edge">
+<title>zdecs_nonempty&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M137.51,-175.37C164.25,-178.62 195.6,-182.44 222.28,-185.68"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="222.04,-187.77 228.25,-186.41 222.55,-183.6 222.04,-187.77"/>
+</g>
+<!-- zdecs_sound -->
+<g id="node7" class="node">
+<title>zdecs_sound</title>
+<g id="a_node7"><a xlink:title="zdecs_sound theory/Reduction.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M116,-136C116,-136 36,-136 36,-136 30,-136 24,-130 24,-124 24,-124 24,-112 24,-112 24,-106 30,-100 36,-100 36,-100 116,-100 116,-100 122,-100 128,-106 128,-112 128,-112 128,-124 128,-124 128,-130 122,-136 116,-136"/>
+<text text-anchor="middle" x="76" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zdecs_sound\n(M1)</text>
+</a>
+</g>
+</g>
+<!-- zdecs_sound&#45;&gt;full_comp_aux_sub -->
+<g id="edge12" class="edge">
+<title>zdecs_sound&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M128.27,-135.09C141.45,-139.48 152,-143 152,-143 152,-143 202,-162.17 240.07,-176.77"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="239.72,-178.88 246.07,-179.07 241.22,-174.96 239.72,-178.88"/>
+</g>
+<!-- R_Gc_disjoint -->
+<g id="node8" class="node">
+<title>R_Gc_disjoint</title>
+<g id="a_node8"><a xlink:title="R_Gc_disjoint theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="486.38" cy="-418" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="486.38" y="-415.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">R_Gc_disjoint</text>
+</a>
+</g>
+</g>
+<!-- Star_red_sub_context -->
+<g id="node9" class="node">
+<title>Star_red_sub_context</title>
+<g id="a_node9"><a xlink:title="Star_red_sub_context theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="283.4" cy="-468" rx="66.29" ry="18"/>
+<text text-anchor="middle" x="283.4" y="-465.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Star_red_sub_context</text>
+</a>
+</g>
+</g>
+<!-- es_count_plug_mono -->
+<g id="node10" class="node">
+<title>es_count_plug_mono</title>
+<g id="a_node10"><a xlink:title="es_count_plug_mono theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="76" cy="-518" rx="64.83" ry="18"/>
+<text text-anchor="middle" x="76" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">es_count_plug_mono</text>
+</a>
+</g>
+</g>
+<!-- red_gc_measure -->
+<g id="node14" class="node">
+<title>red_gc_measure</title>
+<g id="a_node14"><a xlink:title="red_gc_measure theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="283.4" cy="-518" rx="52.58" ry="18"/>
+<text text-anchor="middle" x="283.4" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_gc_measure</text>
+</a>
+</g>
+</g>
+<!-- es_count_plug_mono&#45;&gt;red_gc_measure -->
+<g id="edge16" class="edge">
+<title>es_count_plug_mono&#45;&gt;red_gc_measure</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M140.86,-518C167.57,-518 198.39,-518 224.44,-518"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="224.61,-520.1 230.61,-518 224.61,-515.9 224.61,-520.1"/>
+</g>
+<!-- plus_red_sub_full_comp -->
+<g id="node13" class="node">
+<title>plus_red_sub_full_comp</title>
+<g id="a_node13"><a xlink:title="plus_red_sub_full_comp theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="486.38" cy="-193" rx="71.68" ry="18"/>
+<text text-anchor="middle" x="486.38" y="-190.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">plus_red_sub_full_comp</text>
+</a>
+</g>
+</g>
+<!-- full_comp_aux_sub&#45;&gt;plus_red_sub_full_comp -->
+<g id="edge13" class="edge">
+<title>full_comp_aux_sub&#45;&gt;plus_red_sub_full_comp</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M342.53,-193C363.09,-193 386.54,-193 408.42,-193"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="408.54,-195.1 414.54,-193 408.54,-190.9 408.54,-195.1"/>
+</g>
+<!-- occurs_zfill -->
+<g id="node12" class="node">
+<title>occurs_zfill</title>
+<g id="a_node12"><a xlink:title="occurs_zfill theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="76" cy="-418" rx="38.21" ry="18"/>
+<text text-anchor="middle" x="76" y="-415.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">occurs_zfill</text>
+</a>
+</g>
+</g>
+<!-- zfill_occurs0 -->
+<g id="node23" class="node">
+<title>zfill_occurs0</title>
+<g id="a_node23"><a xlink:title="zfill_occurs0 theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="283.4" cy="-418" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="283.4" y="-415.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">zfill_occurs0</text>
+</a>
+</g>
+</g>
+<!-- occurs_zfill&#45;&gt;zfill_occurs0 -->
+<g id="edge2" class="edge">
+<title>occurs_zfill&#45;&gt;zfill_occurs0</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M114.2,-418C148.18,-418 198.6,-418 235.37,-418"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="235.6,-420.1 241.6,-418 235.6,-415.9 235.6,-420.1"/>
+</g>
+<!-- plus_red_sub_full_comp&#45;&gt;star_red_sub_full_comp -->
+<g id="edge15" class="edge">
+<title>plus_red_sub_full_comp&#45;&gt;star_red_sub_full_comp</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M551.13,-200.77C574.01,-203.56 599.9,-206.72 623.22,-209.56"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="623.2,-211.68 629.41,-210.32 623.7,-207.51 623.2,-211.68"/>
+</g>
+<!-- red_gc_terminates -->
+<g id="node15" class="node">
+<title>red_gc_terminates</title>
+<g id="a_node15"><a xlink:title="red_gc_terminates theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="486.38" cy="-518" rx="58.13" ry="18"/>
+<text text-anchor="middle" x="486.38" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_gc_terminates</text>
+</a>
+</g>
+</g>
+<!-- red_gc_measure&#45;&gt;red_gc_terminates -->
+<g id="edge17" class="edge">
+<title>red_gc_measure&#45;&gt;red_gc_terminates</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M336.15,-518C362.21,-518 394.04,-518 421.6,-518"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="421.78,-520.1 427.78,-518 421.78,-515.9 421.78,-520.1"/>
+</g>
+<!-- red_gc_to_red_sub -->
+<g id="node16" class="node">
+<title>red_gc_to_red_sub</title>
+<g id="a_node16"><a xlink:title="red_gc_to_red_sub theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="283.4" cy="-18" rx="58.13" ry="18"/>
+<text text-anchor="middle" x="283.4" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_gc_to_red_sub</text>
+</a>
+</g>
+</g>
+<!-- red_sub_context -->
+<g id="node17" class="node">
+<title>red_sub_context</title>
+<g id="a_node17"><a xlink:title="red_sub_context theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="76" cy="-468" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="76" y="-465.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_sub_context</text>
+</a>
+</g>
+</g>
+<!-- red_sub_context&#45;&gt;Star_red_sub_context -->
+<g id="edge4" class="edge">
+<title>red_sub_context&#45;&gt;Star_red_sub_context</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M129.36,-468C154.11,-468 184.05,-468 210.85,-468"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="210.88,-470.1 216.88,-468 210.88,-465.9 210.88,-470.1"/>
+</g>
+<!-- red_sub_gc -->
+<g id="node18" class="node">
+<title>red_sub_gc</title>
+<g id="a_node18"><a xlink:title="red_sub_gc theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="76" cy="-18" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="76" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_sub_gc</text>
+</a>
+</g>
+</g>
+<!-- red_sub_gc&#45;&gt;full_comp_aux_sub -->
+<g id="edge9" class="edge">
+<title>red_sub_gc&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M108.37,-28.46C128.92,-35.31 152,-43 152,-43 152,-43 226.9,-129.16 263.17,-170.88"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="261.66,-172.35 267.18,-175.5 264.83,-169.59 261.66,-172.35"/>
+</g>
+<!-- red_sub_gc&#45;&gt;red_gc_to_red_sub -->
+<g id="edge18" class="edge">
+<title>red_sub_gc&#45;&gt;red_gc_to_red_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M115.63,-18C144.62,-18 184.95,-18 218.69,-18"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="218.91,-20.1 224.91,-18 218.91,-15.9 218.91,-20.1"/>
+</g>
+<!-- red_sub_r -->
+<g id="node19" class="node">
+<title>red_sub_r</title>
+<g id="a_node19"><a xlink:title="red_sub_r theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="76" cy="-68" rx="35.44" ry="18"/>
+<text text-anchor="middle" x="76" y="-65.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_sub_r</text>
+</a>
+</g>
+</g>
+<!-- red_sub_r&#45;&gt;full_comp_aux_sub -->
+<g id="edge10" class="edge">
+<title>red_sub_r&#45;&gt;full_comp_aux_sub</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M106.21,-77.74C127.27,-84.76 152,-93 152,-93 152,-93 217.44,-143.19 255.6,-172.45"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="254.53,-174.28 260.57,-176.26 257.09,-170.94 254.53,-174.28"/>
+</g>
+<!-- red_sub_root_to_red1 -->
+<g id="node20" class="node">
+<title>red_sub_root_to_red1</title>
+<g id="a_node20"><a xlink:title="red_sub_root_to_red1 theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="76" cy="-568" rx="65.64" ry="18"/>
+<text text-anchor="middle" x="76" y="-565.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_sub_root_to_red1</text>
+</a>
+</g>
+</g>
+<!-- red_sub_to_red1 -->
+<g id="node21" class="node">
+<title>red_sub_to_red1</title>
+<g id="a_node21"><a xlink:title="red_sub_to_red1 theory/Subsystem.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="283.4" cy="-568" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="283.4" y="-565.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red_sub_to_red1</text>
+</a>
+</g>
+</g>
+<!-- red_sub_root_to_red1&#45;&gt;red_sub_to_red1 -->
+<g id="edge1" class="edge">
+<title>red_sub_root_to_red1&#45;&gt;red_sub_to_red1</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M141.99,-568C168.32,-568 198.49,-568 224.1,-568"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="224.17,-570.1 230.17,-568 224.17,-565.9 224.17,-570.1"/>
+</g>
+<!-- zfill_occurs0&#45;&gt;R_Gc_disjoint -->
+<g id="edge3" class="edge">
+<title>zfill_occurs0&#45;&gt;R_Gc_disjoint</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M325.07,-418C357.14,-418 401.99,-418 436.1,-418"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="436.34,-420.1 442.34,-418 436.34,-415.9 436.34,-420.1"/>
+</g>
+</g>
+</svg>

graphs/dependency-M4.dot

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+digraph deps {
+  rankdir=LR;
+  bgcolor="transparent";
+  splines=polyline;
+  concentrate=true;
+  nodesep=0.2;
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+  node [fontname=Inter, fontsize=9];
+  edge [fontname=Inter, fontsize=8, color="#8f9780", fontcolor="#c9d1bb", arrowsize=0.6, penwidth=0.8];
+  graph [fontname=Inter, fontsize=12, labelloc=t, fontcolor="#c9d1bb", label=<<B>Theorem dependencies M4, 6 results</B>>];
+  par_context [label="par_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="par_context theory/Parallel.v proved"];
+  par_red1_iff [label="par_red1_iff", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="par_red1_iff theory/Parallel.v proved"];
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+  par_red1_iff -> par_to_red1_or_eq;
+  par_red1_iff -> red1_or_eq_to_par;
+}

graphs/dependency-M4.svg

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+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.42.4 (0)
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+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 211)">
+<title>deps</title>
+<text text-anchor="start" x="8" y="-194.4" font-family="Inter" font-weight="bold" font-size="12.00" fill="#c9d1bb">Theorem dependencies M4, 6 results</text>
+<!-- par_context -->
+<g id="node1" class="node">
+<title>par_context</title>
+<g id="a_node1"><a xlink:title="par_context theory/Parallel.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="43.26" cy="-18" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="43.26" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">par_context</text>
+</a>
+</g>
+</g>
+<!-- par_red1_iff -->
+<g id="node2" class="node">
+<title>par_red1_iff</title>
+<g id="a_node2"><a xlink:title="par_red1_iff theory/Parallel.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="43.26" cy="-68" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="43.26" y="-65.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">par_red1_iff</text>
+</a>
+</g>
+</g>
+<!-- par_to_red1_or_eq -->
+<g id="node4" class="node">
+<title>par_to_red1_or_eq</title>
+<g id="a_node4"><a xlink:title="par_to_red1_or_eq theory/Parallel.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="208.72" cy="-93" rx="57.97" ry="18"/>
+<text text-anchor="middle" x="208.72" y="-90.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">par_to_red1_or_eq</text>
+</a>
+</g>
+</g>
+<!-- par_red1_iff&#45;&gt;par_to_red1_or_eq -->
+<g id="edge1" class="edge">
+<title>par_red1_iff&#45;&gt;par_to_red1_or_eq</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M81.77,-73.74C102.1,-76.85 127.66,-80.75 150.43,-84.24"/>
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+<!-- red1_or_eq_to_par -->
+<g id="node6" class="node">
+<title>red1_or_eq_to_par</title>
+<g id="a_node6"><a xlink:title="red1_or_eq_to_par theory/Parallel.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="208.72" cy="-43" rx="57.97" ry="18"/>
+<text text-anchor="middle" x="208.72" y="-40.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red1_or_eq_to_par</text>
+</a>
+</g>
+</g>
+<!-- par_red1_iff&#45;&gt;red1_or_eq_to_par -->
+<g id="edge2" class="edge">
+<title>par_red1_iff&#45;&gt;red1_or_eq_to_par</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M81.77,-62.26C102.1,-59.15 127.66,-55.25 150.43,-51.76"/>
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+<!-- par_reflexive -->
+<g id="node3" class="node">
+<title>par_reflexive</title>
+<g id="a_node3"><a xlink:title="par_reflexive theory/Parallel.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="43.26" cy="-118" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="43.26" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">par_reflexive</text>
+</a>
+</g>
+</g>
+<!-- red1_in_par -->
+<g id="node5" class="node">
+<title>red1_in_par</title>
+<g id="a_node5"><a xlink:title="red1_in_par theory/Parallel.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="43.26" cy="-168" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="43.26" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">red1_in_par</text>
+</a>
+</g>
+</g>
+</g>
+</svg>

graphs/dependency-M5.dot

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+digraph deps {
+  rankdir=LR;
+  bgcolor="transparent";
+  splines=polyline;
+  concentrate=true;
+  nodesep=0.2;
+  ranksep=0.9;
+  node [fontname=Inter, fontsize=9];
+  edge [fontname=Inter, fontsize=8, color="#8f9780", fontcolor="#c9d1bb", arrowsize=0.6, penwidth=0.8];
+  graph [fontname=Inter, fontsize=12, labelloc=t, fontcolor="#c9d1bb", label=<<B>Theorem dependencies M5, 69 results</B>>];
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+  mfvs_open_m [label="mfvs_open_m", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="mfvs_open_m theory/Metaterm.v proved"];
+  mred_B_intro [label="mred_B_intro", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="mred_B_intro theory/Metaterm.v proved"];
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+  mred_term_context [label="mred_term_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="mred_term_context theory/Metaterm.v proved"];
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+  of_trm_injective [label="of_trm_injective", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="of_trm_injective theory/Metaterm.v proved"];
+  of_trm_lift [label="of_trm_lift", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="of_trm_lift theory/Metaterm.v proved"];
+  of_trm_moccurs [label="of_trm_moccurs", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="of_trm_moccurs theory/Metaterm.v proved"];
+  of_trm_moccurs0 [label="of_trm_moccurs0", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="of_trm_moccurs0 theory/Metaterm.v proved"];
+  of_trm_open [label="of_trm_open", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="of_trm_open theory/Metaterm.v proved"];
+  of_trm_plug [label="of_trm_plug", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="of_trm_plug theory/Metaterm.v proved"];
+  PlusN_to_StarN [label="PlusN_to_StarN", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="PlusN_to_StarN theory/NamedEs.v proved"];
+  StarN_one_trans [label="StarN_one_trans", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="StarN_one_trans theory/NamedEs.v proved"];
+  StarN_subred_context [label="StarN_subred_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="StarN_subred_context theory/NamedEs.v proved"];
+  StarN_trans [label="StarN_trans", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="StarN_trans theory/NamedEs.v proved"];
+  nred_stable_source [label="nred_stable_source", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_stable_source theory/NamedEs.v proved"];
+  nred_stable_target [label="nred_stable_target", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_stable_target theory/NamedEs.v proved"];
+  subred_Es_l [label="subred_Es_l", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_Es_l theory/NamedEs.v proved"];
+  subred_Es_lr [label="subred_Es_lr", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_Es_lr theory/NamedEs.v proved"];
+  subred_Es_r [label="subred_Es_r", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_Es_r theory/NamedEs.v proved"];
+  subred_context [label="subred_context", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_context theory/NamedEs.v proved"];
+  subred_context_rule [label="subred_context_rule", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_context_rule theory/NamedEs.v proved"];
+  subred_intro [label="subred_intro", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_intro theory/NamedEs.v proved"];
+  subred_nred [label="subred_nred", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_nred theory/NamedEs.v proved"];
+  subred_of_Es_nred_Es [label="subred_of_Es_nred_Es", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_of_Es_nred_Es theory/NamedEs.v proved"];
+  A1_C_related [label="A1_C_related", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="A1_C_related theory/NamedMeasure.v proved"];
+  A1_Mx_differ [label="A1_Mx_differ", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="A1_Mx_differ theory/NamedMeasure.v proved"];
+  A1_not_invariant [label="A1_not_invariant", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="A1_not_invariant theory/NamedMeasure.v proved"];
+  A1_s_differ [label="A1_s_differ", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="A1_s_differ theory/NamedMeasure.v proved"];
+  Gc_special_nondecrease [label="Gc_special_nondecrease", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Gc_special_nondecrease theory/NamedMeasure.v proved"];
+  Mx_fresh_fails [label="Mx_fresh_fails", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Mx_fresh_fails theory/NamedMeasure.v proved"];
+  Mx_ge_0 [label="Mx_ge_0", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Mx_ge_0 theory/NamedMeasure.v proved"];
+  current_sm_not_terminating [label="current_sm_not_terminating", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="current_sm_not_terminating theory/NamedMeasure.v proved"];
+  current_subred_not_terminating [label="current_subred_not_terminating", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="current_subred_not_terminating theory/NamedMeasure.v proved"];
+  head_has_pos [label="head_has_pos", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="head_has_pos theory/NamedMeasure.v proved"];
+  le_mul_of_one_le [label="le_mul_of_one_le", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="le_mul_of_one_le theory/NamedMeasure.v proved"];
+  nfv_annot_bound_example [label="nfv_annot_bound_example", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nfv_annot_bound_example theory/NamedMeasure.v proved"];
+  nocc_le_Mx [label="nocc_le_Mx", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nocc_le_Mx theory/NamedMeasure.v proved"];
+  nred_Gc_special [label="nred_Gc_special", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_Gc_special theory/NamedMeasure.v proved"];
+  nred_R_selfloop [label="nred_R_selfloop", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_R_selfloop theory/NamedMeasure.v proved"];
+  s_metavar_empty [label="s_metavar_empty", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="s_metavar_empty theory/NamedMeasure.v proved"];
+  subred_selfloop [label="subred_selfloop", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="subred_selfloop theory/NamedMeasure.v proved"];
+  Es_ctx_any [label="Es_ctx_any", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Es_ctx_any theory/NamedMeta.v proved"];
+  Es_fv [label="Es_fv", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Es_fv theory/NamedMeta.v proved"];
+  Es_fv_both [label="Es_fv_both", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Es_fv_both theory/NamedMeta.v proved"];
+  Es_refl_any [label="Es_refl_any", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Es_refl_any theory/NamedMeta.v proved"];
+  Es_sym_any [label="Es_sym_any", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Es_sym_any theory/NamedMeta.v proved"];
+  Es_trans_any [label="Es_trans_any", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="Es_trans_any theory/NamedMeta.v proved"];
+  eqC_in_Es [label="eqC_in_Es", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="eqC_in_Es theory/NamedMeta.v proved"];
+  filter_neq_in [label="filter_neq_in", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="filter_neq_in theory/NamedMeta.v proved"];
+  in_filter_neq [label="in_filter_neq", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="in_filter_neq theory/NamedMeta.v proved"];
+  in_filter_neq_iff [label="in_filter_neq_iff", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="in_filter_neq_iff theory/NamedMeta.v proved"];
+  isubst_meta_in [label="isubst_meta_in", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="isubst_meta_in theory/NamedMeta.v proved"];
+  isubst_meta_notin [label="isubst_meta_notin", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="isubst_meta_notin theory/NamedMeta.v proved"];
+  nplug_esub_fv [label="nplug_esub_fv", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nplug_esub_fv theory/NamedMeta.v proved"];
+  nplug_fv_mono [label="nplug_fv_mono", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nplug_fv_mono theory/NamedMeta.v proved"];
+  nplug_fv_upper [label="nplug_fv_upper", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nplug_fv_upper theory/NamedMeta.v proved"];
+  nred_R_intro [label="nred_R_intro", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_R_intro theory/NamedMeta.v proved"];
+  nred_core_fv [label="nred_core_fv", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_core_fv theory/NamedMeta.v proved"];
+  nred_fv [label="nred_fv", shape=ellipse, style="filled", fillcolor="transparent", color="#a9b492", penwidth=1.0, fontcolor="#c9d1bb", tooltip="nred_fv theory/NamedMeta.v proved"];
+  of_trm_lift -> of_trm_open;
+  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;
+  subred_intro -> subred_nred;
+  subred_intro -> subred_Es_l;
+  subred_intro -> subred_Es_r;
+  subred_Es_l -> subred_Es_lr;
+  subred_Es_r -> subred_Es_lr;
+  subred_intro -> subred_context;
+  subred_context -> subred_context_rule;
+  subred_nred -> subred_context_rule;
+  subred_intro -> subred_of_Es_nred_Es;
+  subred_intro -> nred_stable_source;
+  subred_intro -> nred_stable_target;
+  subred_context -> StarN_subred_context;
+  head_has_pos -> nocc_le_Mx;
+  le_mul_of_one_le -> nocc_le_Mx;
+  nfv_annot_bound_example -> Mx_fresh_fails;
+  A1_C_related -> A1_not_invariant;
+  A1_s_differ -> A1_not_invariant;
+  nred_R_selfloop -> current_sm_not_terminating;
+  nred_R_selfloop -> subred_selfloop;
+  subred_nred -> subred_selfloop;
+  subred_selfloop -> current_subred_not_terminating;
+  filter_neq_in -> nplug_fv_mono;
+  in_filter_neq -> nplug_fv_mono;
+  filter_neq_in -> nplug_fv_upper;
+  in_filter_neq -> nplug_fv_upper;
+  filter_neq_in -> nred_core_fv;
+  in_filter_neq -> nred_core_fv;
+  nplug_fv_mono -> nred_core_fv;
+  nplug_fv_upper -> nred_core_fv;
+  filter_neq_in -> nplug_esub_fv;
+  in_filter_neq -> nplug_esub_fv;
+  filter_neq_in -> nred_fv;
+  in_filter_neq -> nred_fv;
+  nplug_esub_fv -> nred_fv;
+  nplug_fv_mono -> nred_fv;
+  nplug_fv_upper -> nred_fv;
+  in_filter_neq_iff -> Es_fv_both;
+  nplug_fv_mono -> Es_fv_both;
+  Es_fv_both -> Es_fv;
+}

