From 88fd0a8e56bb65e9fe096ad515aa583298e1cd04 Mon Sep 17 00:00:00 2001
From: sneeker <s@cpl0.zip>
Date: Tue, 22 Sep 2026 09:20:00 +0200
Subject: [PATCH] initial commit: confluence and normalisation

---
 .gitignore            |  11 +
 README.md             |  15 ++
 dune-project          |   4 +
 graphs/dependency.dot | 110 ++++++++++
 graphs/dependency.png | Bin 0 -> 163675 bytes
 graphs/dependency.svg | 493 ++++++++++++++++++++++++++++++++++++++++++
 graphs/reduction.dot  |  20 ++
 graphs/reduction.png  | Bin 0 -> 29263 bytes
 graphs/reduction.svg  |  93 ++++++++
 graphs/rules.dot      |  19 ++
 graphs/rules.png      | Bin 0 -> 44541 bytes
 graphs/rules.svg      | 100 +++++++++
 graphs/theorems.json  | 272 +++++++++++++++++++++++
 scripts/audit.sh      |  52 +++++
 scripts/gen_graphs.py | 180 +++++++++++++++
 theory/ExecReducer.v  | 272 +++++++++++++++++++++++
 theory/dune           |   3 +
 17 files changed, 1644 insertions(+)
 create mode 100644 .gitignore
 create mode 100644 README.md
 create mode 100644 dune-project
 create mode 100644 graphs/dependency.dot
 create mode 100644 graphs/dependency.png
 create mode 100644 graphs/dependency.svg
 create mode 100644 graphs/reduction.dot
 create mode 100644 graphs/reduction.png
 create mode 100644 graphs/reduction.svg
 create mode 100644 graphs/rules.dot
 create mode 100644 graphs/rules.png
 create mode 100644 graphs/rules.svg
 create mode 100644 graphs/theorems.json
 create mode 100755 scripts/audit.sh
 create mode 100755 scripts/gen_graphs.py
 create mode 100644 theory/ExecReducer.v
 create mode 100644 theory/dune

diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..e8eff97
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,11 @@
+_build/
+*.vo
+*.vok
+*.vos
+*.glob
+*.aux
+*.cmi
+*.cmo
+*.cmx
+__pycache__/
+*.pyc
diff --git a/README.md b/README.md
new file mode 100644
index 0000000..408c869
--- /dev/null
+++ b/README.md
@@ -0,0 +1,15 @@
+This was (mostly) me being bored in a Monday evening and in an attempt to formalise in Rocq the confluence and normalisation results that Delia Kesner and Shane Ó Conchúir obtain for the lambda calculus of Milner with partial substitutions which is available as arXiv:2312.13270.
+
+## 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:
+
+![Theorem dependency graph](graphs/dependency.svg)
+
+The rule sketch gives the transition system generated by the rules B and Gc and R:
+
+![Reduction rules](graphs/rules.svg)
+
+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)
diff --git a/dune-project b/dune-project
new file mode 100644
index 0000000..0eacd23
--- /dev/null
+++ b/dune-project
@@ -0,0 +1,4 @@
+(lang dune 3.21)
+(name lambda_sub)
+(using rocq 0.11)
+(version 0.1.0)
diff --git a/graphs/dependency.dot b/graphs/dependency.dot
new file mode 100644
index 0000000..8efbe20
--- /dev/null
+++ b/graphs/dependency.dot
@@ -0,0 +1,110 @@
+digraph theorem_deps {
+  rankdir=LR;
+  bgcolor="white";
+  node [fontname=Inter, fontsize=10, fontcolor="#333333"];
+  edge [fontname=Inter, fontsize=8, color="#333333"];
+  graph [fontname=Inter, fontsize=12, labelloc=t, label="Milner's Lambda-Calculus with Partial Substitutions - theorem dependency (M0)"];
+  subgraph cluster_M0 {
+    label="M0";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    exec_step [label="exec_step", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#333333", tooltip="exec_step theory/ExecReducer.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+    exec_examples [label="exec_examples", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#333333", tooltip="exec_examples theory/ExecReducer.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  }
+  subgraph cluster_M1 {
+    label="M1";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    m1_syntax [label="M1 syntax/binding", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m1_syntax (section 2) https://arxiv.org/html/2312.13270v1", shape=ellipse];
+    m1_open_close [label="open/close laws", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m1_open_close (section 2) https://arxiv.org/html/2312.13270v1", shape=ellipse];
+    m1_red1 [label="red1 (B, Gc, R)", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m1_red1 (section 2) https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  }
+  subgraph cluster_M2 {
+    label="M2";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    lemma_2_1_fv_preserved [label="Lemma 2.1 fv preserved", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="lemma_2_1_fv_preserved (section 2) https://arxiv.org/html/2312.13270v1", shape=box];
+    lemma_2_2_full_comp [label="Lemma 2.2 full composition", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="lemma_2_2_full_comp (section 2) https://arxiv.org/html/2312.13270v1", shape=box];
+    lemma_2_3_beta_sim [label="Lemma 2.3 β-simulation", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="lemma_2_3_beta_sim (section 2) https://arxiv.org/html/2312.13270v1", shape=box];
+  }
+  subgraph cluster_M3 {
+    label="M3";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    m3_measure [label="sub measure", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m3_measure (section A) https://arxiv.org/html/2312.13270v1", shape=ellipse];
+    m3_termination [label="{R,Gc} termination", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m3_termination (section 3.1) https://arxiv.org/html/2312.13270v1", shape=box];
+    m3_local_confluence [label="{R,Gc} local confluence", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m3_local_confluence (section 3.1) https://arxiv.org/html/2312.13270v1", shape=box];
+    lemma_3_2_sub_nf_unique [label="Lemma 3.2 sub NF unique", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="lemma_3_2_sub_nf_unique (section 3.1) https://arxiv.org/html/2312.13270v1", shape=box];
+  }
+  subgraph cluster_M4 {
+    label="M4";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    m4_par [label="parallel reduction par", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m4_par (section 3.2) https://arxiv.org/html/2312.13270v1", shape=ellipse];
+    m4_par_diamond [label="par diamond", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m4_par_diamond (section 3.2) https://arxiv.org/html/2312.13270v1", shape=box];
+    theorem_confluence_terms [label="Confluence on terms", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="theorem_confluence_terms (section 3.2) https://arxiv.org/html/2312.13270v1", shape=box];
+  }
+  subgraph cluster_M5 {
+    label="M5";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    m5_mvar [label="XΔ metaterms", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m5_mvar (section 3.3) https://arxiv.org/html/2312.13270v1", shape=ellipse];
+    m5_rx [label="RX rule", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m5_rx (section 3.3) https://arxiv.org/html/2312.13270v1", shape=box];
+    m5_eq_C [label="=C setoid", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m5_eq_C (section 3.3) https://arxiv.org/html/2312.13270v1", shape=box];
+    m5_coherence [label="RX + =C coherence", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m5_coherence (section 3.3) https://arxiv.org/html/2312.13270v1", shape=box];
+    theorem_confluence_metaterms [label="Confluence on metaterms", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="theorem_confluence_metaterms (section 3.3) https://arxiv.org/html/2312.13270v1", shape=box];
+  }
+  subgraph cluster_M6 {
+    label="M6";
+    style="rounded";
+    color="#bbbbbb";
+    fontcolor="#333333";
+    m6_translation [label="translation to λβp", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m6_translation (section 4) https://arxiv.org/html/2312.13270v1", shape=box];
+    m6_sn [label="SN for simply typed", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m6_sn (section 6) https://arxiv.org/html/2312.13270v1", shape=box];
+    m6_intersection [label="intersection typing", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m6_intersection (section 6) https://arxiv.org/html/2312.13270v1", shape=box];
+    m6_typable_sn [label="typable → SN", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m6_typable_sn (section 6) https://arxiv.org/html/2312.13270v1", shape=box];
+    m6_sn_typable [label="SN → typable", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="m6_sn_typable (section 6) https://arxiv.org/html/2312.13270v1", shape=box];
+    corollary_6_21_psn [label="Cor 6.21 PSN", color="#777777", fillcolor="#f3f3f3", style="dashed", fontcolor="#333333", tooltip="corollary_6_21_psn (section 6) https://arxiv.org/html/2312.13270v1", shape=box];
+  }
+  exec_step -> exec_examples;
+  exec_examples -> m1_syntax;
+  m1_syntax -> m1_open_close;
+  m1_syntax -> m1_red1;
+  m1_open_close -> m1_red1;
+  m1_red1 -> lemma_2_1_fv_preserved;
+  m1_red1 -> lemma_2_2_full_comp;
+  m1_red1 -> lemma_2_3_beta_sim;
+  lemma_2_2_full_comp -> lemma_2_3_beta_sim;
+  m1_red1 -> m3_measure;
+  m3_measure -> m3_termination;
+  m3_termination -> m3_local_confluence;
+  m1_red1 -> m3_local_confluence;
+  m3_local_confluence -> lemma_3_2_sub_nf_unique;
+  m1_red1 -> m4_par;
+  lemma_3_2_sub_nf_unique -> m4_par;
+  m4_par -> m4_par_diamond;
+  m4_par_diamond -> theorem_confluence_terms;
+  lemma_2_3_beta_sim -> theorem_confluence_terms;
+  m1_syntax -> m5_mvar;
+  m5_mvar -> m5_rx;
+  m1_red1 -> m5_rx;
+  m5_mvar -> m5_eq_C;
+  m5_rx -> m5_coherence;
+  m5_eq_C -> m5_coherence;
+  lemma_3_2_sub_nf_unique -> m5_coherence;
+  m5_coherence -> theorem_confluence_metaterms;
+  m4_par_diamond -> theorem_confluence_metaterms;
+  m1_red1 -> m6_translation;
+  m6_translation -> m6_sn;
+  m6_sn -> m6_intersection;
+  m6_intersection -> m6_typable_sn;
+  m6_typable_sn -> m6_sn_typable;
+  m6_sn_typable -> corollary_6_21_psn;
+  theorem_confluence_terms -> corollary_6_21_psn;
+}
diff --git a/graphs/dependency.png b/graphs/dependency.png
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+<g id="clust7" class="cluster">
+<title>cluster_M6</title>
+<path fill="white" stroke="#bbbbbb" d="M1075.78,-442.64C1075.78,-442.64 1972.35,-442.64 1972.35,-442.64 1978.35,-442.64 1984.35,-448.64 1984.35,-454.64 1984.35,-454.64 1984.35,-503.64 1984.35,-503.64 1984.35,-509.64 1978.35,-515.64 1972.35,-515.64 1972.35,-515.64 1075.78,-515.64 1075.78,-515.64 1069.78,-515.64 1063.78,-509.64 1063.78,-503.64 1063.78,-503.64 1063.78,-454.64 1063.78,-454.64 1063.78,-448.64 1069.78,-442.64 1075.78,-442.64"/>
+<text text-anchor="middle" x="1524.07" y="-502.04" font-family="Inter" font-size="12.00" fill="#333333">M6</text>
+</g>
+<!-- exec_step -->
+<g id="node1" class="node">
+<title>exec_step</title>
+<g id="a_node1"><a xlink:title="exec_step theory/ExecReducer.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="53.67" cy="-226.64" rx="37.85" ry="18"/>
+<text text-anchor="middle" x="53.67" y="-224.14" font-family="Inter" font-size="10.00" fill="#333333">exec_step</text>
+</a>
+</g>
+</g>
+<!-- exec_examples -->
+<g id="node2" class="node">
+<title>exec_examples</title>
+<g id="a_node2"><a xlink:title="exec_examples theory/ExecReducer.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="179.15" cy="-226.64" rx="51.61" ry="18"/>
+<text text-anchor="middle" x="179.15" y="-224.14" font-family="Inter" font-size="10.00" fill="#333333">exec_examples</text>
+</a>
+</g>
+</g>
+<!-- exec_step&#45;&gt;exec_examples -->
+<g id="edge1" class="edge">
+<title>exec_step&#45;&gt;exec_examples</title>
+<path fill="none" stroke="#333333" d="M91.39,-226.64C99.46,-226.64 108.22,-226.64 116.97,-226.64"/>
+<polygon fill="#333333" stroke="#333333" points="117.18,-230.14 127.18,-226.64 117.18,-223.14 117.18,-230.14"/>
+</g>
+<!-- m1_syntax -->
+<g id="node3" class="node">
+<title>m1_syntax</title>
+<g id="a_node3"><a xlink:title="m1_syntax (section 2) https://arxiv.org/html/2312.13270v1">
+<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="326.41" cy="-226.64" rx="59.41" ry="18"/>
+<text text-anchor="middle" x="326.41" y="-224.14" font-family="Inter" font-size="10.00" fill="#333333">M1 syntax/binding</text>
+</a>
+</g>
+</g>
+<!-- exec_examples&#45;&gt;m1_syntax -->
+<g id="edge2" class="edge">
+<title>exec_examples&#45;&gt;m1_syntax</title>
+<path fill="none" stroke="#333333" d="M230.96,-226.64C239.31,-226.64 248.08,-226.64 256.78,-226.64"/>
+<polygon fill="#333333" stroke="#333333" points="256.92,-230.14 266.92,-226.64 256.92,-223.14 256.92,-230.14"/>
+</g>
+<!-- m1_open_close -->
+<g id="node4" class="node">
+<title>m1_open_close</title>
+<g id="a_node4"><a xlink:title="m1_open_close (section 2) https://arxiv.org/html/2312.13270v1">
+<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="477.2" cy="-206.64" rx="55.17" ry="18"/>
+<text text-anchor="middle" x="477.2" y="-204.14" font-family="Inter" font-size="10.00" fill="#333333">open/close laws</text>
+</a>
+</g>
+</g>
+<!-- m1_syntax&#45;&gt;m1_open_close -->
+<g id="edge3" class="edge">
+<title>m1_syntax&#45;&gt;m1_open_close</title>
+<path fill="none" stroke="#333333" d="M381.12,-219.42C392.21,-217.93 403.97,-216.35 415.29,-214.83"/>
+<polygon fill="#333333" stroke="#333333" points="416.09,-218.25 425.54,-213.45 415.16,-211.31 416.09,-218.25"/>
+</g>
+<!-- m1_red1 -->
+<g id="node5" class="node">
+<title>m1_red1</title>
+<g id="a_node5"><a xlink:title="m1_red1 (section 2) https://arxiv.org/html/2312.13270v1">
+<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="619.16" cy="-206.64" rx="50.75" ry="18"/>
+<text text-anchor="middle" x="619.16" y="-204.14" font-family="Inter" font-size="10.00" fill="#333333">red1 (B, Gc, R)</text>
+</a>
+</g>
+</g>
+<!-- m1_syntax&#45;&gt;m1_red1 -->
+<g id="edge4" class="edge">
+<title>m1_syntax&#45;&gt;m1_red1</title>
+<path fill="none" stroke="#333333" d="M380.86,-233.88C422.44,-238.21 481.47,-241.5 532.53,-233.64 546.53,-231.48 561.37,-227.46 574.7,-223.16"/>
+<polygon fill="#333333" stroke="#333333" points="576.19,-226.35 584.56,-219.85 573.97,-219.72 576.19,-226.35"/>
+</g>
+<!-- m5_mvar -->
+<g id="node16" class="node">
+<title>m5_mvar</title>
+<g id="a_node16"><a xlink:title="m5_mvar (section 3.3) https://arxiv.org/html/2312.13270v1">
+<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="1292.78" cy="-363.64" rx="48.72" ry="18"/>
+<text text-anchor="middle" x="1292.78" y="-361.14" font-family="Inter" font-size="10.00" fill="#333333">XΔ metaterms</text>
+</a>
+</g>
+</g>
+<!-- m1_syntax&#45;&gt;m5_mvar -->
+<g id="edge20" class="edge">
+<title>m1_syntax&#45;&gt;m5_mvar</title>
+<path fill="none" stroke="#333333" d="M347.8,-243.48C393.28,-279.27 507.7,-359.64 618.16,-359.64 618.16,-359.64 618.16,-359.64 955.78,-359.64 1053.24,-359.64 1166.59,-361.34 1233.59,-362.53"/>
+<polygon fill="#333333" stroke="#333333" points="1233.87,-366.03 1243.93,-362.71 1234,-359.03 1233.87,-366.03"/>
+</g>
+<!-- m1_open_close&#45;&gt;m1_red1 -->
+<g id="edge5" class="edge">
+<title>m1_open_close&#45;&gt;m1_red1</title>