graphs/dependency-M5.svg

@@ -0,0 +1,913 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+ "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<!-- Generated by graphviz version 2.42.4 (0)
+ -->
+<!-- Title: deps Pages: 1 -->
+<svg width="854pt" height="2210pt"
+ viewBox="0.00 0.00 854.25 2210.00" 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 2206)">
+<title>deps</title>
+<text text-anchor="start" x="293.63" y="-2189.4" font-family="Inter" font-weight="bold" font-size="12.00" fill="#c9d1bb">Theorem dependencies M5, 69 results</text>
+<!-- lemma_2_2_full_comp -->
+<g id="node1" class="node">
+<title>lemma_2_2_full_comp</title>
+<g id="a_node1"><a xlink:title="lemma_2_2_full_comp (section 2.2) theory/Metatheory.v proved">
+<path fill="transparent" stroke="#8f9780" stroke-width="0.9" stroke-dasharray="5,2" d="M140.67,-86C140.67,-86 18.67,-86 18.67,-86 12.67,-86 6.67,-80 6.67,-74 6.67,-74 6.67,-62 6.67,-62 6.67,-56 12.67,-50 18.67,-50 18.67,-50 140.67,-50 140.67,-50 146.67,-50 152.67,-56 152.67,-62 152.67,-62 152.67,-74 152.67,-74 152.67,-80 146.67,-86 140.67,-86"/>
+<text text-anchor="middle" x="79.67" y="-65.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">lemma_2_2_full_comp\n(M2)</text>
+</a>
+</g>
+</g>
+<!-- mred_full_comp_pure -->
+<g id="node8" class="node">
+<title>mred_full_comp_pure</title>
+<g id="a_node8"><a xlink:title="mred_full_comp_pure theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-43" rx="65.64" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-40.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_full_comp_pure</text>
+</a>
+</g>
+</g>
+<!-- lemma_2_2_full_comp&#45;&gt;mred_full_comp_pure -->
+<g id="edge6" class="edge">
+<title>lemma_2_2_full_comp&#45;&gt;mred_full_comp_pure</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M152.82,-59.4C176.09,-56.62 201.75,-53.56 224.53,-50.85"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="224.86,-52.92 230.56,-50.13 224.36,-48.75 224.86,-52.92"/>
+</g>
+<!-- Plus_mred_of_Plus_red1 -->
+<g id="node2" class="node">
+<title>Plus_mred_of_Plus_red1</title>
+<g id="a_node2"><a xlink:title="Plus_mred_of_Plus_red1 theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-18" rx="71.68" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-15.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Plus_mred_of_Plus_red1</text>
+</a>
+</g>
+</g>
+<!-- Plus_mred_of_Plus_red1&#45;&gt;mred_full_comp_pure -->
+<g id="edge5" class="edge">
+<title>Plus_mred_of_Plus_red1&#45;&gt;mred_full_comp_pure</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M144.71,-25.64C170.02,-28.65 199.08,-32.12 224.56,-35.16"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="224.4,-37.25 230.61,-35.88 224.9,-33.08 224.4,-37.25"/>
+</g>
+<!-- mfvs_close_m -->
+<g id="node3" class="node">
+<title>mfvs_close_m</title>
+<g id="a_node3"><a xlink:title="mfvs_close_m theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-118" rx="45.72" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mfvs_close_m</text>
+</a>
+</g>
+</g>
+<!-- mfvs_lift_m -->
+<g id="node4" class="node">
+<title>mfvs_lift_m</title>
+<g id="a_node4"><a xlink:title="mfvs_lift_m theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-168" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mfvs_lift_m</text>
+</a>
+</g>
+</g>
+<!-- mfvs_open_m -->
+<g id="node5" class="node">
+<title>mfvs_open_m</title>
+<g id="a_node5"><a xlink:title="mfvs_open_m theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-168" rx="45.72" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mfvs_open_m</text>
+</a>
+</g>
+</g>
+<!-- mfvs_lift_m&#45;&gt;mfvs_open_m -->
+<g id="edge2" class="edge">
+<title>mfvs_lift_m&#45;&gt;mfvs_open_m</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M119.13,-168C152.87,-168 202.21,-168 239.26,-168"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="239.56,-170.1 245.56,-168 239.56,-165.9 239.56,-170.1"/>
+</g>
+<!-- mred_B_intro -->
+<g id="node6" class="node">
+<title>mred_B_intro</title>
+<g id="a_node6"><a xlink:title="mred_B_intro theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-218" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-215.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_B_intro</text>
+</a>
+</g>
+</g>
+<!-- mred_context -->
+<g id="node7" class="node">
+<title>mred_context</title>
+<g id="a_node7"><a xlink:title="mred_context theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-268" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-265.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_context</text>
+</a>
+</g>
+</g>
+<!-- mred_gc_intro -->
+<g id="node9" class="node">
+<title>mred_gc_intro</title>
+<g id="a_node9"><a xlink:title="mred_gc_intro theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-318" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-315.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_gc_intro</text>
+</a>
+</g>
+</g>
+<!-- mred_of_red1 -->
+<g id="node10" class="node">
+<title>mred_of_red1</title>
+<g id="a_node10"><a xlink:title="mred_of_red1 theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-368" rx="45.72" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-365.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_of_red1</text>
+</a>
+</g>
+</g>
+<!-- mred_plus_one -->
+<g id="node11" class="node">
+<title>mred_plus_one</title>
+<g id="a_node11"><a xlink:title="mred_plus_one theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-418" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-415.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_plus_one</text>
+</a>
+</g>
+</g>
+<!-- mred_r_intro -->
+<g id="node12" class="node">
+<title>mred_r_intro</title>
+<g id="a_node12"><a xlink:title="mred_r_intro theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-468" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-465.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_r_intro</text>
+</a>
+</g>
+</g>
+<!-- mred_term_context -->
+<g id="node13" class="node">
+<title>mred_term_context</title>
+<g id="a_node13"><a xlink:title="mred_term_context theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-518" rx="61.4" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">mred_term_context</text>
+</a>
+</g>
+</g>
+<!-- of_trm_close -->
+<g id="node14" class="node">
+<title>of_trm_close</title>
+<g id="a_node14"><a xlink:title="of_trm_close theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-568" rx="42.29" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-565.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_close</text>
+</a>
+</g>
+</g>
+<!-- of_trm_fvs -->
+<g id="node15" class="node">
+<title>of_trm_fvs</title>
+<g id="a_node15"><a xlink:title="of_trm_fvs theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-618" rx="36.25" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-615.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_fvs</text>
+</a>
+</g>
+</g>
+<!-- of_trm_injective -->
+<g id="node16" class="node">
+<title>of_trm_injective</title>
+<g id="a_node16"><a xlink:title="of_trm_injective theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-668" rx="50.61" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-665.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_injective</text>
+</a>
+</g>
+</g>
+<!-- of_trm_lift -->
+<g id="node17" class="node">
+<title>of_trm_lift</title>
+<g id="a_node17"><a xlink:title="of_trm_lift theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-718" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-715.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_lift</text>
+</a>
+</g>
+</g>
+<!-- of_trm_open -->
+<g id="node20" class="node">
+<title>of_trm_open</title>
+<g id="a_node20"><a xlink:title="of_trm_open theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-718" rx="42.29" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-715.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_open</text>
+</a>
+</g>
+</g>
+<!-- of_trm_lift&#45;&gt;of_trm_open -->
+<g id="edge1" class="edge">
+<title>of_trm_lift&#45;&gt;of_trm_open</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M115.76,-718C150.62,-718 204.3,-718 242.92,-718"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="242.99,-720.1 248.99,-718 242.99,-715.9 242.99,-720.1"/>
+</g>
+<!-- of_trm_moccurs -->
+<g id="node18" class="node">
+<title>of_trm_moccurs</title>
+<g id="a_node18"><a xlink:title="of_trm_moccurs theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-768" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-765.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_moccurs</text>
+</a>
+</g>
+</g>
+<!-- of_trm_moccurs0 -->
+<g id="node19" class="node">
+<title>of_trm_moccurs0</title>
+<g id="a_node19"><a xlink:title="of_trm_moccurs0 theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-768" rx="54.7" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-765.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_moccurs0</text>
+</a>
+</g>
+</g>
+<!-- of_trm_moccurs&#45;&gt;of_trm_moccurs0 -->
+<g id="edge4" class="edge">
+<title>of_trm_moccurs&#45;&gt;of_trm_moccurs0</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M131.16,-768C160.92,-768 198.75,-768 230.07,-768"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="230.24,-770.1 236.24,-768 230.24,-765.9 230.24,-770.1"/>
+</g>
+<!-- of_trm_plug -->
+<g id="node21" class="node">
+<title>of_trm_plug</title>
+<g id="a_node21"><a xlink:title="of_trm_plug theory/Metaterm.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-518" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">of_trm_plug</text>
+</a>
+</g>
+</g>
+<!-- of_trm_plug&#45;&gt;mred_term_context -->
+<g id="edge3" class="edge">
+<title>of_trm_plug&#45;&gt;mred_term_context</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M121.1,-518C150.11,-518 189.87,-518 223.55,-518"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="223.77,-520.1 229.77,-518 223.77,-515.9 223.77,-520.1"/>
+</g>
+<!-- PlusN_to_StarN -->
+<g id="node22" class="node">
+<title>PlusN_to_StarN</title>
+<g id="a_node22"><a xlink:title="PlusN_to_StarN theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-831" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-828.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">PlusN_to_StarN</text>
+</a>
+</g>
+</g>
+<!-- StarN_one_trans -->
+<g id="node23" class="node">
+<title>StarN_one_trans</title>
+<g id="a_node23"><a xlink:title="StarN_one_trans theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-906" rx="52.58" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-903.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">StarN_one_trans</text>
+</a>
+</g>
+</g>
+<!-- StarN_subred_context -->
+<g id="node24" class="node">
+<title>StarN_subred_context</title>
+<g id="a_node24"><a xlink:title="StarN_subred_context theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1193" rx="67.6" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1190.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">StarN_subred_context</text>
+</a>
+</g>
+</g>
+<!-- StarN_trans -->
+<g id="node25" class="node">
+<title>StarN_trans</title>
+<g id="a_node25"><a xlink:title="StarN_trans theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1081" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1078.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">StarN_trans</text>
+</a>
+</g>
+</g>
+<!-- nred_stable_source -->
+<g id="node26" class="node">
+<title>nred_stable_source</title>
+<g id="a_node26"><a xlink:title="nred_stable_source theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1018" rx="60.09" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1015.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nred_stable_source</text>
+</a>
+</g>
+</g>
+<!-- nred_stable_target -->
+<g id="node27" class="node">
+<title>nred_stable_target</title>
+<g id="a_node27"><a xlink:title="nred_stable_target theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-968" rx="58.13" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-965.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nred_stable_target</text>
+</a>
+</g>
+</g>
+<!-- subred_Es_l -->
+<g id="node28" class="node">
+<title>subred_Es_l</title>
+<g id="a_node28"><a xlink:title="subred_Es_l theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-918" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-915.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_Es_l</text>
+</a>
+</g>
+</g>
+<!-- subred_Es_lr -->
+<g id="node29" class="node">
+<title>subred_Es_lr</title>
+<g id="a_node29"><a xlink:title="subred_Es_lr theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-893" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-890.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_Es_lr</text>
+</a>
+</g>
+</g>
+<!-- subred_Es_l&#45;&gt;subred_Es_lr -->
+<g id="edge10" class="edge">
+<title>subred_Es_l&#45;&gt;subred_Es_lr</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M330.63,-913.53C366.85,-909.3 421.27,-902.95 459.99,-898.44"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="460.35,-900.51 466.06,-897.73 459.86,-896.34 460.35,-900.51"/>
+</g>
+<!-- subred_Es_r -->
+<g id="node30" class="node">
+<title>subred_Es_r</title>
+<g id="a_node30"><a xlink:title="subred_Es_r theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-868" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-865.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_Es_r</text>
+</a>
+</g>
+</g>
+<!-- subred_Es_r&#45;&gt;subred_Es_lr -->
+<g id="edge11" class="edge">
+<title>subred_Es_r&#45;&gt;subred_Es_lr</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M331.12,-872.53C367.41,-876.76 421.65,-883.09 460.21,-887.59"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="460.05,-889.68 466.25,-888.29 460.54,-885.51 460.05,-889.68"/>
+</g>
+<!-- subred_context -->
+<g id="node31" class="node">
+<title>subred_context</title>
+<g id="a_node31"><a xlink:title="subred_context theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1168" rx="50.46" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1165.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_context</text>
+</a>
+</g>
+</g>
+<!-- subred_context&#45;&gt;StarN_subred_context -->
+<g id="edge18" class="edge">
+<title>subred_context&#45;&gt;StarN_subred_context</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M339.34,-1173.49C368.68,-1176.91 406.92,-1181.37 439.42,-1185.16"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="439.22,-1187.25 445.42,-1185.86 439.71,-1183.08 439.22,-1187.25"/>
+</g>
+<!-- subred_context_rule -->
+<g id="node32" class="node">
+<title>subred_context_rule</title>
+<g id="a_node32"><a xlink:title="subred_context_rule theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1143" rx="62.86" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1140.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_context_rule</text>
+</a>
+</g>
+</g>
+<!-- subred_context&#45;&gt;subred_context_rule -->
+<g id="edge13" class="edge">
+<title>subred_context&#45;&gt;subred_context_rule</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M339.34,-1162.51C369.73,-1158.97 409.65,-1154.31 442.87,-1150.43"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="443.27,-1152.5 448.99,-1149.72 442.79,-1148.33 443.27,-1152.5"/>
+</g>
+<!-- subred_intro -->
+<g id="node33" class="node">
+<title>subred_intro</title>
+<g id="a_node33"><a xlink:title="subred_intro theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-968" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-965.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_intro</text>
+</a>
+</g>
+</g>
+<!-- subred_intro&#45;&gt;nred_stable_source -->
+<g id="edge16" class="edge">
+<title>subred_intro&#45;&gt;nred_stable_source</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M117.19,-976.71C150.49,-984.65 200.07,-996.48 237.65,-1005.44"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="237.25,-1007.5 243.57,-1006.85 238.22,-1003.42 237.25,-1007.5"/>
+</g>
+<!-- subred_intro&#45;&gt;nred_stable_target -->
+<g id="edge17" class="edge">
+<title>subred_intro&#45;&gt;nred_stable_target</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M122.6,-968C152.41,-968 192.99,-968 226.76,-968"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="226.99,-970.1 232.99,-968 226.99,-965.9 226.99,-970.1"/>
+</g>
+<!-- subred_intro&#45;&gt;subred_Es_l -->
+<g id="edge8" class="edge">
+<title>subred_intro&#45;&gt;subred_Es_l</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M117.19,-959.29C154.05,-950.5 210.87,-936.95 249.27,-927.79"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="249.92,-929.79 255.27,-926.36 248.94,-925.71 249.92,-929.79"/>
+</g>
+<!-- subred_intro&#45;&gt;subred_Es_r -->
+<g id="edge9" class="edge">
+<title>subred_intro&#45;&gt;subred_Es_r</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M109.08,-954.64C131.61,-944.04 159.35,-931 159.35,-931 159.35,-931 224.35,-893 224.35,-893 224.35,-893 238.69,-887.57 253.99,-881.77"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="254.79,-883.71 259.65,-879.62 253.3,-879.78 254.79,-883.71"/>
+</g>
+<!-- subred_intro&#45;&gt;subred_context -->
+<g id="edge12" class="edge">
+<title>subred_intro&#45;&gt;subred_context</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M95.84,-984.97C118.71,-1010.54 159.35,-1056 159.35,-1056 159.35,-1056 224.35,-1143 224.35,-1143 224.35,-1143 236.61,-1147.65 250.56,-1152.93"/>
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+</g>
+<!-- subred_nred -->
+<g id="node34" class="node">
+<title>subred_nred</title>
+<g id="a_node34"><a xlink:title="subred_nred theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1118" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1115.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_nred</text>
+</a>
+</g>
+</g>
+<!-- subred_intro&#45;&gt;subred_nred -->
+<g id="edge7" class="edge">
+<title>subred_intro&#45;&gt;subred_nred</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M99.16,-984.09C137.97,-1017.85 224.35,-1093 224.35,-1093 224.35,-1093 238.02,-1098.18 252.91,-1103.82"/>
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+</g>
+<!-- subred_of_Es_nred_Es -->
+<g id="node35" class="node">
+<title>subred_of_Es_nred_Es</title>
+<g id="a_node35"><a xlink:title="subred_of_Es_nred_Es theory/NamedEs.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-818" rx="66.95" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-815.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_of_Es_nred_Es</text>
+</a>
+</g>
+</g>
+<!-- subred_intro&#45;&gt;subred_of_Es_nred_Es -->
+<g id="edge15" class="edge">
+<title>subred_intro&#45;&gt;subred_of_Es_nred_Es</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M109.08,-954.64C131.61,-944.04 159.35,-931 159.35,-931 159.35,-931 224.35,-843 224.35,-843 224.35,-843 234.02,-839.34 245.95,-834.82"/>
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+<!-- subred_nred&#45;&gt;subred_context_rule -->
+<g id="edge14" class="edge">
+<title>subred_nred&#45;&gt;subred_context_rule</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M332.62,-1122.7C363.68,-1126.33 407.21,-1131.4 442.96,-1135.58"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="442.87,-1137.68 449.08,-1136.29 443.36,-1133.51 442.87,-1137.68"/>