+<path fill="none" stroke="#333333" d="M532.71,-206.64C541,-206.64 549.6,-206.64 558.01,-206.64"/>
+<polygon fill="#333333" stroke="#333333" points="558.13,-210.14 568.13,-206.64 558.13,-203.14 558.13,-210.14"/>
+</g>
+<!-- lemma_2_1_fv_preserved -->
+<g id="node6" class="node">
+<title>lemma_2_1_fv_preserved</title>
+<g id="a_node6"><a xlink:title="lemma_2_1_fv_preserved (section 2) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="844.78,-121.64 714.78,-121.64 714.78,-85.64 844.78,-85.64 844.78,-121.64"/>
+<text text-anchor="middle" x="779.78" y="-101.14" font-family="Inter" font-size="10.00" fill="#333333">Lemma 2.1 fv preserved</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;lemma_2_1_fv_preserved -->
+<g id="edge6" class="edge">
+<title>m1_red1&#45;&gt;lemma_2_1_fv_preserved</title>
+<path fill="none" stroke="#333333" d="M643.48,-190.76C658.84,-180.38 679.42,-166.57 697.78,-154.64 711.73,-145.58 727.15,-135.81 740.76,-127.27"/>
+<polygon fill="#333333" stroke="#333333" points="742.79,-130.13 749.41,-121.85 739.08,-124.19 742.79,-130.13"/>
+</g>
+<!-- lemma_2_2_full_comp -->
+<g id="node7" class="node">
+<title>lemma_2_2_full_comp</title>
+<g id="a_node7"><a xlink:title="lemma_2_2_full_comp (section 2) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="853.78,-67.64 705.78,-67.64 705.78,-31.64 853.78,-31.64 853.78,-67.64"/>
+<text text-anchor="middle" x="779.78" y="-47.14" font-family="Inter" font-size="10.00" fill="#333333">Lemma 2.2 full composition</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;lemma_2_2_full_comp -->
+<g id="edge7" class="edge">
+<title>m1_red1&#45;&gt;lemma_2_2_full_comp</title>
+<path fill="none" stroke="#333333" d="M626.18,-188.76C636.48,-160.84 660.09,-106.94 697.78,-76.64 699.34,-75.38 700.96,-74.19 702.63,-73.05"/>
+<polygon fill="#333333" stroke="#333333" points="704.6,-75.95 711.35,-67.78 700.98,-69.95 704.6,-75.95"/>
+</g>
+<!-- lemma_2_3_beta_sim -->
+<g id="node8" class="node">
+<title>lemma_2_3_beta_sim</title>
+<g id="a_node8"><a xlink:title="lemma_2_3_beta_sim (section 2) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1019.78,-67.64 889.78,-67.64 889.78,-31.64 1019.78,-31.64 1019.78,-67.64"/>
+<text text-anchor="middle" x="954.78" y="-47.14" font-family="Inter" font-size="10.00" fill="#333333">Lemma 2.3 β&#45;simulation</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;lemma_2_3_beta_sim -->
+<g id="edge8" class="edge">
+<title>m1_red1&#45;&gt;lemma_2_3_beta_sim</title>
+<path fill="none" stroke="#333333" d="M621.5,-188.43C625.43,-149.91 640.58,-60.59 697.78,-22.64 760.49,18.97 851.47,-5.84 906.39,-27.7"/>
+<polygon fill="#333333" stroke="#333333" points="905.3,-31.03 915.88,-31.59 907.95,-24.56 905.3,-31.03"/>
+</g>
+<!-- m3_measure -->
+<g id="node9" class="node">
+<title>m3_measure</title>
+<g id="a_node9"><a xlink:title="m3_measure (section A) https://arxiv.org/html/2312.13270v1">
+<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="779.78" cy="-252.64" rx="45.83" ry="18"/>
+<text text-anchor="middle" x="779.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">sub measure</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;m3_measure -->
+<g id="edge10" class="edge">
+<title>m1_red1&#45;&gt;m3_measure</title>
+<path fill="none" stroke="#333333" d="M658.96,-217.89C681.19,-224.34 709.23,-232.47 732.59,-239.24"/>
+<polygon fill="#333333" stroke="#333333" points="731.7,-242.63 742.28,-242.05 733.65,-235.91 731.7,-242.63"/>
+</g>
+<!-- m3_local_confluence -->
+<g id="node11" class="node">
+<title>m3_local_confluence</title>
+<g id="a_node11"><a xlink:title="m3_local_confluence (section 3.1) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1185.78,-270.64 1055.78,-270.64 1055.78,-234.64 1185.78,-234.64 1185.78,-270.64"/>
+<text text-anchor="middle" x="1120.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">{R,Gc} local confluence</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;m3_local_confluence -->
+<g id="edge13" class="edge">
+<title>m1_red1&#45;&gt;m3_local_confluence</title>
+<path fill="none" stroke="#333333" d="M669.82,-205.16C746.98,-203.71 900.09,-204.24 1027.78,-225.64 1037.16,-227.21 1046.94,-229.43 1056.47,-231.92"/>
+<polygon fill="#333333" stroke="#333333" points="1055.72,-235.34 1066.29,-234.6 1057.56,-228.59 1055.72,-235.34"/>
+</g>
+<!-- m4_par -->
+<g id="node13" class="node">
+<title>m4_par</title>
+<g id="a_node13"><a xlink:title="m4_par (section 3.2) https://arxiv.org/html/2312.13270v1">
+<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="1468.07" cy="-222.64" rx="68.07" ry="18"/>
+<text text-anchor="middle" x="1468.07" y="-220.14" font-family="Inter" font-size="10.00" fill="#333333">parallel reduction par</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;m4_par -->
+<g id="edge15" class="edge">
+<title>m1_red1&#45;&gt;m4_par</title>
+<path fill="none" stroke="#333333" d="M664.23,-198.39C727.9,-187.22 849.43,-168.64 953.78,-168.64 953.78,-168.64 953.78,-168.64 1121.78,-168.64 1223.5,-168.64 1340.07,-191.8 1408.42,-207.78"/>
+<polygon fill="#333333" stroke="#333333" points="1408.01,-211.28 1418.55,-210.18 1409.63,-204.47 1408.01,-211.28"/>
+</g>
+<!-- m5_rx -->
+<g id="node17" class="node">
+<title>m5_rx</title>
+<g id="a_node17"><a xlink:title="m5_rx (section 3.3) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1495.07,-351.64 1441.07,-351.64 1441.07,-315.64 1495.07,-315.64 1495.07,-351.64"/>
+<text text-anchor="middle" x="1468.07" y="-331.14" font-family="Inter" font-size="10.00" fill="#333333">RX rule</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;m5_rx -->
+<g id="edge22" class="edge">
+<title>m1_red1&#45;&gt;m5_rx</title>
+<path fill="none" stroke="#333333" d="M628.47,-224.44C640.27,-247.38 664.22,-286.37 697.78,-303.64 799.1,-355.78 839.83,-317.64 953.78,-317.64 953.78,-317.64 953.78,-317.64 1121.78,-317.64 1234.39,-317.64 1367.43,-326.23 1430.45,-330.83"/>
+<polygon fill="#333333" stroke="#333333" points="1430.43,-334.33 1440.66,-331.58 1430.95,-327.35 1430.43,-334.33"/>
+</g>
+<!-- m6_translation -->
+<g id="node21" class="node">
+<title>m6_translation</title>
+<g id="a_node21"><a xlink:title="m6_translation (section 4) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1169.78,-486.64 1071.78,-486.64 1071.78,-450.64 1169.78,-450.64 1169.78,-486.64"/>
+<text text-anchor="middle" x="1120.78" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">translation to λβp</text>
+</a>
+</g>
+</g>
+<!-- m1_red1&#45;&gt;m6_translation -->
+<g id="edge29" class="edge">
+<title>m1_red1&#45;&gt;m6_translation</title>
+<path fill="none" stroke="#333333" d="M626.69,-224.75C637.39,-251.88 661.18,-302.97 697.78,-331.64 808.02,-417.99 974.51,-450.79 1061.76,-462.55"/>
+<polygon fill="#333333" stroke="#333333" points="1061.38,-466.03 1071.75,-463.85 1062.28,-459.08 1061.38,-466.03"/>
+</g>
+<!-- lemma_2_2_full_comp&#45;&gt;lemma_2_3_beta_sim -->
+<g id="edge9" class="edge">
+<title>lemma_2_2_full_comp&#45;&gt;lemma_2_3_beta_sim</title>
+<path fill="none" stroke="#333333" d="M853.9,-49.64C862.36,-49.64 870.99,-49.64 879.46,-49.64"/>
+<polygon fill="#333333" stroke="#333333" points="879.7,-53.14 889.7,-49.64 879.7,-46.14 879.7,-53.14"/>
+</g>
+<!-- theorem_confluence_terms -->
+<g id="node15" class="node">
+<title>theorem_confluence_terms</title>
+<g id="a_node15"><a xlink:title="theorem_confluence_terms (section 3.2) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1848.85,-240.64 1731.85,-240.64 1731.85,-204.64 1848.85,-204.64 1848.85,-240.64"/>
+<text text-anchor="middle" x="1790.35" y="-220.14" font-family="Inter" font-size="10.00" fill="#333333">Confluence on terms</text>
+</a>
+</g>
+</g>
+<!-- lemma_2_3_beta_sim&#45;&gt;theorem_confluence_terms -->
+<g id="edge19" class="edge">
+<title>lemma_2_3_beta_sim&#45;&gt;theorem_confluence_terms</title>
+<path fill="none" stroke="#333333" d="M1019.91,-62.16C1087.32,-74.22 1196.58,-90.64 1291.78,-90.64 1291.78,-90.64 1291.78,-90.64 1469.07,-90.64 1581.25,-90.64 1699.98,-160.71 1756.17,-198.75"/>
+<polygon fill="#333333" stroke="#333333" points="1754.42,-201.79 1764.64,-204.56 1758.38,-196.02 1754.42,-201.79"/>
+</g>
+<!-- m3_termination -->
+<g id="node10" class="node">
+<title>m3_termination</title>
+<g id="a_node10"><a xlink:title="m3_termination (section 3.1) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1006.78,-270.64 902.78,-270.64 902.78,-234.64 1006.78,-234.64 1006.78,-270.64"/>
+<text text-anchor="middle" x="954.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">{R,Gc} termination</text>
+</a>
+</g>
+</g>
+<!-- m3_measure&#45;&gt;m3_termination -->
+<g id="edge11" class="edge">
+<title>m3_measure&#45;&gt;m3_termination</title>
+<path fill="none" stroke="#333333" d="M825.78,-252.64C846.09,-252.64 870.41,-252.64 892.36,-252.64"/>
+<polygon fill="#333333" stroke="#333333" points="892.59,-256.14 902.58,-252.64 892.58,-249.14 892.59,-256.14"/>
+</g>
+<!-- m3_termination&#45;&gt;m3_local_confluence -->
+<g id="edge12" class="edge">
+<title>m3_termination&#45;&gt;m3_local_confluence</title>
+<path fill="none" stroke="#333333" d="M1007.22,-252.64C1019.3,-252.64 1032.42,-252.64 1045.29,-252.64"/>
+<polygon fill="#333333" stroke="#333333" points="1045.53,-256.14 1055.53,-252.64 1045.53,-249.14 1045.53,-256.14"/>
+</g>
+<!-- lemma_3_2_sub_nf_unique -->
+<g id="node12" class="node">
+<title>lemma_3_2_sub_nf_unique</title>
+<g id="a_node12"><a xlink:title="lemma_3_2_sub_nf_unique (section 3.1) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1363.78,-270.64 1221.78,-270.64 1221.78,-234.64 1363.78,-234.64 1363.78,-270.64"/>
+<text text-anchor="middle" x="1292.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">Lemma 3.2 sub NF unique</text>
+</a>
+</g>
+</g>
+<!-- m3_local_confluence&#45;&gt;lemma_3_2_sub_nf_unique -->
+<g id="edge14" class="edge">
+<title>m3_local_confluence&#45;&gt;lemma_3_2_sub_nf_unique</title>
+<path fill="none" stroke="#333333" d="M1186.05,-252.64C1194.43,-252.64 1203.1,-252.64 1211.7,-252.64"/>
+<polygon fill="#333333" stroke="#333333" points="1211.74,-256.14 1221.74,-252.64 1211.74,-249.14 1211.74,-256.14"/>
+</g>
+<!-- lemma_3_2_sub_nf_unique&#45;&gt;m4_par -->
+<g id="edge16" class="edge">
+<title>lemma_3_2_sub_nf_unique&#45;&gt;m4_par</title>
+<path fill="none" stroke="#333333" d="M1364.26,-240.44C1376.15,-238.38 1388.46,-236.25 1400.24,-234.21"/>
+<polygon fill="#333333" stroke="#333333" points="1401.19,-237.6 1410.45,-232.44 1400,-230.7 1401.19,-237.6"/>
+</g>
+<!-- m5_coherence -->
+<g id="node19" class="node">
+<title>m5_coherence</title>
+<g id="a_node19"><a xlink:title="m5_coherence (section 3.3) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1684.35,-351.64 1572.35,-351.64 1572.35,-315.64 1684.35,-315.64 1684.35,-351.64"/>
+<text text-anchor="middle" x="1628.35" y="-331.14" font-family="Inter" font-size="10.00" fill="#333333">RX + =C coherence</text>
+</a>
+</g>
+</g>
+<!-- lemma_3_2_sub_nf_unique&#45;&gt;m5_coherence -->
+<g id="edge26" class="edge">
+<title>lemma_3_2_sub_nf_unique&#45;&gt;m5_coherence</title>
+<path fill="none" stroke="#333333" d="M1363.89,-267.68C1423.47,-280.57 1504.3,-298.33 1536.35,-306.64 1544.81,-308.83 1553.67,-311.28 1562.41,-313.77"/>
+<polygon fill="#333333" stroke="#333333" points="1561.6,-317.18 1572.18,-316.6 1563.55,-310.46 1561.6,-317.18"/>
+</g>
+<!-- m4_par_diamond -->
+<g id="node14" class="node">
+<title>m4_par_diamond</title>
+<g id="a_node14"><a xlink:title="m4_par_diamond (section 3.2) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1665.85,-240.64 1590.85,-240.64 1590.85,-204.64 1665.85,-204.64 1665.85,-240.64"/>
+<text text-anchor="middle" x="1628.35" y="-220.14" font-family="Inter" font-size="10.00" fill="#333333">par diamond</text>
+</a>
+</g>
+</g>
+<!-- m4_par&#45;&gt;m4_par_diamond -->
+<g id="edge17" class="edge">
+<title>m4_par&#45;&gt;m4_par_diamond</title>
+<path fill="none" stroke="#333333" d="M1536.65,-222.64C1551.38,-222.64 1566.66,-222.64 1580.38,-222.64"/>
+<polygon fill="#333333" stroke="#333333" points="1580.54,-226.14 1590.54,-222.64 1580.54,-219.14 1580.54,-226.14"/>
+</g>
+<!-- m4_par_diamond&#45;&gt;theorem_confluence_terms -->
+<g id="edge18" class="edge">
+<title>m4_par_diamond&#45;&gt;theorem_confluence_terms</title>
+<path fill="none" stroke="#333333" d="M1666.06,-222.64C1682.56,-222.64 1702.58,-222.64 1721.64,-222.64"/>
+<polygon fill="#333333" stroke="#333333" points="1721.8,-226.14 1731.8,-222.64 1721.8,-219.14 1721.8,-226.14"/>
+</g>
+<!-- theorem_confluence_metaterms -->
+<g id="node20" class="node">
+<title>theorem_confluence_metaterms</title>
+<g id="a_node20"><a xlink:title="theorem_confluence_metaterms (section 3.3) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1860.35,-351.64 1720.35,-351.64 1720.35,-315.64 1860.35,-315.64 1860.35,-351.64"/>
+<text text-anchor="middle" x="1790.35" y="-331.14" font-family="Inter" font-size="10.00" fill="#333333">Confluence on metaterms</text>
+</a>
+</g>
+</g>
+<!-- m4_par_diamond&#45;&gt;theorem_confluence_metaterms -->
+<g id="edge28" class="edge">
+<title>m4_par_diamond&#45;&gt;theorem_confluence_metaterms</title>
+<path fill="none" stroke="#333333" d="M1655.41,-240.72C1682.49,-259.51 1725.01,-289 1754.86,-309.71"/>
+<polygon fill="#333333" stroke="#333333" points="1753.09,-312.74 1763.3,-315.57 1757.08,-306.99 1753.09,-312.74"/>
+</g>
+<!-- corollary_6_21_psn -->
+<g id="node26" class="node">
+<title>corollary_6_21_psn</title>
+<g id="a_node26"><a xlink:title="corollary_6_21_psn (section 6) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1976.35,-486.64 1896.35,-486.64 1896.35,-450.64 1976.35,-450.64 1976.35,-486.64"/>
+<text text-anchor="middle" x="1936.35" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">Cor 6.21 PSN</text>
+</a>
+</g>
+</g>
+<!-- theorem_confluence_terms&#45;&gt;corollary_6_21_psn -->
+<g id="edge35" class="edge">
+<title>theorem_confluence_terms&#45;&gt;corollary_6_21_psn</title>
+<path fill="none" stroke="#333333" d="M1812.59,-240.84C1829.48,-256.21 1852.79,-279.51 1868.35,-303.64 1896.91,-347.9 1917.13,-406.33 1927.46,-440.47"/>
+<polygon fill="#333333" stroke="#333333" points="1924.16,-441.64 1930.35,-450.24 1930.87,-439.65 1924.16,-441.64"/>
+</g>
+<!-- m5_mvar&#45;&gt;m5_rx -->
+<g id="edge21" class="edge">
+<title>m5_mvar&#45;&gt;m5_rx</title>
+<path fill="none" stroke="#333333" d="M1337.51,-356.07C1366.4,-351.07 1403.82,-344.59 1431,-339.88"/>
+<polygon fill="#333333" stroke="#333333" points="1431.63,-343.33 1440.88,-338.17 1430.43,-336.43 1431.63,-343.33"/>
+</g>
+<!-- m5_eq_C -->
+<g id="node18" class="node">
+<title>m5_eq_C</title>
+<g id="a_node18"><a xlink:title="m5_eq_C (section 3.3) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1499.07,-405.64 1437.07,-405.64 1437.07,-369.64 1499.07,-369.64 1499.07,-405.64"/>
+<text text-anchor="middle" x="1468.07" y="-385.14" font-family="Inter" font-size="10.00" fill="#333333">=C setoid</text>
+</a>
+</g>
+</g>