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+<!-- subred_selfloop -->
+<g id="node52" class="node">
+<title>subred_selfloop</title>
+<g id="a_node52"><a xlink:title="subred_selfloop theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1093" rx="50.46" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1090.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">subred_selfloop</text>
+</a>
+</g>
+</g>
+<!-- subred_nred&#45;&gt;subred_selfloop -->
+<g id="edge26" class="edge">
+<title>subred_nred&#45;&gt;subred_selfloop</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M332.62,-1113.3C366.66,-1109.33 415.68,-1103.61 453.01,-1099.25"/>
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+<!-- A1_C_related -->
+<g id="node36" class="node">
+<title>A1_C_related</title>
+<g id="a_node36"><a xlink:title="A1_C_related theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1243" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1240.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">A1_C_related</text>
+</a>
+</g>
+</g>
+<!-- A1_not_invariant -->
+<g id="node38" class="node">
+<title>A1_not_invariant</title>
+<g id="a_node38"><a xlink:title="A1_not_invariant theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1218" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1215.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">A1_not_invariant</text>
+</a>
+</g>
+</g>
+<!-- A1_C_related&#45;&gt;A1_not_invariant -->
+<g id="edge22" class="edge">
+<title>A1_C_related&#45;&gt;A1_not_invariant</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M121.59,-1238.12C153.92,-1234.27 199.37,-1228.85 234.95,-1224.6"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="235.31,-1226.67 241.02,-1223.88 234.81,-1222.5 235.31,-1226.67"/>
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+<!-- A1_Mx_differ -->
+<g id="node37" class="node">
+<title>A1_Mx_differ</title>
+<g id="a_node37"><a xlink:title="A1_Mx_differ theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1293" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1290.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">A1_Mx_differ</text>
+</a>
+</g>
+</g>
+<!-- A1_s_differ -->
+<g id="node39" class="node">
+<title>A1_s_differ</title>
+<g id="a_node39"><a xlink:title="A1_s_differ theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1193" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1190.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">A1_s_differ</text>
+</a>
+</g>
+</g>
+<!-- A1_s_differ&#45;&gt;A1_not_invariant -->
+<g id="edge23" class="edge">
+<title>A1_s_differ&#45;&gt;A1_not_invariant</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M116.24,-1197.24C148.6,-1201.1 196.89,-1206.86 234.41,-1211.33"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="234.6,-1213.47 240.81,-1212.1 235.1,-1209.3 234.6,-1213.47"/>
+</g>
+<!-- Gc_special_nondecrease -->
+<g id="node40" class="node">
+<title>Gc_special_nondecrease</title>
+<g id="a_node40"><a xlink:title="Gc_special_nondecrease theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1343" rx="72.99" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1340.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Gc_special_nondecrease</text>
+</a>
+</g>
+</g>
+<!-- Mx_fresh_fails -->
+<g id="node41" class="node">
+<title>Mx_fresh_fails</title>
+<g id="a_node41"><a xlink:title="Mx_fresh_fails theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1393" rx="45.07" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1390.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Mx_fresh_fails</text>
+</a>
+</g>
+</g>
+<!-- Mx_ge_0 -->
+<g id="node42" class="node">
+<title>Mx_ge_0</title>
+<g id="a_node42"><a xlink:title="Mx_ge_0 theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1443" rx="32.16" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1440.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Mx_ge_0</text>
+</a>
+</g>
+</g>
+<!-- current_sm_not_terminating -->
+<g id="node43" class="node">
+<title>current_sm_not_terminating</title>
+<g id="a_node43"><a xlink:title="current_sm_not_terminating theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1043" rx="84.09" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1040.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">current_sm_not_terminating</text>
+</a>
+</g>
+</g>
+<!-- current_subred_not_terminating -->
+<g id="node44" class="node">
+<title>current_subred_not_terminating</title>
+<g id="a_node44"><a xlink:title="current_subred_not_terminating theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="751.57" cy="-1093" rx="94.87" ry="18"/>
+<text text-anchor="middle" x="751.57" y="-1090.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">current_subred_not_terminating</text>
+</a>
+</g>
+</g>
+<!-- head_has_pos -->
+<g id="node45" class="node">
+<title>head_has_pos</title>
+<g id="a_node45"><a xlink:title="head_has_pos theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1543" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1540.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">head_has_pos</text>
+</a>
+</g>
+</g>
+<!-- nocc_le_Mx -->
+<g id="node48" class="node">
+<title>nocc_le_Mx</title>
+<g id="a_node48"><a xlink:title="nocc_le_Mx theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1518" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1515.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nocc_le_Mx</text>
+</a>
+</g>
+</g>
+<!-- head_has_pos&#45;&gt;nocc_le_Mx -->
+<g id="edge19" class="edge">
+<title>head_has_pos&#45;&gt;nocc_le_Mx</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M124.11,-1537.82C160.17,-1533.52 211.18,-1527.44 247.16,-1523.15"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="247.54,-1525.22 253.24,-1522.42 247.04,-1521.05 247.54,-1525.22"/>
+</g>
+<!-- le_mul_of_one_le -->
+<g id="node46" class="node">
+<title>le_mul_of_one_le</title>
+<g id="a_node46"><a xlink:title="le_mul_of_one_le theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1493" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1490.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">le_mul_of_one_le</text>
+</a>
+</g>
+</g>
+<!-- le_mul_of_one_le&#45;&gt;nocc_le_Mx -->
+<g id="edge20" class="edge">
+<title>le_mul_of_one_le&#45;&gt;nocc_le_Mx</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M130.09,-1498.89C165.53,-1503.12 212.84,-1508.76 246.84,-1512.82"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="246.83,-1514.93 253.03,-1513.55 247.33,-1510.76 246.83,-1514.93"/>
+</g>
+<!-- nfv_annot_bound_example -->
+<g id="node47" class="node">
+<title>nfv_annot_bound_example</title>
+<g id="a_node47"><a xlink:title="nfv_annot_bound_example theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1393" rx="79.85" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1390.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nfv_annot_bound_example</text>
+</a>
+</g>
+</g>
+<!-- nfv_annot_bound_example&#45;&gt;Mx_fresh_fails -->
+<g id="edge21" class="edge">
+<title>nfv_annot_bound_example&#45;&gt;Mx_fresh_fails</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M159.63,-1393C186.6,-1393 216,-1393 240.07,-1393"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="240.07,-1395.1 246.07,-1393 240.07,-1390.9 240.07,-1395.1"/>
+</g>
+<!-- nred_Gc_special -->
+<g id="node49" class="node">
+<title>nred_Gc_special</title>
+<g id="a_node49"><a xlink:title="nred_Gc_special theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1593" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1590.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nred_Gc_special</text>
+</a>
+</g>
+</g>
+<!-- nred_R_selfloop -->
+<g id="node50" class="node">
+<title>nred_R_selfloop</title>
+<g id="a_node50"><a xlink:title="nred_R_selfloop theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1068" rx="50.46" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1065.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nred_R_selfloop</text>
+</a>
+</g>
+</g>
+<!-- nred_R_selfloop&#45;&gt;current_sm_not_terminating -->
+<g id="edge24" class="edge">
+<title>nred_R_selfloop&#45;&gt;current_sm_not_terminating</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M339.34,-1062.51C364.99,-1059.52 397.43,-1055.74 426.9,-1052.3"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="427.43,-1054.35 433.14,-1051.57 426.94,-1050.18 427.43,-1054.35"/>
+</g>
+<!-- nred_R_selfloop&#45;&gt;subred_selfloop -->
+<g id="edge25" class="edge">
+<title>nred_R_selfloop&#45;&gt;subred_selfloop</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M339.34,-1073.49C373.11,-1077.43 418.67,-1082.74 453.7,-1086.83"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="453.47,-1088.92 459.68,-1087.53 453.96,-1084.75 453.47,-1088.92"/>
+</g>
+<!-- s_metavar_empty -->
+<g id="node51" class="node">
+<title>s_metavar_empty</title>
+<g id="a_node51"><a xlink:title="s_metavar_empty theory/NamedMeasure.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1643" rx="56.01" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1640.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">s_metavar_empty</text>
+</a>
+</g>
+</g>
+<!-- subred_selfloop&#45;&gt;current_subred_not_terminating -->
+<g id="edge27" class="edge">
+<title>subred_selfloop&#45;&gt;current_subred_not_terminating</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M557.85,-1093C584.53,-1093 618.51,-1093 650.47,-1093"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="650.84,-1095.1 656.84,-1093 650.84,-1090.9 650.84,-1095.1"/>
+</g>
+<!-- Es_ctx_any -->
+<g id="node53" class="node">
+<title>Es_ctx_any</title>
+<g id="a_node53"><a xlink:title="Es_ctx_any theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1693" rx="38.87" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1690.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Es_ctx_any</text>
+</a>
+</g>
+</g>
+<!-- Es_fv -->
+<g id="node54" class="node">
+<title>Es_fv</title>
+<g id="a_node54"><a xlink:title="Es_fv theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="751.57" cy="-1880" rx="27" ry="18"/>
+<text text-anchor="middle" x="751.57" y="-1877.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Es_fv</text>
+</a>
+</g>
+</g>
+<!-- Es_fv_both -->
+<g id="node55" class="node">
+<title>Es_fv_both</title>
+<g id="a_node55"><a xlink:title="Es_fv_both theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1880" rx="37.56" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1877.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Es_fv_both</text>
+</a>
+</g>
+</g>
+<!-- Es_fv_both&#45;&gt;Es_fv -->
+<g id="edge45" class="edge">
+<title>Es_fv_both&#45;&gt;Es_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M545.34,-1880C592.44,-1880 673.35,-1880 718.22,-1880"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="718.3,-1882.1 724.3,-1880 718.3,-1877.9 718.3,-1882.1"/>
+</g>
+<!-- Es_refl_any -->
+<g id="node56" class="node">
+<title>Es_refl_any</title>
+<g id="a_node56"><a xlink:title="Es_refl_any theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1863" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1860.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Es_refl_any</text>
+</a>
+</g>
+</g>
+<!-- Es_sym_any -->
+<g id="node57" class="node">
+<title>Es_sym_any</title>
+<g id="a_node57"><a xlink:title="Es_sym_any theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1913" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1910.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Es_sym_any</text>
+</a>
+</g>
+</g>
+<!-- Es_trans_any -->
+<g id="node58" class="node">
+<title>Es_trans_any</title>
+<g id="a_node58"><a xlink:title="Es_trans_any theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1963" rx="43.76" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1960.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">Es_trans_any</text>
+</a>
+</g>
+</g>
+<!-- eqC_in_Es -->
+<g id="node59" class="node">
+<title>eqC_in_Es</title>
+<g id="a_node59"><a xlink:title="eqC_in_Es theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-2013" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-2010.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">eqC_in_Es</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in -->
+<g id="node60" class="node">
+<title>filter_neq_in</title>
+<g id="a_node60"><a xlink:title="filter_neq_in theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1813" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1810.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">filter_neq_in</text>
+</a>
+</g>
+</g>
+<!-- nplug_esub_fv -->
+<g id="node65" class="node">
+<title>nplug_esub_fv</title>
+<g id="a_node65"><a xlink:title="nplug_esub_fv theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1693" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1690.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nplug_esub_fv</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in&#45;&gt;nplug_esub_fv -->
+<g id="edge36" class="edge">
+<title>filter_neq_in&#45;&gt;nplug_esub_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M105.57,-1798.76C128.62,-1785.58 159.35,-1768 159.35,-1768 159.35,-1768 224.35,-1718 224.35,-1718 224.35,-1718 237,-1713.2 251.23,-1707.81"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="252.08,-1709.74 256.95,-1705.65 250.59,-1705.81 252.08,-1709.74"/>
+</g>
+<!-- nplug_fv_mono -->
+<g id="node66" class="node">
+<title>nplug_fv_mono</title>
+<g id="a_node66"><a xlink:title="nplug_fv_mono theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1853" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1850.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nplug_fv_mono</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in&#45;&gt;nplug_fv_mono -->
+<g id="edge28" class="edge">
+<title>filter_neq_in&#45;&gt;nplug_fv_mono</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M118.16,-1820.15C152.48,-1826.7 203.5,-1836.44 241.1,-1843.61"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="240.72,-1845.67 247.01,-1844.74 241.51,-1841.55 240.72,-1845.67"/>
+</g>
+<!-- nplug_fv_upper -->
+<g id="node67" class="node">
+<title>nplug_fv_upper</title>
+<g id="a_node67"><a xlink:title="nplug_fv_upper theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="291.32" cy="-1743" rx="50.46" ry="18"/>
+<text text-anchor="middle" x="291.32" y="-1740.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nplug_fv_upper</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in&#45;&gt;nplug_fv_upper -->
+<g id="edge30" class="edge">
+<title>filter_neq_in&#45;&gt;nplug_fv_upper</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M112.97,-1802.22C149.21,-1790.12 208.12,-1770.45 248.01,-1757.13"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="248.73,-1759.1 253.75,-1755.21 247.4,-1755.12 248.73,-1759.1"/>
+</g>
+<!-- nred_core_fv -->
+<g id="node69" class="node">
+<title>nred_core_fv</title>
+<g id="a_node69"><a xlink:title="nred_core_fv theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1825" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1822.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nred_core_fv</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in&#45;&gt;nred_core_fv -->
+<g id="edge32" class="edge">
+<title>filter_neq_in&#45;&gt;nred_core_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M113.58,-1823.46C135.14,-1830.31 159.35,-1838 159.35,-1838 159.35,-1838 224.35,-1878 224.35,-1878 224.35,-1878 358.29,-1878 358.29,-1878 358.29,-1878 424.7,-1854.27 468.64,-1838.56"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="469.47,-1840.5 474.41,-1836.5 468.05,-1836.54 469.47,-1840.5"/>
+</g>
+<!-- nred_fv -->
+<g id="node70" class="node">
+<title>nred_fv</title>
+<g id="a_node70"><a xlink:title="nred_fv theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="507.59" cy="-1743" rx="28.74" ry="18"/>
+<text text-anchor="middle" x="507.59" y="-1740.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">nred_fv</text>
+</a>
+</g>
+</g>
+<!-- filter_neq_in&#45;&gt;nred_fv -->
+<g id="edge38" class="edge">
+<title>filter_neq_in&#45;&gt;nred_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M118.98,-1806.79C195.14,-1794.45 358.29,-1768 358.29,-1768 358.29,-1768 429.7,-1755.96 473.41,-1748.59"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="473.82,-1750.65 479.39,-1747.59 473.12,-1746.51 473.82,-1750.65"/>
+</g>
+<!-- in_filter_neq -->
+<g id="node61" class="node">
+<title>in_filter_neq</title>
+<g id="a_node61"><a xlink:title="in_filter_neq theory/NamedMeta.v proved">
+<ellipse fill="transparent" stroke="#a9b492" cx="79.67" cy="-1743" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="79.67" y="-1740.8" font-family="Inter" font-size="9.00" fill="#c9d1bb">in_filter_neq</text>
+</a>
+</g>
+</g>
+<!-- in_filter_neq&#45;&gt;nplug_esub_fv -->
+<g id="edge37" class="edge">
+<title>in_filter_neq&#45;&gt;nplug_esub_fv</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M116.71,-1734.4C152.16,-1725.95 206.4,-1713.01 244.81,-1703.85"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="245.48,-1705.85 250.83,-1702.42 244.51,-1701.77 245.48,-1705.85"/>
+</g>
+<!-- in_filter_neq&#45;&gt;nplug_fv_mono -->
+<g id="edge29" class="edge">
+<title>in_filter_neq&#45;&gt;nplug_fv_mono</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M106.8,-1756.71C144.72,-1776.61 215.03,-1813.5 256.75,-1835.38"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="255.9,-1837.31 262.19,-1838.24 257.85,-1833.59 255.9,-1837.31"/>
+</g>
+<!-- in_filter_neq&#45;&gt;nplug_fv_upper -->
+<g id="edge31" class="edge">
+<title>in_filter_neq&#45;&gt;nplug_fv_upper</title>
+<path fill="none" stroke="#8f9780" stroke-width="0.8" d="M121.59,-1743C153.92,-1743 199.37,-1743 234.95,-1743"/>
+<polygon fill="#8f9780" stroke="#8f9780" stroke-width="0.8" points="235.02,-1745.1 241.02,-1743 235.02,-1740.9 235.02,-1745.1"/>
+</g>
+<!-- in_filter_neq&#45;&gt;nred_core_fv -->
+<g id="edge33" class="edge">
+<title>in_filter_neq&#45;&gt;nred_core_fv</title>
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+<!-- in_filter_neq&#45;&gt;nred_fv -->
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+<title>in_filter_neq_iff</title>
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+<!-- isubst_meta_in -->
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+<!-- nplug_esub_fv&#45;&gt;nred_fv -->
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+<!-- nplug_fv_mono&#45;&gt;Es_fv_both -->
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+<!-- nplug_fv_mono&#45;&gt;nred_core_fv -->
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+<title>nplug_fv_mono&#45;&gt;nred_core_fv</title>
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+<!-- nplug_fv_mono&#45;&gt;nred_fv -->
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+<!-- nplug_fv_upper&#45;&gt;nred_core_fv -->
+<g id="edge35" class="edge">
+<title>nplug_fv_upper&#45;&gt;nred_core_fv</title>
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+<!-- nplug_fv_upper&#45;&gt;nred_fv -->
+<g id="edge42" class="edge">
+<title>nplug_fv_upper&#45;&gt;nred_fv</title>
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+<!-- 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 proved">
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+<!-- M1&#45;&gt;M2 -->
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graphs/dependency.dot

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-  fvs_zplug_lift [label="fvs_zplug_lift", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="fvs_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  plug_fv_mono [label="plug_fv_mono", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="plug_fv_mono theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red1_root_fv [label="red1_root_fv", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red1_root_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red1r_fv [label="red1r_fv", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red1r_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  lemma_2_1_fv_preserved [label="lemma_2_1_fv_preserved", color="#38761d", fillcolor="#d9ead3", style="rounded,filled", fontcolor="#222222", tooltip="lemma_2_1_fv_preserved (section 2.1) theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=box];
-  lemma_2_3_beta_sim [label="lemma_2_3_beta_sim", color="#38761d", fillcolor="#d9ead3", style="rounded,filled", fontcolor="#222222", tooltip="lemma_2_3_beta_sim (section 2.3) theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=box];
-  Plus_trans [label="Plus_trans", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  Plus_step_trans [label="Plus_step_trans", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_step_trans theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  par_red1_iff [label="par_red1_iff", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="par_red1_iff theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red1_in_par [label="red1_in_par", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red1_in_par theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  par_context [label="par_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="par_context theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  par_to_red1_or_eq [label="par_to_red1_or_eq", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="par_to_red1_or_eq theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red1_or_eq_to_par [label="red1_or_eq_to_par", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red1_or_eq_to_par theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  par_reflexive [label="par_reflexive", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="par_reflexive theory/Parallel.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  gen_list_length [label="gen_list_length", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="gen_list_length theory/Random.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  check_term_true [label="check_term_true", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="check_term_true theory/Random.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  check_all_true [label="check_all_true", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="check_all_true theory/Random.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  gen_sound [label="gen_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="gen_sound theory/Random.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  gen_complete [label="gen_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="gen_complete theory/Random.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  gen_fv_preserved [label="gen_fv_preserved", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="gen_fv_preserved theory/Random.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  zdecs_sound [label="zdecs_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="zdecs_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  zdecs_complete [label="zdecs_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="zdecs_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  positions_at [label="positions_at", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="positions_at theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  root_steps_sound [label="root_steps_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="root_steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  root_steps_complete [label="root_steps_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="root_steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  steps_sound [label="steps_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  steps_sound_red1 [label="steps_sound_red1", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="steps_sound_red1 theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  positions_hole [label="positions_hole", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="positions_hole theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  root_steps_in_steps [label="root_steps_in_steps", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="root_steps_in_steps theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  plug_comp [label="plug_comp", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="plug_comp theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  positions_comp [label="positions_comp", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="positions_comp theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  at_ctx_comp [label="at_ctx_comp", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="at_ctx_comp theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  steps_complete [label="steps_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  has_red_f [label="has_red_f", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="has_red_f theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  has_red_spec [label="has_red_spec", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="has_red_spec theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red1_dec [label="red1_dec", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red1_dec theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  nf_iff_steps_nil [label="nf_iff_steps_nil", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nf_iff_steps_nil theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  normal_form_iff_nf [label="normal_form_iff_nf", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="normal_form_iff_nf theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  nf_dec [label="nf_dec", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nf_dec theory/Reduction.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  open_rec_app [label="open_rec_app", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_app theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  open_rec_lam [label="open_rec_lam", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  open_rec_esub [label="open_rec_esub", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  close_rec_app [label="close_rec_app", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="close_rec_app theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  close_rec_lam [label="close_rec_lam", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="close_rec_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  close_rec_esub [label="close_rec_esub", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="close_rec_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  subst_app [label="subst_app", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_app theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  subst_lam [label="subst_lam", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_lam theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  subst_esub [label="subst_esub", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_esub theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  subst_notin [label="subst_notin", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_notin theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  subst_lc_self [label="subst_lc_self", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_lc_self theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  subst_other [label="subst_other", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_other theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  open_rec_lc_atom [label="open_rec_lc_atom", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_lc_atom theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  open_rec_lc [label="open_rec_lc", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_lc theory/Substitution.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_sub_root_to_red1 [label="red_sub_root_to_red1", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_sub_root_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_sub_to_red1 [label="red_sub_to_red1", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_sub_to_red1 theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_sub_context [label="red_sub_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_sub_context theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_sub_gc [label="red_sub_gc", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_sub_gc theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_sub_r [label="red_sub_r", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_sub_r theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  occurs_zfill [label="occurs_zfill", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_zfill theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  zfill_occurs0 [label="zfill_occurs0", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="zfill_occurs0 theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  R_Gc_disjoint [label="R_Gc_disjoint", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="R_Gc_disjoint theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  Star_red_sub_context [label="Star_red_sub_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Star_red_sub_context theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  full_comp_aux_sub [label="full_comp_aux_sub", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="full_comp_aux_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  plus_red_sub_full_comp [label="plus_red_sub_full_comp", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="plus_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  star_red_sub_full_comp [label="star_red_sub_full_comp", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="star_red_sub_full_comp theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  es_count_plug_mono [label="es_count_plug_mono", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="es_count_plug_mono theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_gc_measure [label="red_gc_measure", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_gc_measure theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_gc_terminates [label="red_gc_terminates", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_gc_terminates theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  red_gc_to_red_sub [label="red_gc_to_red_sub", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="red_gc_to_red_sub theory/Subsystem.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  corpus_sound [label="corpus_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="corpus_sound theory/Tests.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  corpus_complete [label="corpus_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="corpus_complete theory/Tests.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  corpus_decides [label="corpus_decides", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="corpus_decides theory/Tests.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  corpus_fv_preserved [label="corpus_fv_preserved", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
-  lift_lc -> lift_0_lc;
-  fvs_lift -> open_rec_fvs;
-  open_rec_bvar -> open_rec_fvs;
-  open_rec_bvar_neq -> open_rec_fvs;
-  open_rec_bvar -> open_close;
-  lift_0_lc -> subst_fvar_self;
-  close_rec_fvs -> subst_fvs;
-  open_rec_fvs -> subst_fvs;
-  in_comb -> size_all_terms;
-  pow2_pos -> size_all_terms;
-  check_all_true -> enumerate_check;
-  steps_sound -> enumerate_sound;
-  steps_complete -> enumerate_complete;
-  occurs_count_zero -> occurs_count_pos;
-  occurs_count_false -> occurs_count_zfill_plug;
-  occurs_lift_self -> occurs_count_zfill_plug;
-  occurs_lift_self -> open_rec_zplug_lift;
-  open_rec_occurs_false -> open_rec_zplug_lift;
-  occurs_count_zero -> full_comp_aux;
-  occurs_count_zfill_plug -> full_comp_aux;
-  open_rec_occurs_false -> full_comp_aux;
-  open_rec_zplug_lift -> full_comp_aux;
-  zdecs_nonempty -> full_comp_aux;
-  zdecs_sound -> full_comp_aux;
-  full_comp_aux -> lemma_2_2_full_comp;
-  fvs_lift -> fvs_zplug_lift;
-  fvs_zplug_lift -> red1_root_fv;
-  plug_fv_mono -> red1r_fv;
-  red1_root_fv -> red1r_fv;
-  red1r_fv -> lemma_2_1_fv_preserved;
-  Plus_red1_context -> lemma_2_3_beta_sim;
-  lemma_2_2_full_comp -> lemma_2_3_beta_sim;
-  par_red1_iff -> par_to_red1_or_eq;
-  par_red1_iff -> red1_or_eq_to_par;
-  has_red_spec -> check_term_true;
-  steps_sound_red1 -> check_term_true;
-  check_term_true -> check_all_true;
-  steps_sound -> gen_sound;
-  steps_complete -> gen_complete;
-  lemma_2_1_fv_preserved -> gen_fv_preserved;
-  steps_sound_red1 -> gen_fv_preserved;
-  zdecs_sound -> root_steps_sound;
-  zdecs_complete -> root_steps_complete;
-  positions_at -> steps_sound;
-  root_steps_sound -> steps_sound;
-  steps_sound -> steps_sound_red1;
-  positions_hole -> root_steps_in_steps;
-  at_ctx_comp -> steps_complete;
-  plug_comp -> steps_complete;
-  positions_comp -> steps_complete;
-  root_steps_complete -> steps_complete;
-  root_steps_in_steps -> steps_complete;
-  has_red_f -> has_red_spec;
-  steps_complete -> has_red_spec;
-  steps_sound -> has_red_spec;
-  has_red_spec -> red1_dec;
-  steps_complete -> nf_iff_steps_nil;
-  steps_sound_red1 -> nf_iff_steps_nil;
-  nf_iff_steps_nil -> normal_form_iff_nf;
-  nf_iff_steps_nil -> nf_dec;
-  close_rec_notin -> subst_notin;
-  open_rec_occurs_false -> subst_notin;
-  subst_fvar_self -> subst_lc_self;
-  subst_fvar_other -> subst_other;
-  open_rec_lc_atom -> open_rec_lc;
-  red_sub_root_to_red1 -> red_sub_to_red1;
-  occurs_zfill -> zfill_occurs0;
-  zfill_occurs0 -> R_Gc_disjoint;
-  red_sub_context -> Star_red_sub_context;
-  occurs_count_zero -> full_comp_aux_sub;
-  occurs_count_zfill_plug -> full_comp_aux_sub;
-  open_rec_occurs_false -> full_comp_aux_sub;
-  open_rec_zplug_lift -> full_comp_aux_sub;
-  red_sub_gc -> full_comp_aux_sub;
-  red_sub_r -> full_comp_aux_sub;
-  zdecs_nonempty -> full_comp_aux_sub;
-  zdecs_sound -> full_comp_aux_sub;
-  full_comp_aux_sub -> plus_red_sub_full_comp;
-  Plus_to_Star -> star_red_sub_full_comp;
-  plus_red_sub_full_comp -> star_red_sub_full_comp;
-  es_count_plug_mono -> red_gc_measure;
-  red_gc_measure -> red_gc_terminates;
-  red_sub_gc -> red_gc_to_red_sub;
-  steps_sound -> corpus_sound;
-  steps_complete -> corpus_complete;
-  has_red_spec -> corpus_decides;
-  lemma_2_1_fv_preserved -> corpus_fv_preserved;
-  steps_sound_red1 -> corpus_fv_preserved;
-}

graphs/dune

@@ -6,9 +6,24 @@
   (source_tree ../theory))
  (targets
   theorems.json
-  dependency.dot
+  dependency-overview.dot
-  dependency.svg
+  dependency-overview.svg
-  dependency.png
+  dependency-overview.png
+  dependency-M1.dot
+  dependency-M1.svg
+  dependency-M1.png
+  dependency-M2.dot
+  dependency-M2.svg
+  dependency-M2.png
+  dependency-M3.dot
+  dependency-M3.svg
+  dependency-M3.png
+  dependency-M4.dot
+  dependency-M4.svg
+  dependency-M4.png
+  dependency-M5.dot
+  dependency-M5.svg
+  dependency-M5.png
   rules.dot
   rules.svg
   rules.png

graphs/index.txt

@@ -107,3 +107,72 @@ proved	M4	infra	par_reflexive	theory/Parallel.v
 proved	M4	infra	par_to_red1_or_eq	theory/Parallel.v
 proved	M4	infra	red1_in_par	theory/Parallel.v
 proved	M4	infra	red1_or_eq_to_par	theory/Parallel.v
+proved	M5	infra	A1_C_related	theory/NamedMeasure.v
+proved	M5	infra	A1_Mx_differ	theory/NamedMeasure.v
+proved	M5	infra	A1_not_invariant	theory/NamedMeasure.v
+proved	M5	infra	A1_s_differ	theory/NamedMeasure.v
+proved	M5	infra	Es_ctx_any	theory/NamedMeta.v
+proved	M5	infra	Es_fv	theory/NamedMeta.v
+proved	M5	infra	Es_fv_both	theory/NamedMeta.v
+proved	M5	infra	Es_refl_any	theory/NamedMeta.v
+proved	M5	infra	Es_sym_any	theory/NamedMeta.v
+proved	M5	infra	Es_trans_any	theory/NamedMeta.v
+proved	M5	infra	Gc_special_nondecrease	theory/NamedMeasure.v
+proved	M5	infra	Mx_fresh_fails	theory/NamedMeasure.v
+proved	M5	infra	Mx_ge_0	theory/NamedMeasure.v
+proved	M5	infra	PlusN_to_StarN	theory/NamedEs.v
+proved	M5	infra	Plus_mred_of_Plus_red1	theory/Metaterm.v
+proved	M5	infra	StarN_one_trans	theory/NamedEs.v
+proved	M5	infra	StarN_subred_context	theory/NamedEs.v
+proved	M5	infra	StarN_trans	theory/NamedEs.v
+proved	M5	infra	current_sm_not_terminating	theory/NamedMeasure.v
+proved	M5	infra	current_subred_not_terminating	theory/NamedMeasure.v
+proved	M5	infra	eqC_in_Es	theory/NamedMeta.v
+proved	M5	infra	filter_neq_in	theory/NamedMeta.v
+proved	M5	infra	head_has_pos	theory/NamedMeasure.v
+proved	M5	infra	in_filter_neq	theory/NamedMeta.v
+proved	M5	infra	in_filter_neq_iff	theory/NamedMeta.v
+proved	M5	infra	isubst_meta_in	theory/NamedMeta.v
+proved	M5	infra	isubst_meta_notin	theory/NamedMeta.v
+proved	M5	infra	le_mul_of_one_le	theory/NamedMeasure.v
+proved	M5	infra	mfvs_close_m	theory/Metaterm.v
+proved	M5	infra	mfvs_lift_m	theory/Metaterm.v
+proved	M5	infra	mfvs_open_m	theory/Metaterm.v
+proved	M5	infra	mred_B_intro	theory/Metaterm.v
+proved	M5	infra	mred_context	theory/Metaterm.v
+proved	M5	infra	mred_full_comp_pure	theory/Metaterm.v
+proved	M5	infra	mred_gc_intro	theory/Metaterm.v
+proved	M5	infra	mred_of_red1	theory/Metaterm.v
+proved	M5	infra	mred_plus_one	theory/Metaterm.v
+proved	M5	infra	mred_r_intro	theory/Metaterm.v
+proved	M5	infra	mred_term_context	theory/Metaterm.v
+proved	M5	infra	nfv_annot_bound_example	theory/NamedMeasure.v
+proved	M5	infra	nocc_le_Mx	theory/NamedMeasure.v
+proved	M5	infra	nplug_esub_fv	theory/NamedMeta.v
+proved	M5	infra	nplug_fv_mono	theory/NamedMeta.v
+proved	M5	infra	nplug_fv_upper	theory/NamedMeta.v
+proved	M5	infra	nred_Gc_special	theory/NamedMeasure.v
+proved	M5	infra	nred_R_intro	theory/NamedMeta.v
+proved	M5	infra	nred_R_selfloop	theory/NamedMeasure.v
+proved	M5	infra	nred_core_fv	theory/NamedMeta.v
+proved	M5	infra	nred_fv	theory/NamedMeta.v
+proved	M5	infra	nred_stable_source	theory/NamedEs.v
+proved	M5	infra	nred_stable_target	theory/NamedEs.v
+proved	M5	infra	of_trm_close	theory/Metaterm.v
+proved	M5	infra	of_trm_fvs	theory/Metaterm.v
+proved	M5	infra	of_trm_injective	theory/Metaterm.v
+proved	M5	infra	of_trm_lift	theory/Metaterm.v
+proved	M5	infra	of_trm_moccurs	theory/Metaterm.v
+proved	M5	infra	of_trm_moccurs0	theory/Metaterm.v
+proved	M5	infra	of_trm_open	theory/Metaterm.v
+proved	M5	infra	of_trm_plug	theory/Metaterm.v
+proved	M5	infra	s_metavar_empty	theory/NamedMeasure.v
+proved	M5	infra	subred_Es_l	theory/NamedEs.v
+proved	M5	infra	subred_Es_lr	theory/NamedEs.v
+proved	M5	infra	subred_Es_r	theory/NamedEs.v
+proved	M5	infra	subred_context	theory/NamedEs.v
+proved	M5	infra	subred_context_rule	theory/NamedEs.v
+proved	M5	infra	subred_intro	theory/NamedEs.v
+proved	M5	infra	subred_nred	theory/NamedEs.v
+proved	M5	infra	subred_of_Es_nred_Es	theory/NamedEs.v
+proved	M5	infra	subred_selfloop	theory/NamedMeasure.v

graphs/reduction.dot

@@ -1,20 +1,20 @@
 digraph reduction_dup {
   rankdir=LR;
-  bgcolor="white";
+  bgcolor="transparent";
-  node [fontname=Inter, fontsize=10];
+  node [fontname=Inter, fontsize=10, fontcolor="#c9d1bb"];
-  edge [fontname=Inter, fontsize=9, color="#333333"];
+  edge [fontname=Inter, fontsize=9, color="#8f9780", fontcolor="#c9d1bb"];
-  graph [labelloc=t, fontcolor="#333333", label="(lambda x. x x) y reduction graph"];
+  graph [labelloc=t, fontcolor="#c9d1bb", label="(lambda x. x x) y reduction graph"];
-  n0 [label="(lambda x. x x) y", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n0 [label="(lambda x. x x) y", shape=box, style="filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  n1 [label="(x x)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n1 [label="(x x)[x/y]", shape=box, style="filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  n2 [label="(y x)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n2 [label="(y x)[x/y]", shape=box, style="filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  n2b [label="(x y)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n2b [label="(x y)[x/y]", shape=box, style="filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  n3 [label="(y y)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n3 [label="(y y)[x/y]", shape=box, style="filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  nf [label="y y (normal form)", shape=doublecircle, style=filled, fillcolor="#d9ead3", color="#38761d", fontcolor="#333333"];
+  nf [label="y y (normal form)", shape=doublecircle, style=filled, fillcolor="#2a3a22", color="#8fb071", fontcolor="#c9d1bb"];
-  n0 -> n1 [label="B", color="#990000"];
+  n0 -> n1 [label="B", color="#dfe7cc"];
-  n1 -> n2 [label="R", color="#674ea7"];
+  n1 -> n2 [label="R", color="#a9b492"];
-  n1 -> n2b [label="R", color="#674ea7"];
+  n1 -> n2b [label="R", color="#a9b492"];
-  n2 -> n3 [label="R", color="#674ea7"];
+  n2 -> n3 [label="R", color="#a9b492"];
-  n2b -> n3 [label="R", color="#674ea7"];
+  n2b -> n3 [label="R", color="#a9b492"];
-  n3 -> nf [label="Gc", color="#38761d"];
+  n3 -> nf [label="Gc", color="#8fb071"];
   {rank=same; n2; n2b;}
 }