+<!-- m5_mvar&#45;&gt;m5_eq_C -->
+<g id="edge23" class="edge">
+<title>m5_mvar&#45;&gt;m5_eq_C</title>
+<path fill="none" stroke="#333333" d="M1338.86,-369.88C1366.13,-373.66 1400.59,-378.43 1426.92,-382.08"/>
+<polygon fill="#333333" stroke="#333333" points="1426.55,-385.56 1436.94,-383.47 1427.51,-378.63 1426.55,-385.56"/>
+</g>
+<!-- m5_rx&#45;&gt;m5_coherence -->
+<g id="edge24" class="edge">
+<title>m5_rx&#45;&gt;m5_coherence</title>
+<path fill="none" stroke="#333333" d="M1495.2,-333.64C1513.41,-333.64 1538.54,-333.64 1562,-333.64"/>
+<polygon fill="#333333" stroke="#333333" points="1562.28,-337.14 1572.28,-333.64 1562.28,-330.14 1562.28,-337.14"/>
+</g>
+<!-- m5_eq_C&#45;&gt;m5_coherence -->
+<g id="edge25" class="edge">
+<title>m5_eq_C&#45;&gt;m5_coherence</title>
+<path fill="none" stroke="#333333" d="M1499.21,-377.35C1517.81,-371.01 1542.37,-362.63 1565,-354.91"/>
+<polygon fill="#333333" stroke="#333333" points="1566.23,-358.19 1574.56,-351.65 1563.97,-351.57 1566.23,-358.19"/>
+</g>
+<!-- m5_coherence&#45;&gt;theorem_confluence_metaterms -->
+<g id="edge27" class="edge">
+<title>m5_coherence&#45;&gt;theorem_confluence_metaterms</title>
+<path fill="none" stroke="#333333" d="M1684.41,-333.64C1692.73,-333.64 1701.45,-333.64 1710.15,-333.64"/>
+<polygon fill="#333333" stroke="#333333" points="1710.34,-337.14 1720.34,-333.64 1710.34,-330.14 1710.34,-337.14"/>
+</g>
+<!-- m6_sn -->
+<g id="node22" class="node">
+<title>m6_sn</title>
+<g id="a_node22"><a xlink:title="m6_sn (section 6) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1347.78,-486.64 1237.78,-486.64 1237.78,-450.64 1347.78,-450.64 1347.78,-486.64"/>
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+</a>
+</g>
+</g>
+<!-- m6_translation&#45;&gt;m6_sn -->
+<g id="edge30" class="edge">
+<title>m6_translation&#45;&gt;m6_sn</title>
+<path fill="none" stroke="#333333" d="M1170.04,-468.64C1187.93,-468.64 1208.51,-468.64 1227.6,-468.64"/>
+<polygon fill="#333333" stroke="#333333" points="1227.72,-472.14 1237.72,-468.64 1227.72,-465.14 1227.72,-472.14"/>
+</g>
+<!-- m6_intersection -->
+<g id="node23" class="node">
+<title>m6_intersection</title>
+<g id="a_node23"><a xlink:title="m6_intersection (section 6) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1520.07,-486.64 1416.07,-486.64 1416.07,-450.64 1520.07,-450.64 1520.07,-486.64"/>
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+</a>
+</g>
+</g>
+<!-- m6_sn&#45;&gt;m6_intersection -->
+<g id="edge31" class="edge">
+<title>m6_sn&#45;&gt;m6_intersection</title>
+<path fill="none" stroke="#333333" d="M1348.14,-468.64C1366.39,-468.64 1386.88,-468.64 1405.64,-468.64"/>
+<polygon fill="#333333" stroke="#333333" points="1405.86,-472.14 1415.86,-468.64 1405.86,-465.14 1405.86,-472.14"/>
+</g>
+<!-- m6_typable_sn -->
+<g id="node24" class="node">
+<title>m6_typable_sn</title>
+<g id="a_node24"><a xlink:title="m6_typable_sn (section 6) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1668.85,-486.64 1587.85,-486.64 1587.85,-450.64 1668.85,-450.64 1668.85,-486.64"/>
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+</a>
+</g>
+</g>
+<!-- m6_intersection&#45;&gt;m6_typable_sn -->
+<g id="edge32" class="edge">
+<title>m6_intersection&#45;&gt;m6_typable_sn</title>
+<path fill="none" stroke="#333333" d="M1520.46,-468.64C1538.8,-468.64 1559.43,-468.64 1577.63,-468.64"/>
+<polygon fill="#333333" stroke="#333333" points="1577.75,-472.14 1587.75,-468.64 1577.75,-465.14 1577.75,-472.14"/>
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+<!-- m6_sn_typable -->
+<g id="node25" class="node">
+<title>m6_sn_typable</title>
+<g id="a_node25"><a xlink:title="m6_sn_typable (section 6) https://arxiv.org/html/2312.13270v1">
+<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1830.85,-486.64 1749.85,-486.64 1749.85,-450.64 1830.85,-450.64 1830.85,-486.64"/>
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+</a>
+</g>
+</g>
+<!-- m6_typable_sn&#45;&gt;m6_sn_typable -->
+<g id="edge33" class="edge">
+<title>m6_typable_sn&#45;&gt;m6_sn_typable</title>
+<path fill="none" stroke="#333333" d="M1668.9,-468.64C1690.25,-468.64 1716.81,-468.64 1739.55,-468.64"/>
+<polygon fill="#333333" stroke="#333333" points="1739.7,-472.14 1749.7,-468.64 1739.7,-465.14 1739.7,-472.14"/>
+</g>
+<!-- m6_sn_typable&#45;&gt;corollary_6_21_psn -->
+<g id="edge34" class="edge">
+<title>m6_sn_typable&#45;&gt;corollary_6_21_psn</title>
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+</g>
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diff --git a/graphs/reduction.dot b/graphs/reduction.dot
new file mode 100644
index 0000000..2115012
--- /dev/null
+++ b/graphs/reduction.dot
@@ -0,0 +1,20 @@
+digraph reduction_dup {
+  rankdir=LR;
+  bgcolor="white";
+  node [fontname=Inter, fontsize=10];
+  edge [fontname=Inter, fontsize=9, color="#333333"];
+  graph [labelloc=t, fontcolor="#333333", 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"];
+  n1 [label="(x x)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n2 [label="(y x)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n2b [label="(x y)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  n3 [label="(y y)[x/y]", shape=box, style="filled", fillcolor="#f5f5f5", color="#333333", fontcolor="#333333"];
+  nf [label="y y (normal form)", shape=doublecircle, style=filled, fillcolor="#d9ead3", color="#38761d", fontcolor="#333333"];
+  n0 -> n1 [label="B", color="#990000"];
+  n1 -> n2 [label="R", color="#674ea7"];
+  n1 -> n2b [label="R", color="#674ea7"];
+  n2 -> n3 [label="R", color="#674ea7"];
+  n2b -> n3 [label="R", color="#674ea7"];
+  n3 -> nf [label="Gc", color="#38761d"];
+  {rank=same; n2; n2b;}
+}
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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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+<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>
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+</g>
+<!-- n0&#45;&gt;n1 -->
+<g id="edge1" class="edge">
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+<g id="edge3" class="edge">
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diff --git a/graphs/rules.dot b/graphs/rules.dot
new file mode 100644
index 0000000..c0054d6
--- /dev/null
+++ b/graphs/rules.dot
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+  start -> es [label="ESub"];
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diff --git a/graphs/theorems.json b/graphs/theorems.json
new file mode 100644
index 0000000..f36a280
--- /dev/null
+++ b/graphs/theorems.json
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+{
+  "paper": {
+    "title": "Milner's Lambda-Calculus with Partial Substitutions",
+    "authors": ["Delia Kesner", "Shane Ó Conchúir"],
+    "url": "https://arxiv.org/abs/2312.13270",
+    "html": "https://arxiv.org/html/2312.13270v1"
+  },
+  "milestones": ["M0", "M1", "M2", "M3", "M4", "M5", "M6"],
+  "current": "M0",
+  "nodes": [
+    {
+      "id": "exec_step",
+      "label": "exec_step",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/ExecReducer.v",
+      "depends": []
+    },
+    {
+      "id": "exec_examples",
+      "label": "exec_examples",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/ExecReducer.v",
+      "depends": ["exec_step"]
+    },
+    {
+      "id": "m1_syntax",
+      "label": "M1 syntax/binding",
+      "kind": "infra",
+      "status": "planned",
+      "milestone": "M1",
+      "section": "2",
+      "file": "",
+      "depends": ["exec_examples"]