graphs/reduction.svg

@@ -4,90 +4,89 @@
 <!-- Generated by graphviz version 2.42.4 (0)
  -->
 <!-- Title: reduction_dup Pages: 1 -->
-<svg width="580pt" height="154pt"
+<svg width="601pt" height="160pt"
- viewBox="0.00 0.00 580.38 154.38" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+ viewBox="0.00 0.00 601.26 160.26" 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(4 150.38)">
+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 156.26)">
 <title>reduction_dup</title>
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-150.38 576.38,-150.38 576.38,4 -4,4"/>
+<text text-anchor="middle" x="296.63" y="-137.06" font-family="Times,serif" font-size="14.00" fill="#c9d1bb">(lambda x. x x) y reduction graph</text>
-<text text-anchor="middle" x="286.19" y="-131.18" font-family="Times,serif" font-size="14.00" fill="#333333">(lambda x. x x) y reduction graph</text>
 <!-- n0 -->
 <g id="node1" class="node">
 <title>n0</title>
-<polygon fill="#f5f5f5" stroke="#333333" points="94,-79.69 0,-79.69 0,-43.69 94,-43.69 94,-79.69"/>
+<polygon fill="transparent" stroke="#a9b492" points="100,-82.63 0,-82.63 0,-46.63 100,-46.63 100,-82.63"/>
-<text text-anchor="middle" x="47" y="-59.19" font-family="Inter" font-size="10.00" fill="#333333">(lambda x. x x) y</text>
+<text text-anchor="middle" x="50" y="-62.13" font-family="Inter" font-size="10.00" fill="#c9d1bb">(lambda x. x x) y</text>
 </g>
 <!-- n1 -->
 <g id="node2" class="node">
 <title>n1</title>
-<polygon fill="#f5f5f5" stroke="#333333" points="196,-79.69 137,-79.69 137,-43.69 196,-43.69 196,-79.69"/>
+<polygon fill="transparent" stroke="#a9b492" points="205,-82.63 143,-82.63 143,-46.63 205,-46.63 205,-82.63"/>
-<text text-anchor="middle" x="166.5" y="-59.19" font-family="Inter" font-size="10.00" fill="#333333">(x x)[x/y]</text>
+<text text-anchor="middle" x="174" y="-62.13" font-family="Inter" font-size="10.00" fill="#c9d1bb">(x x)[x/y]</text>
 </g>
 <!-- n0&#45;&gt;n1 -->
 <g id="edge1" class="edge">
 <title>n0&#45;&gt;n1</title>
-<path fill="none" stroke="#990000" d="M94.1,-61.69C104.9,-61.69 116.3,-61.69 126.75,-61.69"/>
+<path fill="none" stroke="#dfe7cc" d="M100.27,-64.63C111.08,-64.63 122.41,-64.63 132.81,-64.63"/>
-<polygon fill="#990000" stroke="#990000" points="126.81,-65.19 136.81,-61.69 126.81,-58.19 126.81,-65.19"/>
+<polygon fill="#dfe7cc" stroke="#dfe7cc" points="132.86,-68.13 142.86,-64.63 132.86,-61.13 132.86,-68.13"/>
-<text text-anchor="middle" x="115.5" y="-64.49" font-family="Inter" font-size="9.00">B</text>
+<text text-anchor="middle" x="121.5" y="-67.43" font-family="Inter" font-size="9.00" fill="#c9d1bb">B</text>
 </g>
 <!-- n2 -->
 <g id="node3" class="node">
 <title>n2</title>
-<polygon fill="#f5f5f5" stroke="#333333" points="298,-107.69 239,-107.69 239,-71.69 298,-71.69 298,-107.69"/>
+<polygon fill="transparent" stroke="#a9b492" points="310,-110.63 248,-110.63 248,-74.63 310,-74.63 310,-110.63"/>
-<text text-anchor="middle" x="268.5" y="-87.19" font-family="Inter" font-size="10.00" fill="#333333">(y x)[x/y]</text>
+<text text-anchor="middle" x="279" y="-90.13" font-family="Inter" font-size="10.00" fill="#c9d1bb">(y x)[x/y]</text>
 </g>
 <!-- n1&#45;&gt;n2 -->
 <g id="edge2" class="edge">
 <title>n1&#45;&gt;n2</title>
-<path fill="none" stroke="#674ea7" d="M196.16,-69.71C206.41,-72.58 218.13,-75.87 229.08,-78.93"/>
+<path fill="none" stroke="#a9b492" d="M205.07,-72.81C215.34,-75.6 227,-78.77 237.94,-81.74"/>
-<polygon fill="#674ea7" stroke="#674ea7" points="228.27,-82.34 238.84,-81.66 230.15,-75.6 228.27,-82.34"/>
+<polygon fill="#a9b492" stroke="#a9b492" points="237.15,-85.15 247.72,-84.4 238.98,-78.4 237.15,-85.15"/>
-<text text-anchor="middle" x="217.5" y="-78.49" font-family="Inter" font-size="9.00">R</text>
+<text text-anchor="middle" x="226.5" y="-81.43" font-family="Inter" font-size="9.00" fill="#c9d1bb">R</text>
 </g>
 <!-- n2b -->
 <g id="node4" class="node">
 <title>n2b</title>
-<polygon fill="#f5f5f5" stroke="#333333" points="298,-53.69 239,-53.69 239,-17.69 298,-17.69 298,-53.69"/>
+<polygon fill="transparent" stroke="#a9b492" points="310,-56.63 248,-56.63 248,-20.63 310,-20.63 310,-56.63"/>
-<text text-anchor="middle" x="268.5" y="-33.19" font-family="Inter" font-size="10.00" fill="#333333">(x y)[x/y]</text>
+<text text-anchor="middle" x="279" y="-36.13" font-family="Inter" font-size="10.00" fill="#c9d1bb">(x y)[x/y]</text>
 </g>
 <!-- n1&#45;&gt;n2b -->
 <g id="edge3" class="edge">
 <title>n1&#45;&gt;n2b</title>
-<path fill="none" stroke="#674ea7" d="M196.16,-54.24C206.41,-51.57 218.13,-48.52 229.08,-45.68"/>
+<path fill="none" stroke="#a9b492" d="M205.07,-57.04C215.34,-54.45 227,-51.51 237.94,-48.75"/>
-<polygon fill="#674ea7" stroke="#674ea7" points="230.04,-49.04 238.84,-43.14 228.28,-42.27 230.04,-49.04"/>
+<polygon fill="#a9b492" stroke="#a9b492" points="238.88,-52.12 247.72,-46.28 237.16,-45.33 238.88,-52.12"/>
-<text text-anchor="middle" x="217.5" y="-51.49" font-family="Inter" font-size="9.00">R</text>
+<text text-anchor="middle" x="226.5" y="-54.43" font-family="Inter" font-size="9.00" fill="#c9d1bb">R</text>
 </g>
 <!-- n3 -->
 <g id="node5" class="node">
 <title>n3</title>
-<polygon fill="#f5f5f5" stroke="#333333" points="400,-79.69 341,-79.69 341,-43.69 400,-43.69 400,-79.69"/>
+<polygon fill="transparent" stroke="#a9b492" points="415,-82.63 353,-82.63 353,-46.63 415,-46.63 415,-82.63"/>
-<text text-anchor="middle" x="370.5" y="-59.19" font-family="Inter" font-size="10.00" fill="#333333">(y y)[x/y]</text>
+<text text-anchor="middle" x="384" y="-62.13" font-family="Inter" font-size="10.00" fill="#c9d1bb">(y y)[x/y]</text>
 </g>
 <!-- n2&#45;&gt;n3 -->
 <g id="edge4" class="edge">
 <title>n2&#45;&gt;n3</title>
-<path fill="none" stroke="#674ea7" d="M298.16,-81.67C308.41,-78.79 320.13,-75.51 331.08,-72.45"/>
+<path fill="none" stroke="#a9b492" d="M310.07,-84.46C320.34,-81.67 332,-78.5 342.94,-75.52"/>
-<polygon fill="#674ea7" stroke="#674ea7" points="332.15,-75.78 340.84,-69.71 330.27,-69.04 332.15,-75.78"/>
+<polygon fill="#a9b492" stroke="#a9b492" points="343.98,-78.87 352.72,-72.86 342.15,-72.11 343.98,-78.87"/>
-<text text-anchor="middle" x="319.5" y="-78.49" font-family="Inter" font-size="9.00">R</text>
+<text text-anchor="middle" x="331.5" y="-81.43" font-family="Inter" font-size="9.00" fill="#c9d1bb">R</text>
 </g>
 <!-- n2b&#45;&gt;n3 -->
 <g id="edge5" class="edge">
 <title>n2b&#45;&gt;n3</title>
-<path fill="none" stroke="#674ea7" d="M298.16,-43.14C308.41,-45.8 320.13,-48.85 331.08,-51.7"/>
+<path fill="none" stroke="#a9b492" d="M310.07,-46.22C320.34,-48.82 332,-51.76 342.94,-54.52"/>
-<polygon fill="#674ea7" stroke="#674ea7" points="330.28,-55.11 340.84,-54.24 332.04,-48.33 330.28,-55.11"/>
+<polygon fill="#a9b492" stroke="#a9b492" points="342.16,-57.93 352.72,-56.99 343.88,-51.15 342.16,-57.93"/>
-<text text-anchor="middle" x="319.5" y="-51.49" font-family="Inter" font-size="9.00">R</text>
+<text text-anchor="middle" x="331.5" y="-54.43" font-family="Inter" font-size="9.00" fill="#c9d1bb">R</text>
 </g>
 <!-- nf -->
 <g id="node6" class="node">
 <title>nf</title>
-<ellipse fill="#d9ead3" stroke="#38761d" cx="510.69" cy="-61.69" rx="57.87" ry="57.87"/>
+<ellipse fill="#2a3a22" stroke="#8fb071" cx="528.63" cy="-64.63" rx="60.76" ry="60.76"/>
-<ellipse fill="none" stroke="#38761d" cx="510.69" cy="-61.69" rx="61.88" ry="61.88"/>
+<ellipse fill="none" stroke="#8fb071" cx="528.63" cy="-64.63" rx="64.76" ry="64.76"/>
-<text text-anchor="middle" x="510.69" y="-59.19" font-family="Inter" font-size="10.00" fill="#333333">y y (normal form)</text>
+<text text-anchor="middle" x="528.63" y="-62.13" font-family="Inter" font-size="10.00" fill="#c9d1bb">y y (normal form)</text>
 </g>
 <!-- n3&#45;&gt;nf -->
 <g id="edge6" class="edge">
 <title>n3&#45;&gt;nf</title>
-<path fill="none" stroke="#38761d" d="M400.13,-61.69C411.41,-61.69 424.91,-61.69 438.51,-61.69"/>
+<path fill="none" stroke="#8fb071" d="M415.25,-64.63C426.59,-64.63 440.03,-64.63 453.6,-64.63"/>
-<polygon fill="#38761d" stroke="#38761d" points="438.9,-65.19 448.9,-61.69 438.9,-58.19 438.9,-65.19"/>
+<polygon fill="#8fb071" stroke="#8fb071" points="453.97,-68.13 463.97,-64.63 453.97,-61.13 453.97,-68.13"/>
-<text text-anchor="middle" x="424.5" y="-64.49" font-family="Inter" font-size="9.00">Gc</text>
+<text text-anchor="middle" x="439.5" y="-67.43" font-family="Inter" font-size="9.00" fill="#c9d1bb">Gc</text>
 </g>
 </g>
 </svg>

graphs/rules.dot

@@ -1,19 +1,19 @@
 digraph lambda_sub_cfg {
   rankdir=TB;
-  bgcolor="white";
+  bgcolor="transparent";
-  node [fontname=Inter, fontsize=10];
+  node [fontname=Inter, fontsize=10, fontcolor="#c9d1bb"];
-  edge [fontname=Inter, fontsize=9, color="#333333"];
+  edge [fontname=Inter, fontsize=9, color="#8f9780", fontcolor="#c9d1bb"];
-  start [label="term", shape=oval, style=filled, fillcolor="#f3f3f3", color="#333333"];
+  start [label="term", shape=oval, style=filled, fillcolor="transparent", color="#a9b492"];
-  beta [label="(lambda x. t) u", shape=box, style="rounded,filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  beta [label="(lambda x. t) u", shape=box, style="rounded,filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  es [label="t[x/u]", shape=box, style="rounded,filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  es [label="t[x/u]", shape=box, style="rounded,filled", fillcolor="transparent", color="#a9b492", fontcolor="#c9d1bb"];
-  pure [label="pure term", shape=oval, style=filled, fillcolor="#f3f3f3", color="#333333"];
+  pure [label="pure term", shape=oval, style=filled, fillcolor="transparent", color="#a9b492"];
-  nf [label="normal form", shape=doublecircle, style=filled, fillcolor="#d9ead3", color="#38761d"];
+  nf [label="normal form", shape=doublecircle, style=filled, fillcolor="#2a3a22", color="#8fb071"];
   start -> beta [label="App(Lam,_)"];
   start -> es [label="ESub"];
   start -> pure [label="BVar/FVar/Lam"];
-  beta -> es [label="B", color="#990000"];
+  beta -> es [label="B", color="#dfe7cc"];
-  es -> es [label="R (one occurrence)", color="#674ea7"];
+  es -> es [label="R (one occurrence)", color="#a9b492"];
-  es -> pure [label="Gc (x not in fv)", color="#38761d"];
+  es -> pure [label="Gc (x not in fv)", color="#8fb071"];
-  pure -> pure [label="B under context", color="#990000"];
+  pure -> pure [label="B under context", color="#dfe7cc"];
-  es -> nf [label="R/Gc normalisation", color="#674ea7"];
+  es -> nf [label="R/Gc normalisation", color="#a9b492"];
 }

graphs/rules.svg

@@ -4,97 +4,96 @@
 <!-- Generated by graphviz version 2.42.4 (0)
  -->
 <!-- Title: lambda_sub_cfg Pages: 1 -->
-<svg width="322pt" height="349pt"
+<svg width="338pt" height="353pt"
- viewBox="0.00 0.00 321.65 349.12" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+ viewBox="0.00 0.00 337.59 352.65" 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(4 345.12)">
+<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 348.65)">
 <title>lambda_sub_cfg</title>
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-345.12 317.65,-345.12 317.65,4 -4,4"/>
 <!-- start -->
 <g id="node1" class="node">
 <title>start</title>
-<ellipse fill="#f3f3f3" stroke="#333333" cx="152.56" cy="-323.12" rx="27" ry="18"/>
+<ellipse fill="transparent" stroke="#a9b492" cx="161.33" cy="-326.65" rx="27" ry="18"/>
-<text text-anchor="middle" x="152.56" y="-320.62" font-family="Inter" font-size="10.00">term</text>
+<text text-anchor="middle" x="161.33" y="-324.15" font-family="Inter" font-size="10.00" fill="#c9d1bb">term</text>
 </g>
 <!-- beta -->
 <g id="node2" class="node">
 <title>beta</title>
-<path fill="#f5f5f5" stroke="#333333" d="M104.56,-259.12C104.56,-259.12 44.56,-259.12 44.56,-259.12 38.56,-259.12 32.56,-253.12 32.56,-247.12 32.56,-247.12 32.56,-235.12 32.56,-235.12 32.56,-229.12 38.56,-223.12 44.56,-223.12 44.56,-223.12 104.56,-223.12 104.56,-223.12 110.56,-223.12 116.56,-229.12 116.56,-235.12 116.56,-235.12 116.56,-247.12 116.56,-247.12 116.56,-253.12 110.56,-259.12 104.56,-259.12"/>
+<path fill="transparent" stroke="#a9b492" d="M112.83,-262.65C112.83,-262.65 47.83,-262.65 47.83,-262.65 41.83,-262.65 35.83,-256.65 35.83,-250.65 35.83,-250.65 35.83,-238.65 35.83,-238.65 35.83,-232.65 41.83,-226.65 47.83,-226.65 47.83,-226.65 112.83,-226.65 112.83,-226.65 118.83,-226.65 124.83,-232.65 124.83,-238.65 124.83,-238.65 124.83,-250.65 124.83,-250.65 124.83,-256.65 118.83,-262.65 112.83,-262.65"/>
-<text text-anchor="middle" x="74.56" y="-238.62" font-family="Inter" font-size="10.00" fill="#333333">(lambda x. t) u</text>
+<text text-anchor="middle" x="80.33" y="-242.15" font-family="Inter" font-size="10.00" fill="#c9d1bb">(lambda x. t) u</text>
 </g>
 <!-- start&#45;&gt;beta -->
 <g id="edge1" class="edge">
 <title>start&#45;&gt;beta</title>
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graphs/theorems.json

@@ -12,9 +12,10 @@
     "M1",
     "M2",
     "M3",
-    "M4"
+    "M4",
+    "M5"
   ],
-  "current": "M4",
+  "current": "M5",
   "nodes": [
     {
       "id": "fvs_lift",
@@ -288,6 +289,217 @@
         "steps_complete"
       ]
     },
+    {
+      "id": "of_trm_fvs",
+      "label": "of_trm_fvs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_lift",
+      "label": "of_trm_lift",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_open",
+      "label": "of_trm_open",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_lift"
+      ]
+    },
+    {
+      "id": "of_trm_close",
+      "label": "of_trm_close",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_injective",
+      "label": "of_trm_injective",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_lift_m",
+      "label": "mfvs_lift_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_close_m",
+      "label": "mfvs_close_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_open_m",
+      "label": "mfvs_open_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "mfvs_lift_m"
+      ]
+    },
+    {
+      "id": "of_trm_plug",
+      "label": "of_trm_plug",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_of_red1",
+      "label": "mred_of_red1",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_context",
+      "label": "mred_context",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_term_context",
+      "label": "mred_term_context",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_plug"
+      ]
+    },
+    {
+      "id": "of_trm_moccurs",
+      "label": "of_trm_moccurs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_moccurs0",
+      "label": "of_trm_moccurs0",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_moccurs"
+      ]
+    },
+    {
+      "id": "mred_B_intro",
+      "label": "mred_B_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_gc_intro",
+      "label": "mred_gc_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_r_intro",
+      "label": "mred_r_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "Plus_mred_of_Plus_red1",
+      "label": "Plus_mred_of_Plus_red1",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_full_comp_pure",
+      "label": "mred_full_comp_pure",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "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": "M5",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
     {
       "id": "Plus_red1_context",
       "label": "Plus_red1_context",
@@ -497,6 +709,558 @@
       "file": "theory/Metatheory.v",
       "depends": []
     },
+    {
+      "id": "subred_intro",
+      "label": "subred_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": []
+    },
+    {
+      "id": "subred_nred",
+      "label": "subred_nred",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "subred_Es_l",
+      "label": "subred_Es_l",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "subred_Es_r",
+      "label": "subred_Es_r",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "subred_Es_lr",
+      "label": "subred_Es_lr",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_Es_l",
+        "subred_Es_r"
+      ]
+    },
+    {
+      "id": "subred_context",
+      "label": "subred_context",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "subred_context_rule",
+      "label": "subred_context_rule",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_context",
+        "subred_nred"
+      ]
+    },
+    {
+      "id": "subred_of_Es_nred_Es",
+      "label": "subred_of_Es_nred_Es",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "nred_stable_source",
+      "label": "nred_stable_source",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "nred_stable_target",
+      "label": "nred_stable_target",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_intro"
+      ]
+    },
+    {
+      "id": "StarN_trans",
+      "label": "StarN_trans",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": []
+    },
+    {
+      "id": "StarN_one_trans",
+      "label": "StarN_one_trans",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": []
+    },
+    {
+      "id": "PlusN_to_StarN",
+      "label": "PlusN_to_StarN",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": []
+    },
+    {
+      "id": "StarN_subred_context",
+      "label": "StarN_subred_context",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedEs.v",
+      "depends": [
+        "subred_context"
+      ]
+    },
+    {
+      "id": "Mx_ge_0",
+      "label": "Mx_ge_0",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "head_has_pos",
+      "label": "head_has_pos",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "le_mul_of_one_le",
+      "label": "le_mul_of_one_le",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "nocc_le_Mx",
+      "label": "nocc_le_Mx",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": [
+        "head_has_pos",
+        "le_mul_of_one_le"
+      ]
+    },
+    {
+      "id": "nfv_annot_bound_example",
+      "label": "nfv_annot_bound_example",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "Mx_fresh_fails",
+      "label": "Mx_fresh_fails",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": [
+        "nfv_annot_bound_example"
+      ]
+    },
+    {
+      "id": "A1_C_related",
+      "label": "A1_C_related",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "A1_s_differ",
+      "label": "A1_s_differ",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "A1_Mx_differ",
+      "label": "A1_Mx_differ",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "A1_not_invariant",
+      "label": "A1_not_invariant",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": [
+        "A1_C_related",
+        "A1_s_differ"
+      ]
+    },
+    {
+      "id": "Gc_special_nondecrease",
+      "label": "Gc_special_nondecrease",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "nred_Gc_special",
+      "label": "nred_Gc_special",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "s_metavar_empty",
+      "label": "s_metavar_empty",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "nred_R_selfloop",
+      "label": "nred_R_selfloop",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": []
+    },
+    {
+      "id": "current_sm_not_terminating",
+      "label": "current_sm_not_terminating",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": [
+        "nred_R_selfloop"
+      ]
+    },
+    {
+      "id": "subred_selfloop",
+      "label": "subred_selfloop",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": [
+        "nred_R_selfloop",
+        "subred_nred"
+      ]
+    },
+    {
+      "id": "current_subred_not_terminating",
+      "label": "current_subred_not_terminating",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeasure.v",
+      "depends": [
+        "subred_selfloop"
+      ]
+    },
+    {
+      "id": "isubst_meta_in",
+      "label": "isubst_meta_in",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "isubst_meta_notin",
+      "label": "isubst_meta_notin",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "eqC_in_Es",
+      "label": "eqC_in_Es",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_ctx_any",
+      "label": "Es_ctx_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "nred_R_intro",
+      "label": "nred_R_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_refl_any",
+      "label": "Es_refl_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_sym_any",
+      "label": "Es_sym_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_trans_any",
+      "label": "Es_trans_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "in_filter_neq",
+      "label": "in_filter_neq",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "filter_neq_in",
+      "label": "filter_neq_in",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "nplug_fv_mono",
+      "label": "nplug_fv_mono",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq"
+      ]
+    },
+    {
+      "id": "nplug_fv_upper",
+      "label": "nplug_fv_upper",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq"
+      ]
+    },
+    {
+      "id": "nred_core_fv",
+      "label": "nred_core_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq",
+        "nplug_fv_mono",
+        "nplug_fv_upper"
+      ]
+    },
+    {
+      "id": "nplug_esub_fv",
+      "label": "nplug_esub_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq"
+      ]
+    },
+    {
+      "id": "nred_fv",
+      "label": "nred_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "filter_neq_in",
+        "in_filter_neq",
+        "nplug_esub_fv",
+        "nplug_fv_mono",
+        "nplug_fv_upper"
+      ]
+    },
+    {
+      "id": "in_filter_neq_iff",
+      "label": "in_filter_neq_iff",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "Es_fv_both",
+      "label": "Es_fv_both",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "in_filter_neq_iff",
+        "nplug_fv_mono"
+      ]
+    },
+    {
+      "id": "Es_fv",
+      "label": "Es_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M5",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": [
+        "Es_fv_both"
+      ]
+    },
     {
       "id": "par_red1_iff",
       "label": "par_red1_iff",

scripts/audit.sh

@@ -35,7 +35,7 @@ done < <(find theory -name '*.v' -print0 2>/dev/null)
 echo "== Print Assumptions =="
 WORK="$(mktemp -d ./.audit-work.XXXXXX)"
 cp theory/*.v "$WORK"/
-for f in ExecReducer Binding Reduction Metatheory Closure Subsystem Substitution Parallel Metaterm Tests Random Enumerate; do
+for f in ExecReducer Binding Reduction Metatheory Closure Subsystem Substitution Parallel Metaterm NamedMeta NamedEs NamedMeasure Tests Random Enumerate; do
   echo "-- $f"
   out="$(rocq compile -Q "$WORK" LambdaSub "$WORK/$f.v" 2>&1 || true)"
   echo "$out"

scripts/extract_metadata.py

@@ -25,6 +25,10 @@ MILESTONE = {
     "Substitution": "M2",
     "Subsystem": "M3",
     "Parallel": "M4",
+    "Metaterm": "M5",
+    "NamedMeta": "M5",
+    "NamedEs": "M5",
+    "NamedMeasure": "M5",
     "Tests": "M2",
     "Random": "M2",
     "Enumerate": "M2",

scripts/gen_graphs.py

@@ -9,28 +9,18 @@ ROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
 GRAPH_DIR = os.path.join(ROOT, "graphs")
 META = os.path.join(GRAPH_DIR, "theorems.json")
 
-STATUS_FILL = {
+# Dark sage palette, transparent so the diagrams sit on the page background.
-    "proved": "#d9ead3",
+INK = "#a9b492"
-    "stated": "#fff2cc",
+MUTED = "#8f9780"
-    "planned": "#f3f3f3",
+FONT = "#c9d1bb"
-    "blocked": "#f4cccc",
+FILL = "transparent"
-}
+EXT_FILL = "transparent"
-STATUS_PEN = {
+NF_FILL = "#2a3a22"
-    "proved": "#38761d",
+
-    "stated": "#b45f06",
+# Distinct sage tones for the rule sketches.
-    "planned": "#777777",
+RULE_B = "#dfe7cc"
-    "blocked": "#cc0000",
+RULE_R = "#a9b492"
-}
+RULE_G = "#8fb071"
-STATUS_STYLE = {
-    "proved": "filled",
-    "stated": "filled,dashed",
-    "planned": "dashed",
-    "blocked": "filled,bold",
-}
-INK = "#333333"
-RULE_B = "#990000"
-RULE_R = "#674ea7"
-RULE_G = "#38761d"
 
 
 def q(s):
@@ -46,110 +36,146 @@ def esc_html(s):
     return str(s).replace("&", "&amp;").replace("<", "&lt;").replace(">", "&gt;")
 
 
-def dependency_dot(data):
+def node_line(n, external=False):
-    nodes = data["nodes"]
+    if external:
-    by_id = {n["id"]: n for n in nodes}
+        label = n["id"] + "\\n(" + n["milestone"] + ")"
-    title = data["paper"]["title"] + " - theorem dependency (" + data["current"] + ")"
+        shape = "box"
-    legend = (
+        style = "rounded,dashed,filled"
-        "<B>" + esc_html(title) + "</B>"
+        fill = EXT_FILL
-        "<BR/><FONT POINT-SIZE=\"9\">"
+        color = MUTED
-        "<FONT COLOR=\"#38761d\">proved</FONT>   "
+        fontcolor = FONT
-        "<FONT COLOR=\"#b45f06\">stated</FONT>   "
+        pen = 0.9
-        "<FONT COLOR=\"#777777\">planned</FONT>   "
+    else:
-        "<FONT COLOR=\"#cc0000\">blocked</FONT>"
+        label = n["id"]
-        "</FONT>"
+        shape = "box" if n["kind"] == "paper" else "ellipse"
+        style = "rounded,filled" if n["kind"] == "paper" else "filled"
+        fill = FILL
+        color = INK
+        fontcolor = FONT
+        pen = 1.0
+    tip = n["id"]
+    if n.get("section"):
+        tip += " (section " + n["section"] + ")"
+    if n.get("file"):
+        tip += " " + n["file"]
+    tip += " " + n["status"]
+    return (
+        "  %s [label=%s, shape=%s, style=%s, fillcolor=%s, color=%s, "
+        "penwidth=%.1f, fontcolor=%s, tooltip=%s];"
+        % (n["id"], q(label), shape, q(style), q(fill), q(color), pen, q(fontcolor), q(tip))
     )
-    lines = [
+
-        "digraph theorem_deps {",
+
+def dep_header(title):
+    return [
+        "digraph deps {",
         "  rankdir=LR;",
-        '  bgcolor="white";',
+        '  bgcolor="transparent";',
-        "  splines=spline;",
+        "  splines=polyline;",
-        "  nodesep=0.22;",
+        "  concentrate=true;",
-        "  ranksep=0.6;",
+        "  nodesep=0.2;",
-        '  node [fontname=Inter, fontsize=9, fontcolor="#222222"];',
+        "  ranksep=0.9;",
-        '  edge [fontname=Inter, fontsize=8, color="#9a9a9a", arrowsize=0.6, penwidth=0.8];',
+        '  node [fontname=Inter, fontsize=9];',
-        "  graph [fontname=Inter, fontsize=12, labelloc=t, label=<%s>];" % legend,
+        '  edge [fontname=Inter, fontsize=8, color="%s", fontcolor="%s", arrowsize=0.6, penwidth=0.8];' % (MUTED, FONT),
+        '  graph [fontname=Inter, fontsize=12, labelloc=t, fontcolor="%s", label=<<B>%s</B>>];' % (FONT, esc_html(title)),
     ]
-    for n in nodes:
+
-        fill = STATUS_FILL.get(n["status"], "#f3f3f3")
+
-        pen = STATUS_PEN.get(n["status"], "#777777")
+def dependency_overview_dot(data):
-        style = STATUS_STYLE.get(n["status"], "dashed")
+    by_id = {n["id"]: n for n in data["nodes"]}
-        tip = n["id"]
+    counts = {}
-        if n.get("section"):
+    for n in data["nodes"]:
-            tip += " (section " + n["section"] + ")"
+        counts[n["milestone"]] = counts.get(n["milestone"], 0) + 1
-        if n.get("file"):
+    milestones = [m for m in data["milestones"] if counts.get(m)]
-            tip += " " + n["file"]
+    edges = set()
-        tip += " " + data["paper"]["html"]
+    for n in data["nodes"]:
-        if n["kind"] == "paper":
+        for d in n["depends"]:
-            shape = "box"
-            style = "rounded," + style
-        else:
-            shape = "ellipse"
-        attrs = [
-            "label=%s" % q(n["label"]),
-            "color=%s" % q(pen),
-            "fillcolor=%s" % q(fill),
-            "style=%s" % q(style),
-            "fontcolor=%s" % q("#222222"),
-            "tooltip=%s" % q(tip),
-            "shape=%s" % shape,
-        ]
-        lines.append("  %s [%s];" % (n["id"], ", ".join(attrs)))
-    for n in nodes:
-        for d in n.get("depends", []):
             if d in by_id:
+                a = by_id[d]["milestone"]
+                b = n["milestone"]
+                if a != b and a in counts and b in counts:
+                    edges.add((a, b))
+    lines = dep_header("Theorem dependency by milestone")
+    for m in milestones:
+        label = "%s\\n%d results" % (m, counts[m])
+        lines.append(
+            '  %s [label=%s, shape=box, style="rounded,filled", fillcolor=%s, '
+            "color=%s, fontcolor=%s];" % (m, q(label), q(FILL), q(INK), q(FONT))
+        )
+    for a, b in sorted(edges):
+        lines.append("  %s -> %s;" % (a, b))
+    lines.append("}")
+    return "\n".join(lines) + "\n"
+
+
+def dependency_subset_dot(data, title, keep_ids):
+    by_id = {n["id"]: n for n in data["nodes"]}
+    inside = [n for n in data["nodes"] if n["id"] in keep_ids]
+    ids = {n["id"] for n in inside}
+    ext_ids = sorted(
+        {d for n in inside for d in n["depends"] if d not in ids and d in by_id}
+    )
+    lines = dep_header(title)
+    for d in ext_ids:
+        lines.append(node_line(by_id[d], external=True))
+    for n in sorted(inside, key=lambda x: (x["file"], x["id"])):
+        lines.append(node_line(n))
+    keep = ids | set(ext_ids)
+    for n in inside:
+        for d in n["depends"]:
+            if d in keep:
                 lines.append("  %s -> %s;" % (d, n["id"]))
     lines.append("}")
     return "\n".join(lines) + "\n"
 
 
 def rules_dot():
-    box = 'shape=box, style="rounded,filled", fillcolor="#f5f5f5", color="%s", fontcolor="%s"' % (INK, INK)
+    box = 'shape=box, style="rounded,filled", fillcolor="%s", color="%s", fontcolor="%s"' % (FILL, INK, FONT)
     lines = [
         "digraph lambda_sub_cfg {",
         "  rankdir=TB;",
-        "  bgcolor=\"white\";",
+        '  bgcolor="transparent";',
-        "  node [fontname=Inter, fontsize=10];",
+        '  node [fontname=Inter, fontsize=10, fontcolor="%s"];' % FONT,
-        "  edge [fontname=Inter, fontsize=9, color=\"%s\"];" % INK,
+        '  edge [fontname=Inter, fontsize=9, color="%s", fontcolor="%s"];' % (MUTED, FONT),
-        "  start [label=\"term\", shape=oval, style=filled, fillcolor=\"#f3f3f3\", color=\"%s\"];" % INK,
+        '  start [label="term", shape=oval, style=filled, fillcolor="%s", color="%s"];' % (FILL, INK),
         "  beta [label=\"(lambda x. t) u\", %s];" % box,
         "  es [label=\"t[x/u]\", %s];" % box,
-        "  pure [label=\"pure term\", shape=oval, style=filled, fillcolor=\"#f3f3f3\", color=\"%s\"];" % INK,
+        '  pure [label="pure term", shape=oval, style=filled, fillcolor="%s", color="%s"];' % (FILL, INK),
-        "  nf [label=\"normal form\", shape=doublecircle, style=filled, fillcolor=\"#d9ead3\", color=\"%s\"];" % RULE_G,
+        '  nf [label="normal form", shape=doublecircle, style=filled, fillcolor="%s", color="%s"];' % (NF_FILL, RULE_G),
-        "  start -> beta [label=\"App(Lam,_)\"];",
+        '  start -> beta [label="App(Lam,_)"];',
-        "  start -> es [label=\"ESub\"];",
+        '  start -> es [label="ESub"];',
-        "  start -> pure [label=\"BVar/FVar/Lam\"];",
+        '  start -> pure [label="BVar/FVar/Lam"];',
-        "  beta -> es [label=\"B\", color=\"%s\"];" % RULE_B,
+        '  beta -> es [label="B", color="%s"];' % RULE_B,
-        "  es -> es [label=\"R (one occurrence)\", color=\"%s\"];" % RULE_R,
+        '  es -> es [label="R (one occurrence)", color="%s"];' % RULE_R,
-        "  es -> pure [label=\"Gc (x not in fv)\", color=\"%s\"];" % RULE_G,
+        '  es -> pure [label="Gc (x not in fv)", color="%s"];' % RULE_G,
-        "  pure -> pure [label=\"B under context\", color=\"%s\"];" % RULE_B,
+        '  pure -> pure [label="B under context", color="%s"];' % RULE_B,
-        "  es -> nf [label=\"R/Gc normalisation\", color=\"%s\"];" % RULE_R,
+        '  es -> nf [label="R/Gc normalisation", color="%s"];' % RULE_R,
         "}",
     ]
     return "\n".join(lines) + "\n"
 
 
 def reduction_dot():
-    box = 'shape=box, style="filled", fillcolor="#f5f5f5", color="%s", fontcolor="%s"' % (INK, INK)
+    box = 'shape=box, style="filled", fillcolor="%s", color="%s", fontcolor="%s"' % (FILL, INK, FONT)
     lines = [
         "digraph reduction_dup {",
         "  rankdir=LR;",
-        "  bgcolor=\"white\";",
+        '  bgcolor="transparent";',
-        "  node [fontname=Inter, fontsize=10];",
+        '  node [fontname=Inter, fontsize=10, fontcolor="%s"];' % FONT,
-        "  edge [fontname=Inter, fontsize=9, color=\"%s\"];" % INK,
+        '  edge [fontname=Inter, fontsize=9, color="%s", fontcolor="%s"];' % (MUTED, FONT),
-        "  graph [labelloc=t, fontcolor=\"%s\", label=\"(lambda x. x x) y reduction graph\"];" % INK,
+        '  graph [labelloc=t, fontcolor="%s", label="(lambda x. x x) y reduction graph"];' % FONT,
         "  n0 [label=\"(lambda x. x x) y\", %s];" % box,
         "  n1 [label=\"(x x)[x/y]\", %s];" % box,
         "  n2 [label=\"(y x)[x/y]\", %s];" % box,
         "  n2b [label=\"(x y)[x/y]\", %s];" % box,
         "  n3 [label=\"(y y)[x/y]\", %s];" % box,
-        "  nf [label=\"y y (normal form)\", shape=doublecircle, style=filled, fillcolor=\"#d9ead3\", color=\"%s\", fontcolor=\"%s\"];" % (RULE_G, INK),
+        '  nf [label="y y (normal form)", shape=doublecircle, style=filled, fillcolor="%s", color="%s", fontcolor="%s"];' % (NF_FILL, RULE_G, FONT),
-        "  n0 -> n1 [label=\"B\", color=\"%s\"];" % RULE_B,
+        '  n0 -> n1 [label="B", color="%s"];' % RULE_B,
-        "  n1 -> n2 [label=\"R\", color=\"%s\"];" % RULE_R,
+        '  n1 -> n2 [label="R", color="%s"];' % RULE_R,
-        "  n1 -> n2b [label=\"R\", color=\"%s\"];" % RULE_R,
+        '  n1 -> n2b [label="R", color="%s"];' % RULE_R,
-        "  n2 -> n3 [label=\"R\", color=\"%s\"];" % RULE_R,
+        '  n2 -> n3 [label="R", color="%s"];' % RULE_R,
-        "  n2b -> n3 [label=\"R\", color=\"%s\"];" % RULE_R,
+        '  n2b -> n3 [label="R", color="%s"];' % RULE_R,
-        "  n3 -> nf [label=\"Gc\", color=\"%s\"];" % RULE_G,
+        '  n3 -> nf [label="Gc", color="%s"];' % RULE_G,
         "  {rank=same; n2; n2b;}",
         "}",
     ]
@@ -192,11 +218,22 @@ def main():
         print("wrote", os.path.relpath(args.meta, ROOT))
     else:
         data = load(args.meta)
-    outputs = {
+
-        "dependency.dot": dependency_dot(data),
+    counts = {}
-        "rules.dot": rules_dot(),
+    for n in data["nodes"]:
-        "reduction.dot": reduction_dot(),
+        counts[n["milestone"]] = counts.get(n["milestone"], 0) + 1
-    }
+
+    outputs = {"dependency-overview.dot": dependency_overview_dot(data)}
+    for m in data["milestones"]:
+        if not counts.get(m):
+            continue
+        keep = {n["id"] for n in data["nodes"] if n["milestone"] == m}
+        title = "Theorem dependencies %s, %d results" % (m, counts[m])
+        outputs["dependency-%s.dot" % m] = dependency_subset_dot(data, title, keep)
+
+    outputs["rules.dot"] = rules_dot()
+    outputs["reduction.dot"] = reduction_dot()
+
     ok = True
     for name, text in outputs.items():
         path = os.path.join(args.outdir, name)

theory/Metaterm.v

@@ -1,6 +1,6 @@
 From Stdlib Require Import List Bool Arith Lia PeanoNat.
 Import ListNotations.
-From LambdaSub Require Import ExecReducer Binding.
+From LambdaSub Require Import ExecReducer Binding Reduction Closure Metatheory.
 