+    },
+    {
+      "id": "m1_open_close",
+      "label": "open/close laws",
+      "kind": "infra",
+      "status": "planned",
+      "milestone": "M1",
+      "section": "2",
+      "file": "",
+      "depends": ["m1_syntax"]
+    },
+    {
+      "id": "m1_red1",
+      "label": "red1 (B, Gc, R)",
+      "kind": "infra",
+      "status": "planned",
+      "milestone": "M1",
+      "section": "2",
+      "file": "",
+      "depends": ["m1_syntax", "m1_open_close"]
+    },
+    {
+      "id": "lemma_2_1_fv_preserved",
+      "label": "Lemma 2.1 fv preserved",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M2",
+      "section": "2",
+      "file": "",
+      "depends": ["m1_red1"]
+    },
+    {
+      "id": "lemma_2_2_full_comp",
+      "label": "Lemma 2.2 full composition",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M2",
+      "section": "2",
+      "file": "",
+      "depends": ["m1_red1"]
+    },
+    {
+      "id": "lemma_2_3_beta_sim",
+      "label": "Lemma 2.3 β-simulation",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M2",
+      "section": "2",
+      "file": "",
+      "depends": ["m1_red1", "lemma_2_2_full_comp"]
+    },
+    {
+      "id": "m3_measure",
+      "label": "sub measure",
+      "kind": "infra",
+      "status": "planned",
+      "milestone": "M3",
+      "section": "A",
+      "file": "",
+      "depends": ["m1_red1"]
+    },
+    {
+      "id": "m3_termination",
+      "label": "{R,Gc} termination",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M3",
+      "section": "3.1",
+      "file": "",
+      "depends": ["m3_measure"]
+    },
+    {
+      "id": "m3_local_confluence",
+      "label": "{R,Gc} local confluence",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M3",
+      "section": "3.1",
+      "file": "",
+      "depends": ["m3_termination", "m1_red1"]
+    },
+    {
+      "id": "lemma_3_2_sub_nf_unique",
+      "label": "Lemma 3.2 sub NF unique",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M3",
+      "section": "3.1",
+      "file": "",
+      "depends": ["m3_local_confluence"]
+    },
+    {
+      "id": "m4_par",
+      "label": "parallel reduction par",
+      "kind": "infra",
+      "status": "planned",
+      "milestone": "M4",
+      "section": "3.2",
+      "file": "",
+      "depends": ["m1_red1", "lemma_3_2_sub_nf_unique"]
+    },
+    {
+      "id": "m4_par_diamond",
+      "label": "par diamond",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M4",
+      "section": "3.2",
+      "file": "",
+      "depends": ["m4_par"]
+    },
+    {
+      "id": "theorem_confluence_terms",
+      "label": "Confluence on terms",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M4",
+      "section": "3.2",
+      "file": "",
+      "depends": ["m4_par_diamond", "lemma_2_3_beta_sim"]
+    },
+    {
+      "id": "m5_mvar",
+      "label": "XΔ metaterms",
+      "kind": "infra",
+      "status": "planned",
+      "milestone": "M5",
+      "section": "3.3",
+      "file": "",
+      "depends": ["m1_syntax"]
+    },
+    {
+      "id": "m5_rx",
+      "label": "RX rule",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M5",
+      "section": "3.3",
+      "file": "",
+      "depends": ["m5_mvar", "m1_red1"]
+    },
+    {
+      "id": "m5_eq_C",
+      "label": "=C setoid",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M5",
+      "section": "3.3",
+      "file": "",
+      "depends": ["m5_mvar"]
+    },
+    {
+      "id": "m5_coherence",
+      "label": "RX + =C coherence",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M5",
+      "section": "3.3",
+      "file": "",
+      "depends": ["m5_rx", "m5_eq_C", "lemma_3_2_sub_nf_unique"]
+    },
+    {
+      "id": "theorem_confluence_metaterms",
+      "label": "Confluence on metaterms",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M5",
+      "section": "3.3",
+      "file": "",
+      "depends": ["m5_coherence", "m4_par_diamond"]
+    },
+    {
+      "id": "m6_translation",
+      "label": "translation to λβp",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M6",
+      "section": "4",
+      "file": "",
+      "depends": ["m1_red1"]
+    },
+    {
+      "id": "m6_sn",
+      "label": "SN for simply typed",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M6",
+      "section": "6",
+      "file": "",
+      "depends": ["m6_translation"]
+    },
+    {
+      "id": "m6_intersection",
+      "label": "intersection typing",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M6",
+      "section": "6",
+      "file": "",
+      "depends": ["m6_sn"]
+    },
+    {
+      "id": "m6_typable_sn",
+      "label": "typable → SN",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M6",
+      "section": "6",
+      "file": "",
+      "depends": ["m6_intersection"]
+    },
+    {
+      "id": "m6_sn_typable",
+      "label": "SN → typable",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M6",
+      "section": "6",
+      "file": "",
+      "depends": ["m6_typable_sn"]
+    },
+    {
+      "id": "corollary_6_21_psn",
+      "label": "Cor 6.21 PSN",
+      "kind": "paper",
+      "status": "planned",
+      "milestone": "M6",
+      "section": "6",
+      "file": "",
+      "depends": ["m6_sn_typable", "theorem_confluence_terms"]
+    }
+  ]
+}
diff --git a/scripts/audit.sh b/scripts/audit.sh
new file mode 100755
index 0000000..2c0c386
--- /dev/null
+++ b/scripts/audit.sh
@@ -0,0 +1,52 @@
+#!/usr/bin/env bash
+set -euo pipefail
+ROOT="$(cd "$(dirname "$0")/.." && pwd)"
+cd "$ROOT"
+
+if ! command -v rocq >/dev/null 2>&1; then
+  if command -v opam >/dev/null 2>&1; then
+    eval "$(opam env --switch=rocq-system --set-switch 2>/dev/null || true)"
+  fi
+fi
+
+if ! command -v rocq >/dev/null 2>&1; then
+  echo "rocq not on PATH; run: eval \$(opam env --switch=rocq-system --set-switch)" >&2
+  exit 1
+fi
+
+fail=0
+
+echo "== forbidden tokens =="
+while IFS= read -r -d '' f; do
+  if grep -nE '\b(Admitted|admit|Axiom|Conjecture|native_compute)\b|TODO|FIXME|XXX' "$f"; then
+    echo "forbidden token in $f" >&2
+    fail=1
+  fi
+done < <(find theory -name '*.v' -print0 2>/dev/null)
+
+echo "== forbidden Parameter =="
+while IFS= read -r -d '' f; do
+  if grep -nE '^[[:space:]]*Parameter[[:space:]]' "$f"; then
+    echo "forbidden Parameter in $f" >&2
+    fail=1
+  fi
+done < <(find theory -name '*.v' -print0 2>/dev/null)
+
+echo "== Print Assumptions =="
+mkdir -p /tmp/opencode/audit
+out="$(rocq compile -o /tmp/opencode/audit/ExecReducer.vo theory/ExecReducer.v 2>&1 || true)"
+echo "$out"
+if echo "$out" | grep -q "Axioms:"; then
+  echo "unexpected axioms reported" >&2
+  fail=1
+fi
+if ! echo "$out" | grep -q "Closed under the global context"; then
+  echo "no Closed-under-global-context report found" >&2
+  fail=1
+fi
+
+if [ "$fail" -ne 0 ]; then
+  echo "AUDIT FAIL"
+  exit 1
+fi
+echo "AUDIT OK"
diff --git a/scripts/gen_graphs.py b/scripts/gen_graphs.py
new file mode 100755
index 0000000..a572dd9
--- /dev/null
+++ b/scripts/gen_graphs.py
@@ -0,0 +1,180 @@
+#!/usr/bin/env python3
+import json
+import os
+import subprocess
+import sys
+
+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 = {
+    "proved": "#d9ead3",
+    "stated": "#fff2cc",
+    "planned": "#f3f3f3",
+    "blocked": "#f4cccc",
+}
+STATUS_PEN = {
+    "proved": "#38761d",
+    "stated": "#b45f06",
+    "planned": "#777777",
+    "blocked": "#cc0000",
+}