 Inductive mtrm : Type :=
 | mBVar : nat -> mtrm
@@ -110,3 +110,218 @@ 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.
+
+Inductive mctx : Type :=
+| mGHole : mctx
+| mGAppL : mctx -> mtrm -> mctx
+| mGAppR : mtrm -> mctx -> mctx
+| mGLam : mctx -> mctx
+| mGESubL : mctx -> mtrm -> mctx
+| mGESubR : mtrm -> mctx -> mctx.
+
+Fixpoint mplug (C : mctx) (t : mtrm) : mtrm :=
+  match C with
+  | mGHole => t
+  | mGAppL C1 u => mApp (mplug C1 t) u
+  | mGAppR u C1 => mApp u (mplug C1 t)
+  | mGLam C1 => mLam (mplug C1 t)
+  | mGESubL C1 u => mESub (mplug C1 t) u
+  | mGESubR t1 C1 => mESub t1 (mplug C1 t)
+  end.
+
+Fixpoint mctx_of (C : ctx) : mctx :=
+  match C with
+  | GHole => mGHole
+  | GAppL C1 u => mGAppL (mctx_of C1) (of_trm u)
+  | GAppR u C1 => mGAppR (of_trm u) (mctx_of C1)
+  | GLam C1 => mGLam (mctx_of C1)
+  | GESubL C1 u => mGESubL (mctx_of C1) (of_trm u)
+  | GESubR t1 C1 => mGESubR (of_trm t1) (mctx_of C1)
+  end.
+
+Lemma of_trm_plug : forall C t, of_trm (plug C t) = mplug (mctx_of C) (of_trm t).
+Proof.
+  induction C; intros a; simpl; try reflexivity; rewrite IHC; reflexivity.
+Qed.
+
+Fixpoint moccurs (n : nat) (t : mtrm) : bool :=
+  match t with
+  | mBVar m => Nat.eqb m n
+  | mFVar _ => false
+  | mMVar _ _ => false
+  | mApp a b => moccurs n a || moccurs n b
+  | mLam a => moccurs (S n) a
+  | mESub a b => moccurs (S n) a || moccurs n b
+  end.
+
+Definition moccurs0 (t : mtrm) : bool := moccurs 0 t.
+
+Inductive mzctx : Type :=
+| mZTop : mzctx
+| mZAppL : mzctx -> mtrm -> mzctx
+| mZAppR : mtrm -> mzctx -> mzctx
+| mZLam : mzctx -> mzctx
+| mZESubL : mzctx -> mtrm -> mzctx
+| mZESubR : mtrm -> mzctx -> mzctx.
+
+Fixpoint mzfill (C : mzctx) (k : nat) : mtrm :=
+  match C with
+  | mZTop => mBVar k
+  | mZAppL C1 u => mApp (mzfill C1 k) u
+  | mZAppR u C1 => mApp u (mzfill C1 k)
+  | mZLam C1 => mLam (mzfill C1 (S k))
+  | mZESubL C1 u => mESub (mzfill C1 (S k)) u
+  | mZESubR t1 C1 => mESub t1 (mzfill C1 k)
+  end.
+
+Fixpoint mzplug_lift (C : mzctx) (k : nat) (t : mtrm) : mtrm :=
+  match C with
+  | mZTop => lift_m k t
+  | mZAppL C1 u => mApp (mzplug_lift C1 k t) u
+  | mZAppR u C1 => mApp u (mzplug_lift C1 k t)
+  | mZLam C1 => mLam (mzplug_lift C1 (S k) t)
+  | mZESubL C1 u => mESub (mzplug_lift C1 (S k) t) u
+  | mZESubR t1 C1 => mESub t1 (mzplug_lift C1 k t)
+  end.
+
+Fixpoint mzdecs (t : mtrm) (k : nat) : list mzctx :=
+  match t with
+  | mBVar n => if Nat.eqb n k then [mZTop] else []
+  | mFVar _ => []
+  | mMVar _ _ => []
+  | mApp a b => map (fun C => mZAppL C b) (mzdecs a k) ++ map (fun C => mZAppR a C) (mzdecs b k)
+  | mLam a => map mZLam (mzdecs a (S k))
+  | mESub a b => map (fun C => mZESubL C b) (mzdecs a (S k)) ++ map (fun C => mZESubR a C) (mzdecs b k)
+  end.
+
+Inductive mred : mtrm -> mtrm -> Prop :=
+| mred_emb : forall t t', red1 t t' -> mred (of_trm t) (of_trm t')
+| mred_ctx : forall C t t', mred t t' -> mred (mplug C t) (mplug C t')
+| mred_B : forall body u, mred (mApp (mLam body) u) (mESub body u)
+| mred_gc : forall body u, moccurs0 body = false -> mred (mESub body u) body
+| mred_r : forall C body u, mzfill C 0 = body ->
+    mred (mESub body u) (mESub (mzplug_lift C 0 u) u).
+
+Lemma mred_of_red1 : forall t t', red1 t t' -> mred (of_trm t) (of_trm t').
+Proof. intros. apply mred_emb. exact H. Qed.
+
+Lemma mred_context : forall C t t', mred t t' -> mred (mplug C t) (mplug C t').
+Proof. intros. apply mred_ctx. exact H. Qed.
+
+Lemma mred_term_context : forall C t t',
+  red1 t t' -> mred (of_trm (plug C t)) (of_trm (plug C t')).
+Proof.
+  intros. rewrite of_trm_plug, of_trm_plug.
+  apply mred_ctx. apply mred_emb. exact H.
+Qed.
+
+Print Assumptions of_trm_plug.
+Print Assumptions mred_of_red1.
+Print Assumptions mred_context.
+Print Assumptions mred_term_context.
+
+Lemma of_trm_moccurs : forall t k, moccurs k (of_trm t) = occurs k t.
+Proof.
+  induction t; intros k; simpl; try reflexivity.
+  - rewrite IHt1, IHt2. reflexivity.
+  - rewrite IHt. reflexivity.
+  - rewrite IHt1, IHt2. reflexivity.
+Qed.
+
+Lemma of_trm_moccurs0 : forall t, moccurs0 (of_trm t) = occurs0 t.
+Proof. intros. apply of_trm_moccurs. Qed.
+
+Lemma mred_B_intro : forall body u, mred (mApp (mLam body) u) (mESub body u).
+Proof. intros. apply mred_B. Qed.
+
+Lemma mred_gc_intro : forall body u, moccurs0 body = false -> mred (mESub body u) body.
+Proof. intros. apply mred_gc. exact H. Qed.
+
+Print Assumptions of_trm_moccurs.
+Print Assumptions of_trm_moccurs0.
+
+Lemma mred_r_intro : forall C body u, mzfill C 0 = body ->
+  mred (mESub body u) (mESub (mzplug_lift C 0 u) u).
+Proof. intros. apply mred_r. exact H. Qed.
+
+Print Assumptions mred_r_intro.
+
+Inductive Plus_m (R : mtrm -> mtrm -> Prop) : mtrm -> mtrm -> Prop :=
+| plusm1 : forall t u, R t u -> Plus_m R t u
+| plusmS : forall t u v, R t u -> Plus_m R u v -> Plus_m R t v.
+
+Lemma Plus_mred_of_Plus_red1 : forall a b,
+  Plus red1 a b -> Plus_m mred (of_trm a) (of_trm b).
+Proof.
+  intros a b H. induction H as [a0 b0 HR | a0 b0 u0 HR HP IH].
+  - apply plusm1. apply mred_emb. exact HR.
+  - apply plusmS with (u := of_trm b0).
+    + apply mred_emb. exact HR.
+    + exact IH.
+Qed.
+
+Lemma mred_full_comp_pure : forall body u,
+  Plus_m mred (mESub (of_trm body) (of_trm u)) (of_trm (open_rec 0 u body)).
+Proof.
+  intros body u.
+  change (Plus_m mred (of_trm (ESub body u)) (of_trm (open_rec 0 u body))).
+  apply Plus_mred_of_Plus_red1. apply lemma_2_2_full_comp.
+Qed.
+
+Lemma mred_plus_one : forall a b, red1 a b -> Plus_m mred (of_trm a) (of_trm b).
+Proof. intros. apply plusm1. apply mred_emb. exact H. Qed.
+
+Print Assumptions Plus_mred_of_Plus_red1.
+Print Assumptions mred_full_comp_pure.
+Print Assumptions mred_plus_one.

theory/NamedEs.v

@@ -0,0 +1,173 @@
+From Stdlib Require Import List Bool Arith Lia PeanoNat.
+From Stdlib Require Import Setoid Morphisms.
+Import ListNotations.
+From LambdaSub Require Import NamedMeta.
+
+(* A setoid rewriting layer for the C equivalence Es on named metaterms.
+
+   Es is the congruence generated by the equation C (together with the
+   implicit alpha convention of the paper, which the named core does not
+   quotient). We register Es as an equivalence and as a congruence for the
+   term constructors and for every context, so that setoid rewriting can
+   move along Es. We then define the reduction relation modulo Es (subred)
+   and prove that it is stable on both sides and under every context. *)
+
+Instance Es_Equivalence : Equivalence Es.
+Proof.
+  split.
+  - intro t. apply Es_refl.
+  - intros a b Hab. apply Es_sym. exact Hab.
+  - intros a b c Hab Hbc. apply Es_trans with (u := b); assumption.
+Qed.
+
+Instance Es_NVar_Proper : Proper (eq ==> Es) NVar.
+Proof. intros a b Hab. subst b. apply Es_refl. Qed.
+
+Instance Es_NMVar_Proper : Proper (eq ==> eq ==> Es) NMVar.
+Proof. intros a b Hab c d Hcd. subst b d. apply Es_refl. Qed.
+
+Instance Es_NApp_Proper : Proper (Es ==> Es ==> Es) NApp.
+Proof.
+  intros a b Hab c d Hcd.
+  apply Es_trans with (u := NApp b c).
+  - apply Es_ctx with (C := nAppL nHole c). exact Hab.
+  - apply Es_ctx with (C := nAppR b nHole). exact Hcd.
+Qed.
+
+Instance Es_NLam_Proper (x : atom) : Proper (Es ==> Es) (NLam x).
+Proof. intros a b Hab. apply Es_ctx with (C := nLam x nHole). exact Hab. Qed.
+
+Instance Es_NESub_Proper : Proper (Es ==> eq ==> Es ==> Es) NESub.
+Proof.
+  intros a b Hab x y Hxy c d Hcd. subst y.
+  apply Es_trans with (u := NESub b x c).
+  - apply Es_ctx with (C := nESubL nHole x c). exact Hab.
+  - apply Es_ctx with (C := nESubR b x nHole). exact Hcd.
+Qed.
+
+Instance Es_nplug_Proper (C : nctx) : Proper (Es ==> Es) (nplug C).
+Proof. intros a b Hab. apply Es_ctx. exact Hab. Qed.
+
+Instance Es_nfv_Proper (z : atom) : Proper (Es ==> iff) (fun t => In z (nfv t)).
+Proof. intros a b Hab. apply (Es_fv a b Hab z). Qed.
+
+(* The reduction relation modulo Es, as in the paper:
+   t ->sub t' iff there are s, s' with t =Es s ->sm s' =Es t'. *)
+Definition subred (t t' : ntrm) : Prop :=
+  exists s, exists s', Es t s /\ nred s s' /\ Es s' t'.
+
+Lemma subred_intro : forall t s s' t',
+    Es t s -> nred s s' -> Es s' t' -> subred t t'.
+Proof. intros. unfold subred. exists s, s'. split; [| split]; assumption. Qed.
+
+Lemma subred_nred : forall t u, nred t u -> subred t u.
+Proof.
+  intros t u H. apply subred_intro with (s := t) (s' := u).
+  - apply Es_refl.
+  - exact H.
+  - apply Es_refl.
+Qed.
+
+Lemma subred_Es_l : forall t s u, Es t s -> subred s u -> subred t u.
+Proof.
+  intros t s u Ht [a [b [Ha [Hn Hb]]]].
+  apply subred_intro with (s := a) (s' := b).
+  - apply Es_trans with (u := s); assumption.
+  - exact Hn.
+  - exact Hb.
+Qed.
+
+Lemma subred_Es_r : forall t u v, subred t u -> Es u v -> subred t v.
+Proof.
+  intros t u v [a [b [Ha [Hn Hb]]]] Huv.
+  apply subred_intro with (s := a) (s' := b).
+  - exact Ha.
+  - exact Hn.
+  - apply Es_trans with (u := u); assumption.
+Qed.
+
+Lemma subred_Es_lr : forall t s s' t',
+    Es t s -> subred s s' -> Es s' t' -> subred t t'.
+Proof.
+  intros t s s' t' Hts Hss' Hs't'.
+  apply subred_Es_r with (u := s').
+  - apply subred_Es_l with (s := s); assumption.
+  - exact Hs't'.
+Qed.
+
+Lemma subred_context : forall C t t', subred t t' -> subred (nplug C t) (nplug C t').
+Proof.
+  intros C t t' [a [b [Ha [Hn Hb]]]].
+  apply subred_intro with (s := nplug C a) (s' := nplug C b).
+  - apply Es_ctx. exact Ha.
+  - apply nred_ctx. exact Hn.
+  - apply Es_ctx. exact Hb.
+Qed.
+
+Lemma subred_context_rule : forall C t t', nred t t' -> subred (nplug C t) (nplug C t').
+Proof. intros. apply subred_context. apply subred_nred. exact H. Qed.
+
+Lemma subred_of_Es_nred_Es : forall t s s' t',
+    Es t s -> nred s s' -> Es s' t' -> subred t t'.
+Proof. intros. apply subred_intro with (s := s) (s' := s'); assumption. Qed.
+
+(* The rules are stable modulo Es: rewriting the redex or the reduct by Es
+   gives a valid subred step. *)
+Lemma nred_stable_source : forall t s u, Es t s -> nred s u -> subred t u.
+Proof. intros. apply subred_intro with (s := s) (s' := u); [exact H | exact H0 | apply Es_refl]. Qed.
+
+Lemma nred_stable_target : forall t u v, nred t u -> Es u v -> subred t v.
+Proof. intros. apply subred_intro with (s := t) (s' := u); [apply Es_refl | exact H | exact H0]. Qed.
+
+Print Assumptions Es_Equivalence.
+Print Assumptions Es_NApp_Proper.
+Print Assumptions Es_NLam_Proper.
+Print Assumptions Es_NESub_Proper.
+Print Assumptions Es_nplug_Proper.
+Print Assumptions Es_nfv_Proper.
+Print Assumptions subred_nred.
+Print Assumptions subred_Es_l.
+Print Assumptions subred_Es_r.
+Print Assumptions subred_context.
+Print Assumptions nred_stable_source.
+Print Assumptions nred_stable_target.
+
+Inductive StarN (R : ntrm -> ntrm -> Prop) : ntrm -> ntrm -> Prop :=
+| starN_refl : forall t, StarN R t t
+| starN_step : forall t u v, R t u -> StarN R u v -> StarN R t v.
+
+Inductive PlusN (R : ntrm -> ntrm -> Prop) : ntrm -> ntrm -> Prop :=
+| plusN1 : forall t u, R t u -> PlusN R t u
+| plusNS : forall t u v, R t u -> PlusN R u v -> PlusN R t v.
+
+Lemma StarN_trans : forall R a b c, StarN R a b -> StarN R b c -> StarN R a c.
+Proof.
+  intros R a b c H. induction H; intros Hbc.
+  - exact Hbc.
+  - apply starN_step with (u := u). exact H. apply IHStarN. exact Hbc.
+Qed.
+
+Lemma StarN_one_trans : forall R a b c, R a b -> StarN R b c -> StarN R a c.
+Proof. intros. apply starN_step with (u := b); assumption. Qed.
+
+Lemma PlusN_to_StarN : forall R t u, PlusN R t u -> StarN R t u.
+Proof.
+  intros R t u H. induction H.
+  - apply starN_step with (u := u); [exact H | apply starN_refl].
+  - apply starN_step with (u := u); [exact H | exact IHPlusN].
+Qed.
+
+Lemma StarN_subred_context : forall C t t',
+    StarN subred t t' -> StarN subred (nplug C t) (nplug C t').
+Proof.
+  intros C t t' H. induction H.
+  - apply starN_refl.
+  - apply starN_step with (u := nplug C u).
+    + apply subred_context. exact H.
+    + exact IHStarN.
+Qed.
+
+Print Assumptions StarN_trans.
+Print Assumptions StarN_one_trans.
+Print Assumptions PlusN_to_StarN.
+Print Assumptions StarN_subred_context.