+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):
+    return '"' + str(s).replace("\\", "\\\\").replace('"', '\\"') + '"'
+
+
+def load():
+    with open(META, "r", encoding="utf-8") as f:
+        return json.load(f)
+
+
+def dependency_dot(data):
+    nodes = data["nodes"]
+    by_id = {n["id"]: n for n in nodes}
+    title = data["paper"]["title"] + " - theorem dependency (" + data["current"] + ")"
+    lines = [
+        "digraph theorem_deps {",
+        "  rankdir=LR;",
+        "  bgcolor=\"white\";",
+        "  node [fontname=Inter, fontsize=10, fontcolor=\"%s\"];" % INK,
+        "  edge [fontname=Inter, fontsize=8, color=\"%s\"];" % INK,
+        "  graph [fontname=Inter, fontsize=12, labelloc=t, label=%s];" % q(title),
+    ]
+    for ms in data["milestones"]:
+        members = [n for n in nodes if n["milestone"] == ms]
+        if not members:
+            continue
+        lines.append("  subgraph cluster_%s {" % ms)
+        lines.append("    label=%s;" % q(ms))
+        lines.append("    style=\"rounded\";")
+        lines.append("    color=\"#bbbbbb\";")
+        lines.append("    fontcolor=\"%s\";" % INK)
+        for n in members:
+            fill = STATUS_FILL.get(n["status"], "#f3f3f3")
+            pen = STATUS_PEN.get(n["status"], "#777777")
+            style = STATUS_STYLE.get(n["status"], "dashed")
+            tip = n["id"]
+            if n.get("section"):
+                tip += " (section " + n["section"] + ")"
+            if n.get("file"):
+                tip += " " + n["file"]
+            tip += " " + data["paper"]["html"]
+            attrs = [
+                "label=%s" % q(n["label"]),
+                "color=%s" % q(pen),
+                "fillcolor=%s" % q(fill),
+                "style=%s" % q(style),
+                "fontcolor=%s" % q(INK),
+                "tooltip=%s" % q(tip),
+                "shape=%s" % ("box" if n["kind"] == "paper" else "ellipse"),
+            ]
+            lines.append("    %s [%s];" % (n["id"], ", ".join(attrs)))
+        lines.append("  }")
+    for n in nodes:
+        for d in n.get("depends", []):
+            if d in by_id:
+                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)
+    lines = [
+        "digraph lambda_sub_cfg {",
+        "  rankdir=TB;",
+        "  bgcolor=\"white\";",
+        "  node [fontname=Inter, fontsize=10];",
+        "  edge [fontname=Inter, fontsize=9, color=\"%s\"];" % INK,
+        "  start [label=\"term\", shape=oval, style=filled, fillcolor=\"#f3f3f3\", color=\"%s\"];" % 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,
+        "  nf [label=\"normal form\", shape=doublecircle, style=filled, fillcolor=\"#d9ead3\", color=\"%s\"];" % RULE_G,
+        "  start -> beta [label=\"App(Lam,_)\"];",
+        "  start -> es [label=\"ESub\"];",
+        "  start -> pure [label=\"BVar/FVar/Lam\"];",
+        "  beta -> es [label=\"B\", color=\"%s\"];" % RULE_B,
+        "  es -> es [label=\"R (one occurrence)\", color=\"%s\"];" % RULE_R,
+        "  es -> pure [label=\"Gc (x not in fv)\", color=\"%s\"];" % RULE_G,
+        "  pure -> pure [label=\"B under context\", color=\"%s\"];" % RULE_B,
+        "  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)
+    lines = [
+        "digraph reduction_dup {",
+        "  rankdir=LR;",
+        "  bgcolor=\"white\";",
+        "  node [fontname=Inter, fontsize=10];",
+        "  edge [fontname=Inter, fontsize=9, color=\"%s\"];" % INK,
+        "  graph [labelloc=t, fontcolor=\"%s\", label=\"(lambda x. x x) y reduction graph\"];" % INK,
+        "  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),
+        "  n0 -> n1 [label=\"B\", color=\"%s\"];" % RULE_B,
+        "  n1 -> n2 [label=\"R\", color=\"%s\"];" % RULE_R,
+        "  n1 -> n2b [label=\"R\", color=\"%s\"];" % RULE_R,
+        "  n2 -> n3 [label=\"R\", color=\"%s\"];" % RULE_R,
+        "  n2b -> n3 [label=\"R\", color=\"%s\"];" % RULE_R,
+        "  n3 -> nf [label=\"Gc\", color=\"%s\"];" % RULE_G,
+        "  {rank=same; n2; n2b;}",
+        "}",
+    ]
+    return "\n".join(lines) + "\n"
+
+
+def run_dot(path):
+    svg = path[:-4] + ".svg"
+    png = path[:-4] + ".png"
+    ok = True
+    for fmt, out in (("svg", svg), ("png", png)):
+        r = subprocess.run(["dot", "-T" + fmt, path, "-o", out], capture_output=True, text=True)
+        if r.returncode != 0:
+            print(r.stderr, file=sys.stderr)
+            ok = False
+        else:
+            print("wrote", os.path.relpath(out, ROOT))
+    return ok
+
+
+def main():
+    data = load()
+    outputs = {
+        "dependency.dot": dependency_dot(data),
+        "rules.dot": rules_dot(),
+        "reduction.dot": reduction_dot(),
+    }
+    ok = True
+    for name, text in outputs.items():
+        p = os.path.join(GRAPH_DIR, name)
+        with open(p, "w", encoding="utf-8") as f:
+            f.write(text)
+        print("wrote", os.path.relpath(p, ROOT))
+        if not run_dot(p):
+            ok = False
+    return 0 if ok else 1
+
+
+if __name__ == "__main__":
+    sys.exit(main())
diff --git a/theory/ExecReducer.v b/theory/ExecReducer.v
new file mode 100644
index 0000000..01cc7cc
--- /dev/null
+++ b/theory/ExecReducer.v
@@ -0,0 +1,272 @@
+From Stdlib Require Import List Bool PeanoNat Lia.
+Import ListNotations.
+
+Definition atom := nat.
+
+Inductive trm : Type :=
+| BVar : nat -> trm
+| FVar : atom -> trm
+| App : trm -> trm -> trm
+| Lam : trm -> trm
+| ESub : trm -> trm -> trm.
+
+Fixpoint fvs (t : trm) : list atom :=
+  match t with
+  | BVar _ => []
+  | FVar x => [x]
+  | App t1 t2 => fvs t1 ++ fvs t2
+  | Lam t1 => fvs t1
+  | ESub t1 t2 => fvs t1 ++ fvs t2
+  end.
+
+Definition in_fvs (x : atom) (t : trm) : bool :=
+  existsb (Nat.eqb x) (fvs t).
+
+Fixpoint lift (k : nat) (t : trm) : trm :=
+  match t with
+  | BVar n => if Nat.ltb n k then BVar n else BVar (S n)
+  | FVar x => FVar x
+  | App t1 t2 => App (lift k t1) (lift k t2)
+  | Lam t1 => Lam (lift (S k) t1)
+  | ESub t1 t2 => ESub (lift (S k) t1) (lift k t2)
+  end.
+
+Fixpoint open_rec (k : nat) (u : trm) (t : trm) : trm :=
+  match t with
+  | BVar n => if Nat.eqb n k then lift k u else BVar n
+  | FVar x => FVar x
+  | App t1 t2 => App (open_rec k u t1) (open_rec k u t2)
+  | Lam t1 => Lam (open_rec (S k) u t1)
+  | ESub t1 t2 => ESub (open_rec (S k) u t1) (open_rec k u t2)
+  end.
+
+Fixpoint close_rec (x : atom) (k : nat) (t : trm) : trm :=
+  match t with
+  | BVar n => BVar n
+  | FVar y => if Nat.eqb y x then BVar k else FVar y
+  | App t1 t2 => App (close_rec x k t1) (close_rec x k t2)
+  | Lam t1 => Lam (close_rec x (S k) t1)
+  | ESub t1 t2 => ESub (close_rec x (S k) t1) (close_rec x k t2)
+  end.
+
+Definition subst (x : atom) (u t : trm) : trm :=
+  open_rec 0 u (close_rec x 0 t).
+
+Fixpoint occurs (n : nat) (t : trm) : bool :=
+  match t with
+  | BVar m => Nat.eqb m n
+  | FVar _ => false
+  | App t1 t2 => occurs n t1 || occurs n t2
+  | Lam t1 => occurs (S n) t1
+  | ESub t1 t2 => occurs (S n) t1 || occurs n t2
+  end.
+
+Definition occurs0 (t : trm) : bool := occurs 0 t.
+
+Inductive rule : Type := RB | RGc | RR.
+
+Definition rule_eqb (r s : rule) : bool :=
+  match r, s with
+  | RB, RB => true
+  | RGc, RGc => true
+  | RR, RR => true
+  | _, _ => false
+  end.
+
+Inductive zctx : Type :=
+| ZTop : zctx
+| ZAppL : zctx -> trm -> zctx
+| ZAppR : trm -> zctx -> zctx
+| ZLam : zctx -> zctx
+| ZESubL : zctx -> trm -> zctx
+| ZESubR : trm -> zctx -> zctx.