theory/NamedMeasure.v

@@ -0,0 +1,230 @@
+From Stdlib Require Import List Bool Arith Lia PeanoNat.
+From Stdlib Require Import Setoid Morphisms.
+Import ListNotations.
+From LambdaSub Require Import NamedMeta NamedEs.
+
+(* The measure of Appendix A of the paper.
+
+   For a metaterm t the paper defines two functions:
+     Mx(t)  an upper bound on the number of free occurrences of x that can
+            give rise to redexes in sub-reducts of t,
+     s(t)   the size measure used to orient the rules R, RX and Gc.
+   Both discriminate on whether the body of a substitution is a chain
+   X_Delta[x1/u1]...[xn/un] ending in a metavariable and the substituted
+   variable belongs to the annotation Delta. We capture that shape with
+   mhead below. *)
+
+Fixpoint mhead (t : ntrm) : option (list atom) :=
+  match t with
+  | NMVar _ d => Some d
+  | NESub a _ _ => mhead a
+  | _ => None
+  end.
+
+Definition head_has (a : ntrm) (y : atom) : bool :=
+  match mhead a with
+  | Some d => existsb (Nat.eqb y) d
+  | None => false
+  end.
+
+Fixpoint Mx (x : atom) (t : ntrm) : nat :=
+  match t with
+  | NVar y => if Nat.eqb y x then 1 else 0
+  | NApp a b => Mx x a + Mx x b
+  | NLam _ a => Mx x a
+  | NMVar _ d => if existsb (Nat.eqb x) d then 1 else 0
+  | NESub a y u =>
+      if head_has a y
+      then Mx x a + Mx y a * Mx x u
+      else Mx x a + Mx x u + Mx y a * Mx x u
+  end.
+
+Fixpoint s (t : ntrm) : nat :=
+  match t with
+  | NVar _ => 1
+  | NApp a b => s a + s b
+  | NLam _ a => s a
+  | NMVar _ d => length d
+  | NESub a y u =>
+      if head_has a y
+      then s a - 1 + Mx y a * s u
+      else s a + s u + Mx y a * s u
+  end.
+
+(* The counting invariant of the paper: Mx is an upper bound on the number
+   of free occurrences of x. We count free occurrences directly with nocc. *)
+Fixpoint nocc (x : atom) (t : ntrm) : nat :=
+  match t with
+  | NVar y => if Nat.eqb y x then 1 else 0
+  | NApp a b => nocc x a + nocc x b
+  | NLam y a => if Nat.eqb y x then 0 else nocc x a
+  | NESub a y u => (if Nat.eqb y x then 0 else nocc x a) + nocc x u
+  | NMVar _ d => if existsb (Nat.eqb x) d then 1 else 0
+  end.
+
+Lemma Mx_ge_0 : forall x t, 0 <= Mx x t.
+Proof. intros x t. induction t; simpl; lia. Qed.
+
+Lemma head_has_pos : forall a y, head_has a y = true -> 1 <= Mx y a.
+Proof.
+  intros a. induction a as [z | a1 IHa1 a2 _ | z a IHa | a1 IHa1 z u _ | X d]; intros y H.
+  - simpl in H. discriminate H.
+  - simpl in H. discriminate H.
+  - simpl in H. discriminate H.
+  - simpl in H. simpl. destruct (head_has a1 z) eqn:E; pose proof (IHa1 y H); lia.
+  - simpl in H. simpl.
+    assert (Hex : existsb (Nat.eqb y) d = true).
+    { unfold head_has, mhead in H. exact H. }
+    rewrite Hex. lia.
+Qed.
+
+Lemma le_mul_of_one_le : forall n m, 1 <= n -> m <= n * m.
+Proof.
+  intros n m H. rewrite <- (Nat.mul_1_l m) at 1.
+  apply Nat.mul_le_mono_r. exact H.
+Qed.
+
+Lemma nocc_le_Mx : forall x t, nocc x t <= Mx x t.
+Proof.
+  intros x t. induction t as [y | a IHa b IHb | y a IHa | a IHa y u IHu | X d].
+  - simpl. destruct (Nat.eqb y x); lia.
+  - simpl. lia.
+  - simpl. destruct (Nat.eqb y x); lia.
+  - simpl. destruct (head_has a y) eqn:E.
+    + assert (Hpos : 1 <= Mx y a) by (apply head_has_pos; exact E).
+      assert (Hnl : Mx x u <= Mx y a * Mx x u) by (apply le_mul_of_one_le; exact Hpos).
+      destruct (Nat.eqb y x) eqn:Ey; lia.
+    + destruct (Nat.eqb y x) eqn:Ey; lia.
+  - simpl. destruct (existsb (Nat.eqb x) d); lia.
+Qed.
+
+Print Assumptions Mx_ge_0.
+Print Assumptions head_has_pos.
+Print Assumptions le_mul_of_one_le.
+Print Assumptions nocc_le_Mx.
+
+(* ------------------------------------------------------------------ *)
+(* The missing invariants.
+
+   The measure proof needs the reduction rules to preserve freeness, in the
+   sense that a variable that is not free in a term contributes nothing
+   to the measure. Concretely the paper uses
+
+       x notin fv(t)  implies  Mx(x,t) = 0
+
+   for the Gc case and, through Lemma A.2, for the R and RX cases. That
+   property is not true on the raw named syntax: a metavariable may carry
+   an annotation variable that an enclosing substitution binds. The
+   metaterm X_[x][x/y] below is the smallest witness. Its free variables
+   are {y}, it therefore has no free occurrence of x, yet Mx(x, .) = 1. *)
+
+Lemma nfv_annot_bound_example :
+  nfv (NESub (NMVar 9 [0]) 0 (NVar 1)) = [1].
+Proof. reflexivity. Qed.
+
+Lemma Mx_fresh_fails :
+  ~ (forall x t, ~ In x (nfv t) -> Mx x t = 0).
+Proof.
+  intro H.
+  assert (Hn : ~ In 0 (nfv (NESub (NMVar 9 [0]) 0 (NVar 1)))).
+  { rewrite nfv_annot_bound_example. intros [Hc|Hc]; [discriminate Hc | destruct Hc]. }
+  specialize (H 0 (NESub (NMVar 9 [0]) 0 (NVar 1)) Hn).
+  compute in H. discriminate H.
+Qed.
+
+(* Lemma A.1, invariance of Mx and s under the C equation, also needs the
+   invariant. The two terms below are related by C with x = 0, y = 1, y not
+   free in u and x not free in v, yet their sizes and Mx values differ: the
+   occurrences of 1 and 0 that sit inside the metavariable annotation but
+   below the substitutions are counted by the special clauses. *)
+Definition A1_t : ntrm := NMVar 0 [0; 1].
+Definition A1_u : ntrm :=
+  NApp (NESub (NMVar 8 [1]) 1 (NVar 3))
+       (NApp (NApp (NVar 5) (NVar 6)) (NVar 7)).
+Definition A1_v : ntrm := NESub (NMVar 7 [0]) 0 (NVar 4).
+Definition A1_lhs : ntrm := NESub (NESub A1_t 0 A1_u) 1 A1_v.
+Definition A1_rhs : ntrm := NESub (NESub A1_t 1 A1_v) 0 A1_u.
+
+Lemma A1_C_related : Es A1_lhs A1_rhs.
+Proof.
+  apply Es_C with (x := 0) (y := 1).
+  - intro E. discriminate E.
+  - compute. intros [H|[H|[H|[H|H]]]]; try discriminate H; destruct H.
+  - compute. intros [H|H]; [discriminate H | destruct H].
+Qed.
+
+Lemma A1_s_differ : s A1_lhs <> s A1_rhs.
+Proof. compute. discriminate. Qed.
+
+Lemma A1_Mx_differ : Mx 0 A1_lhs <> Mx 0 A1_rhs.
+Proof. compute. discriminate. Qed.
+
+Lemma A1_not_invariant :
+  ~ (forall t u, Es t u -> s t = s u /\ forall z, Mx z t = Mx z u).
+Proof.
+  intro H. destruct (H A1_lhs A1_rhs A1_C_related) as [Hs _].
+  apply A1_s_differ. exact Hs.
+Qed.
+
+(* The corresponding Gc step is legal in the current rule system, but the
+   measure does not decrease: the redex has the same size as its reduct.
+   The body is X_[x][x/y] with x in the annotation, so the special clause
+   of s applies and the (bound) x in the annotation is counted by Mx,
+   cancelling the subtraction of one. *)
+Lemma Gc_special_nondecrease :
+  s (NESub (NESub (NMVar 9 [0]) 0 (NVar 1)) 0 (NVar 2))
+  = s (NESub (NMVar 9 [0]) 0 (NVar 1)).
+Proof. reflexivity. Qed.
+
+Lemma nred_Gc_special :
+  nred (NESub (NESub (NMVar 9 [0]) 0 (NVar 1)) 0 (NVar 2))
+       (NESub (NMVar 9 [0]) 0 (NVar 1)).
+Proof.
+  apply nred_Gc. simpl. intros [Hc|Hc]; [discriminate Hc | destruct Hc].
+Qed.
+
+(* A second, independent, gap: the paper states s(t) >= 1, but a
+   metavariable with empty annotation has size zero. Empty annotations are
+   not excluded by the raw syntax. *)
+Lemma s_metavar_empty : s (NMVar 9 []) = 0.
+Proof. reflexivity. Qed.
+
+(* Rule R as currently stated has no freshness side condition on the
+   substituted term, so it creates a loop, which no terminating measure can
+   orient. The step below is derivable with the empty context. The paper
+   excludes it through the convention that no free and bound variable of a
+   term share a name: x[x/u] with x free in u is not an alpha-canonical
+   metaterm. *)
+Lemma nred_R_selfloop : forall x,
+  nred (NESub (NVar x) x (NVar x)) (NESub (NVar x) x (NVar x)).
+Proof.
+  intro x. apply nred_R with (C := nHole) (phi := [x]).
+  - simpl. left. reflexivity.
+  - intros y Hy. simpl in Hy. destruct Hy as [Hy|[]]. subst y. simpl. left. reflexivity.
+  - reflexivity.
+Qed.
+
+Theorem current_sm_not_terminating : exists t, nred t t.
+Proof. exists (NESub (NVar 0) 0 (NVar 0)). apply nred_R_selfloop. Qed.
+
+(* Consequence for Es: the modulo relation subred inherits the loop, so it
+   is not terminating either without the freshness invariant. *)
+Lemma subred_selfloop : forall x,
+  subred (NESub (NVar x) x (NVar x)) (NESub (NVar x) x (NVar x)).
+Proof. intro x. apply subred_nred. apply nred_R_selfloop. Qed.
+
+Theorem current_subred_not_terminating : exists t, subred t t.
+Proof. exists (NESub (NVar 0) 0 (NVar 0)). apply subred_selfloop. Qed.
+
+Print Assumptions Mx_fresh_fails.
+Print Assumptions A1_C_related.
+Print Assumptions A1_s_differ.
+Print Assumptions A1_Mx_differ.
+Print Assumptions A1_not_invariant.
+Print Assumptions Gc_special_nondecrease.
+Print Assumptions nred_Gc_special.
+Print Assumptions s_metavar_empty.
+Print Assumptions nred_R_selfloop.
+Print Assumptions current_sm_not_terminating.
+Print Assumptions subred_selfloop.
+Print Assumptions current_subred_not_terminating.

theory/NamedMeta.v

@@ -0,0 +1,357 @@
+From Stdlib Require Import List Bool Arith Lia PeanoNat.
+Import ListNotations.
+
+Definition atom := nat.
+
+Inductive ntrm : Type :=
+| NVar : atom -> ntrm
+| NApp : ntrm -> ntrm -> ntrm
+| NLam : atom -> ntrm -> ntrm
+| NESub : ntrm -> atom -> ntrm -> ntrm
+| NMVar : atom -> list atom -> ntrm.
+
+Fixpoint nfv (t : ntrm) : list atom :=
+  match t with
+  | NVar x => [x]
+  | NApp a b => nfv a ++ nfv b
+  | NLam x a => filter (fun y => negb (Nat.eqb y x)) (nfv a)
+  | NESub a x b => filter (fun y => negb (Nat.eqb y x)) (nfv a) ++ nfv b
+  | NMVar _ d => d
+  end.
+
+Definition isubst_meta (X : atom) (d : list atom) (x : atom) (v : ntrm) : ntrm :=
+  if existsb (Nat.eqb x) d then NESub (NMVar X d) x v else NMVar X d.
+
+Lemma isubst_meta_in : forall X d x v, In x d -> isubst_meta X d x v = NESub (NMVar X d) x v.
+Proof.
+  intros X d x v H. unfold isubst_meta.
+  destruct (existsb (Nat.eqb x) d) eqn:E; [reflexivity |].
+  assert (Hex : existsb (Nat.eqb x) d = true)
+    by (apply existsb_exists; exists x; split; [exact H | apply Nat.eqb_refl]).
+  rewrite Hex in E. discriminate E.
+Qed.
+
+Lemma isubst_meta_notin : forall X d x v, ~ In x d -> isubst_meta X d x v = NMVar X d.
+Proof.
+  intros X d x v H. unfold isubst_meta.
+  destruct (existsb (Nat.eqb x) d) eqn:E; [| reflexivity].
+  exfalso. apply existsb_exists in E. destruct E as [y [Hy HE]].
+  apply Nat.eqb_eq in HE. subst y. apply H. exact Hy.
+Qed.
+
+Inductive nctx : Type :=
+| nHole : nctx
+| nAppL : nctx -> ntrm -> nctx
+| nAppR : ntrm -> nctx -> nctx
+| nLam : atom -> nctx -> nctx
+| nESubL : nctx -> atom -> ntrm -> nctx
+| nESubR : ntrm -> atom -> nctx -> nctx.
+
+Fixpoint nplug (C : nctx) (t : ntrm) : ntrm :=
+  match C with
+  | nHole => t
+  | nAppL C1 u => NApp (nplug C1 t) u
+  | nAppR u C1 => NApp u (nplug C1 t)
+  | nLam x C1 => NLam x (nplug C1 t)
+  | nESubL C1 x u => NESub (nplug C1 t) x u
+  | nESubR t1 x C1 => NESub t1 x (nplug C1 t)
+  end.
+
+Fixpoint is_subst_chain (C : nctx) : bool :=
+  match C with
+  | nHole => true
+  | nESubL C1 _ _ => is_subst_chain C1
+  | _ => false
+  end.
+
+Inductive Es : ntrm -> ntrm -> Prop :=
+| Es_refl : forall t, Es t t
+| Es_sym : forall t u, Es t u -> Es u t
+| Es_trans : forall t u v, Es t u -> Es u v -> Es t v
+| Es_C : forall t x u y v,
+    y <> x -> ~ In y (nfv u) -> ~ In x (nfv v) ->
+    Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u)
+| 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)
+| nred_Gc : forall t x u, ~ In x (nfv t) -> nred (NESub t x u) t
+| nred_RX : forall C X d x u phi,
+    In x d ->
+    In x phi ->
+    (forall y, In y (nfv u) -> In y phi) ->
+    is_subst_chain C = false ->
+    nred (NESub (nplug C (NMVar X d)) x u)
+         (NESub (nplug C (NESub (NMVar X d) x u)) x u)
+| nred_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,
+  y <> x -> ~ In y (nfv u) -> ~ In x (nfv v) ->
+  Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u).
+Proof. intros. apply Es_C; assumption. Qed.
+
+Lemma Es_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.
+
+Lemma Es_sym_any : forall t u, Es t u -> Es u t.
+Proof. intros. apply Es_sym. exact H. Qed.
+
+Lemma Es_trans_any : forall t u v, Es t u -> Es u v -> Es t v.
+Proof. intros. apply Es_trans with (u := u); assumption. Qed.
+
+Print Assumptions isubst_meta_in.
+Print Assumptions isubst_meta_notin.
+Print Assumptions Es_ctx_any.
+Print Assumptions nred_R_intro.
+Print Assumptions eqC_in_Es.
+Print Assumptions Es_refl_any.
+Print Assumptions Es_sym_any.
+Print Assumptions Es_trans_any.
+
+Lemma in_filter_neq : forall y x l,
+  In y (filter (fun z => negb (Nat.eqb z x)) l) -> In y l /\ y <> x.
+Proof.
+  intros y x l H. apply filter_In in H. destruct H as [Hl Hp]. split; [exact Hl |].
+  apply Nat.eqb_neq. destruct (Nat.eqb y x) eqn:E; [| reflexivity].
+  simpl in Hp. discriminate Hp.
+Qed.
+
+Lemma filter_neq_in : forall y x l,
+  In y l -> y <> x -> In y (filter (fun z => negb (Nat.eqb z x)) l).
+Proof.
+  intros y x l Hl Hne. apply filter_In. split; [exact Hl |].
+  apply Nat.eqb_neq in Hne. rewrite Hne. reflexivity.
+Qed.
+
+
+Lemma nplug_fv_mono : forall C t s,
+  incl (nfv t) (nfv s) -> incl (nfv (nplug C t)) (nfv (nplug C s)).
+Proof.
+  induction C; intros t s Hincl; unfold incl in *; simpl in *.
+  - apply Hincl.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply (IHC t s Hincl). exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. exact Hy.
+    + apply in_app_iff. right. apply (IHC t s Hincl). exact Hy.
+  - intros y Hy. apply in_filter_neq in Hy as [Hy Hne].
+    apply filter_neq_in.
+    + apply (IHC t s Hincl). exact Hy.
+    + exact Hne.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply in_filter_neq in Hy as [Hy Hne].
+      apply filter_neq_in.
+      * apply (IHC t s Hincl). exact Hy.
+      * exact Hne.
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. exact Hy.
+    + apply in_app_iff. right. apply (IHC t s Hincl). exact Hy.
+Qed.
+
+Lemma nplug_fv_upper : forall C t s,
+  incl (nfv (nplug C t)) (nfv (nplug C s) ++ nfv t).
+Proof.
+  induction C; intros t s; unfold incl in *; simpl in *.
+  - intros y Hy. apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. left. exact Hy.
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply in_app_iff. left. exact Hy.
+    + apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+      * apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_filter_neq in Hy as [Hy Hne].
+    apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_app_iff. left. apply in_app_iff. left. exact Hy.
+    + apply (IHC t s) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply in_app_iff. right. exact Hy.
+      * apply in_app_iff. right. exact Hy.
+Qed.
+
+Inductive nred_core : ntrm -> ntrm -> Prop :=
+| ncore_B : forall t x u, nred_core (NApp (NLam x t) u) (NESub t x u)
+| ncore_Gc : forall t x u, ~ In x (nfv t) -> nred_core (NESub t x u) t
+| ncore_R : forall C x u phi,
+    In x phi -> (forall y, In y (nfv u) -> In y phi) -> cavoid C phi = true ->
+    nred_core (NESub (nplug C (NVar x)) x u) (NESub (nplug C u) x u)
+| ncore_ctx : forall C t t', nred_core t t' -> nred_core (nplug C t) (nplug C t').
+
+Lemma nred_core_fv : forall t t', nred_core t t' -> incl (nfv t') (nfv t).
+Proof.
+  intros t t' H. induction H; simpl; unfold incl in *.
+  - intros y Hy. exact Hy.
+  - intros y Hy. apply in_app_iff. left.
+    apply filter_neq_in; [exact Hy | intro E; apply H; subst; exact Hy].
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (nplug_fv_upper C u (NVar x)) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - apply nplug_fv_mono. exact IHnred_core.
+Qed.
+
+Print Assumptions nplug_fv_mono.
+Print Assumptions nplug_fv_upper.
+Print Assumptions nred_core_fv.
+
+Lemma nplug_esub_fv : forall C X d x u y,
+  y <> x ->
+  In y (nfv (nplug C (NESub (NMVar X d) x u))) ->
+  In y (nfv (nplug C (NMVar X d))) \/ In y (nfv u).
+Proof.
+  induction C; intros X d x u y Hne Hy; simpl in *.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + left. apply in_filter_neq in Hy as [Hy _]. simpl. exact Hy.
+    + right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. left. exact Hy.
+      * right. exact Hy.
+    + left. apply in_app_iff. right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + left. apply in_app_iff. left. exact Hy.
+    + apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. right. exact Hy.
+      * right. exact Hy.
+  - apply in_filter_neq in Hy as [Hy Hz].
+    apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+    + left. apply filter_neq_in; [exact Hy | exact Hz].
+    + right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hz].
+      apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hz].
+      * right. exact Hy.
+    + left. apply in_app_iff. right. exact Hy.
+  - apply in_app_iff in Hy as [Hy|Hy].
+    + left. apply in_app_iff. left. exact Hy.
+    + apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * left. apply in_app_iff. right. exact Hy.
+      * right. exact Hy.
+Qed.
+
+Lemma nred_fv : forall t t', nred t t' -> incl (nfv t') (nfv t).
+Proof.
+  intros t t' H. induction H; simpl; unfold incl in *.
+  - intros y Hy. exact Hy.
+  - intros y Hy. apply in_app_iff. left.
+    apply filter_neq_in; [exact Hy | intro E; apply H; subst; exact Hy].
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (nplug_esub_fv C X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+    + apply in_filter_neq in Hy as [Hy Hne].
+      apply (nplug_fv_upper C u (NVar x)) in Hy. apply in_app_iff in Hy as [Hy|Hy].
+      * apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
+      * apply in_app_iff. right. exact Hy.
+    + apply in_app_iff. right. exact Hy.
+  - apply nplug_fv_mono. exact IHnred.
+Qed.
+
+Print Assumptions nplug_esub_fv.
+Print Assumptions nred_fv.
+
+Lemma in_filter_neq_iff : forall z a l,
+  In z (filter (fun w => negb (Nat.eqb w a)) l) <-> In z l /\ z <> a.
+Proof.
+  intros z a l. split.
+  - intro H. apply filter_In in H. destruct H as [Hl Hp]. split; [exact Hl |].
+    apply Nat.eqb_neq. destruct (Nat.eqb z a) eqn:E; [| reflexivity].
+    simpl in Hp. discriminate Hp.
+  - intros [Hl Hne]. apply filter_In. split; [exact Hl |].
+    apply Nat.eqb_neq in Hne. rewrite Hne. reflexivity.
+Qed.
+
+Lemma Es_fv_both : forall t u, Es t u ->
+  incl (nfv t) (nfv u) /\ incl (nfv u) (nfv t).
+Proof.
+  apply Es_ind.
+  - intros t0. split; unfold incl; auto.
+  - intros t0 u0 H1 IH1. split.
+    + apply (proj2 IH1).
+    + apply (proj1 IH1).
+  - intros t0 u0 v0 H1 H2 IH1 IH2. split.
+    + intros z Hz. apply (proj1 IH2). apply (proj1 H2). exact Hz.
+    + intros z Hz. apply (proj2 H2). apply (proj2 IH2). exact Hz.
+  - intros t0 x0 u0 y0 v0 Hyx Hyu Hxv. split; unfold incl; intros z Hz; simpl in *;
+    apply in_app_iff in Hz as [Hz|Hz].
+    + apply in_filter_neq_iff in Hz as [Hz Hy]. apply in_app_iff in Hz as [Hz|Hz].
+      * apply in_filter_neq_iff in Hz as [Hz Hx].
+        apply in_app_iff. left. apply in_filter_neq_iff. split; [| exact Hx].
+        apply in_app_iff. left. apply in_filter_neq_iff. split; [exact Hz | exact Hy].
+      * apply in_app_iff. right. exact Hz.
+    + apply in_app_iff. left. apply in_filter_neq_iff. split; [| ].
+      * apply in_app_iff. right. exact Hz.
+      * intro E. apply Hxv. subst z. exact Hz.
+    + apply in_filter_neq_iff in Hz as [Hz Hx]. apply in_app_iff in Hz as [Hz|Hz].
+      * apply in_filter_neq_iff in Hz as [Hz Hy].
+        apply in_app_iff. left. apply in_filter_neq_iff. split; [| exact Hy].
+        apply in_app_iff. left. apply in_filter_neq_iff. split; [exact Hz | exact Hx].
+      * apply in_app_iff. right. exact Hz.
+    + apply in_app_iff. left. apply in_filter_neq_iff. split; [| ].
+      * apply in_app_iff. right. exact Hz.
+      * intro E. apply Hyu. subst z. exact Hz.
+  - intros C0 t0 t0' H1 IH1. split; apply nplug_fv_mono.
+    + apply (proj1 IH1).
+    + apply (proj2 IH1).
+Qed.
+
+Lemma Es_fv : forall t u, Es t u -> forall z, In z (nfv t) <-> In z (nfv u).
+Proof.
+  intros t u H z. destruct (Es_fv_both t u H) as [Htu Hut]. split.
+  - apply Htu.
+  - apply Hut.
+Qed.
+
+Print Assumptions in_filter_neq_iff.
+Print Assumptions Es_fv.