+
+Fixpoint zplug (C : zctx) (k : nat) (t : trm) : trm :=
+  match C with
+  | ZTop => t
+  | ZAppL C1 u => App (zplug C1 k t) u
+  | ZAppR u C1 => App u (zplug C1 k t)
+  | ZLam C1 => Lam (zplug C1 (S k) t)
+  | ZESubL C1 u => ESub (zplug C1 (S k) t) u
+  | ZESubR t1 C1 => ESub t1 (zplug C1 k t)
+  end.
+
+Fixpoint zplug_lift (C : zctx) (k : nat) (t : trm) : trm :=
+  match C with
+  | ZTop => lift k t
+  | ZAppL C1 u => App (zplug_lift C1 k t) u
+  | ZAppR u C1 => App u (zplug_lift C1 k t)
+  | ZLam C1 => Lam (zplug_lift C1 (S k) t)
+  | ZESubL C1 u => ESub (zplug_lift C1 (S k) t) u
+  | ZESubR t1 C1 => ESub t1 (zplug_lift C1 k t)
+  end.
+
+Fixpoint zdecs (t : trm) (k : nat) : list zctx :=
+  match t with
+  | BVar n => if Nat.eqb n k then [ZTop] else []
+  | FVar _ => []
+  | App t1 t2 =>
+      map (fun C => ZAppL C t2) (zdecs t1 k)
+      ++ map (fun C => ZAppR t1 C) (zdecs t2 k)
+  | Lam t1 => map ZLam (zdecs t1 (S k))
+  | ESub t1 t2 =>
+      map (fun C => ZESubL C t2) (zdecs t1 (S k))
+      ++ map (fun C => ZESubR t1 C) (zdecs t2 k)
+  end.
+
+Definition root_steps (t : trm) : list (rule * trm) :=
+  match t with
+  | App (Lam body) u => [(RB, ESub body u)]
+  | ESub body u =>
+      (if occurs0 body then [] else [(RGc, body)])
+      ++ map (fun C => (RR, ESub (zplug_lift C 0 u) u)) (zdecs body 0)
+  | _ => []
+  end.
+
+Inductive ctx : Type :=
+| GHole : ctx
+| GAppL : ctx -> trm -> ctx
+| GAppR : trm -> ctx -> ctx
+| GLam : ctx -> ctx
+| GESubL : ctx -> trm -> ctx
+| GESubR : trm -> ctx -> ctx.
+
+Fixpoint plug (C : ctx) (t : trm) : trm :=
+  match C with
+  | GHole => t
+  | GAppL C1 u => App (plug C1 t) u
+  | GAppR u C1 => App u (plug C1 t)
+  | GLam C1 => Lam (plug C1 t)
+  | GESubL C1 u => ESub (plug C1 t) u
+  | GESubR t1 C1 => ESub t1 (plug C1 t)
+  end.
+
+Fixpoint positions (t : trm) : list ctx :=
+  GHole ::
+  match t with
+  | BVar _ => []
+  | FVar _ => []
+  | App t1 t2 =>
+      map (fun C => GAppL C t2) (positions t1)
+      ++ map (fun C => GAppR t1 C) (positions t2)
+  | Lam t1 => map GLam (positions t1)
+  | ESub t1 t2 =>
+      map (fun C => GESubL C t2) (positions t1)
+      ++ map (fun C => GESubR t1 C) (positions t2)
+  end.
+
+Fixpoint at_ctx (C : ctx) (t : trm) : option trm :=
+  match C, t with
+  | GHole, t => Some t
+  | GAppL C1 _, App t1 _ => at_ctx C1 t1
+  | GAppR _ C2, App _ t2 => at_ctx C2 t2
+  | GLam C1, Lam t1 => at_ctx C1 t1
+  | GESubL C1 _, ESub t1 _ => at_ctx C1 t1
+  | GESubR _ C2, ESub _ t2 => at_ctx C2 t2
+  | _, _ => None
+  end.
+
+Definition step_at (C : ctx) (t : trm) : list (rule * trm) :=
+  match at_ctx C t with
+  | None => []
+  | Some s => map (fun p => (fst p, plug C (snd p))) (root_steps s)
+  end.
+
+Definition steps (t : trm) : list (rule * trm) :=
+  flat_map (fun C => step_at C t) (positions t).
+
+Definition has_rule (r : rule) (l : list (rule * trm)) : bool :=
+  existsb (fun p => rule_eqb r (fst p)) l.
+
+Definition count_rule (r : rule) (l : list (rule * trm)) : nat :=
+  length (filter (fun p => rule_eqb r (fst p)) l).
+
+Definition normal_form (t : trm) : bool :=
+  match steps t with [] => true | _ => false end.
+
+Definition ex_id : trm := App (Lam (BVar 0)) (FVar 1).
+Definition ex_dup : trm := App (Lam (App (BVar 0) (BVar 0))) (FVar 1).
+Definition ex_dup_es : trm := ESub (App (BVar 0) (BVar 0)) (FVar 1).
+Definition ex_es : trm := ESub (BVar 0) (FVar 1).
+Definition ex_gc : trm := ESub (FVar 2) (FVar 1).
+Definition ex_nested : trm := ESub (ESub (BVar 0) (FVar 1)) (FVar 2).
+
+Example ex_id_beta : has_rule RB (steps ex_id) = true.
+Proof. reflexivity. Qed.
+
+Example ex_id_one_step : length (steps ex_id) = 1.
+Proof. reflexivity. Qed.
+
+Example ex_dup_beta : has_rule RB (steps ex_dup) = true.
+Proof. reflexivity. Qed.
+
+Example ex_dup_es_two_R : count_rule RR (steps ex_dup_es) = 2.
+Proof. reflexivity. Qed.
+
+Example ex_dup_es_no_Gc : has_rule RGc (steps ex_dup_es) = false.
+Proof. reflexivity. Qed.
+
+Example ex_es_R : has_rule RR (steps ex_es) = true.
+Proof. reflexivity. Qed.
+
+Example ex_es_no_Gc : has_rule RGc (steps ex_es) = false.
+Proof. reflexivity. Qed.
+
+Example ex_gc_gc : has_rule RGc (steps ex_gc) = true.
+Proof. reflexivity. Qed.
+
+Example ex_gc_no_R : has_rule RR (steps ex_gc) = false.
+Proof. reflexivity. Qed.
+
+Example ex_nested_has_gc : has_rule RGc (steps ex_nested) = true.
+Proof. reflexivity. Qed.
+
+Example ex_id_terminates : normal_form (plug GHole (FVar 5)) = true.
+Proof. reflexivity. Qed.
+
+Example fvs_lam : fvs (Lam (App (BVar 0) (FVar 1))) = [1].
+Proof. reflexivity. Qed.
+
+Example fvs_es : fvs (ESub (FVar 0) (FVar 1)) = [0; 1].
+Proof. reflexivity. Qed.
+
+Example occurs0_lam : occurs0 (Lam (BVar 0)) = false.
+Proof. reflexivity. Qed.
+
+Example occurs0_bvar : occurs0 (BVar 0) = true.
+Proof. reflexivity. Qed.
+
+Example open_basic : open_rec 0 (FVar 1) (BVar 0) = FVar 1.
+Proof. reflexivity. Qed.
+
+Example open_under_lam : open_rec 0 (FVar 1) (Lam (BVar 1)) = Lam (FVar 1).
+Proof. reflexivity. Qed.
+
+Example subst_hit : subst 0 (FVar 1) (FVar 0) = FVar 1.
+Proof. reflexivity. Qed.
+
+Example subst_miss : subst 0 (FVar 1) (FVar 2) = FVar 2.
+Proof. reflexivity. Qed.
+
+Example capture_avoid_1 :
+  subst 0 (FVar 1) (Lam (BVar 0)) = Lam (BVar 0).
+Proof. reflexivity. Qed.
+
+Example capture_avoid_2 :
+  subst 0 (FVar 1) (Lam (App (BVar 0) (FVar 0)))
+  = Lam (App (BVar 0) (FVar 1)).
+Proof. reflexivity. Qed.
+
+Example capture_avoid_es :
+  subst 0 (FVar 1) (ESub (FVar 0) (FVar 2))
+  = ESub (FVar 1) (FVar 2).
+Proof. reflexivity. Qed.
+
+Print Assumptions ex_id_beta.
+Print Assumptions ex_dup_es_two_R.
+Print Assumptions ex_gc_gc.
+Print Assumptions ex_nested_has_gc.
+Print Assumptions capture_avoid_1.
+Print Assumptions capture_avoid_2.
+Print Assumptions capture_avoid_es.
+Print Assumptions open_under_lam.
diff --git a/theory/dune b/theory/dune
new file mode 100644
index 0000000..7f43088
--- /dev/null
+++ b/theory/dune
@@ -0,0 +1,3 @@
+(rocq.theory
+ (name LambdaSub)
+ (theories Stdlib))