From 1291234a8a1170327b159edf0242fbc492641c5c Mon Sep 17 00:00:00 2001
From: milner <m@cpl0.zip>
Date: Tue, 22 Sep 2026 12:17:00 +0200
Subject: [PATCH] add(tests): bounded conformance suite for the reducer against
 the relational semantics (golden corpus)

---
 graphs/dependency.dot       |  207 ++++---
 graphs/dependency.png       |  Bin 163675 -> 226999 bytes
 graphs/dependency.svg       | 1080 ++++++++++++++++++++++-------------
 graphs/theorems.json        |  716 ++++++++++++++++-------
 scripts/audit.sh            |    7 +-
 scripts/difftest.sh         |   20 +
 scripts/extract_metadata.py |    2 +-
 scripts/gen_graphs.py       |   79 +--
 theory/Tests.v              |   77 +++
 9 files changed, 1438 insertions(+), 750 deletions(-)
 create mode 100755 scripts/difftest.sh
 create mode 100644 theory/Tests.v

diff --git a/graphs/dependency.dot b/graphs/dependency.dot
index 8efbe20..4a3e429 100644
--- a/graphs/dependency.dot
+++ b/graphs/dependency.dot
@@ -1,110 +1,107 @@
 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;
+  splines=spline;
+  nodesep=0.22;
+  ranksep=0.6;
+  node [fontname=Inter, fontsize=9, fontcolor="#222222"];
+  edge [fontname=Inter, fontsize=8, color="#9a9a9a", arrowsize=0.6, penwidth=0.8];
+  graph [fontname=Inter, fontsize=12, labelloc=t, label=<<B>Milner's Lambda-Calculus with Partial Substitutions - theorem dependency (M2)</B><BR/><FONT POINT-SIZE="9"><FONT COLOR="#38761d">proved</FONT>   <FONT COLOR="#b45f06">stated</FONT>   <FONT COLOR="#777777">planned</FONT>   <FONT COLOR="#cc0000">blocked</FONT></FONT>>];
+  fvs_lift [label="fvs_lift", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="fvs_lift theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  lift_lc [label="lift_lc", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="lift_lc theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  lift_0_lc [label="lift_0_lc", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="lift_0_lc theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  open_rec_bvar [label="open_rec_bvar", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_bvar theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  close_rec_fvar [label="close_rec_fvar", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="close_rec_fvar theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  open_rec_occurs_false [label="open_rec_occurs_false", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_occurs_false theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  close_rec_notin [label="close_rec_notin", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="close_rec_notin theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  close_rec_fvs [label="close_rec_fvs", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="close_rec_fvs theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  open_rec_bvar_neq [label="open_rec_bvar_neq", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_bvar_neq theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  open_rec_fvs [label="open_rec_fvs", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_fvs theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  open_close [label="open_close", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_close theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  subst_fvar_self [label="subst_fvar_self", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_fvar_self theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  subst_fvar_other [label="subst_fvar_other", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_fvar_other theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  subst_fvs [label="subst_fvs", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="subst_fvs theory/Binding.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  Plus_red1_context [label="Plus_red1_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_red1_context theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  occurs_count_zero [label="occurs_count_zero", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_zero theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  occurs_count_false [label="occurs_count_false", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_false theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  occurs_count_pos [label="occurs_count_pos", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_pos theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  occurs_lift_self [label="occurs_lift_self", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_lift_self theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  zdecs_nonempty [label="zdecs_nonempty", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="zdecs_nonempty theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  occurs_count_zfill_plug [label="occurs_count_zfill_plug", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_zfill_plug theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  open_rec_zplug_lift [label="open_rec_zplug_lift", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="open_rec_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  full_comp_aux [label="full_comp_aux", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="full_comp_aux theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  lemma_2_2_full_comp [label="lemma_2_2_full_comp", color="#38761d", fillcolor="#d9ead3", style="rounded,filled", fontcolor="#222222", tooltip="lemma_2_2_full_comp (section 2.2) theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=box];
+  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];
+  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];
+  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;
+  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;
-  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;
+  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_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;
 }
diff --git a/graphs/dependency.png b/graphs/dependency.png
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-<text text-anchor="middle" x="1048.82" y="-286.04" font-family="Inter" font-size="12.00" fill="#333333">M3</text>
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-<title>cluster_M4</title>
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-<text text-anchor="middle" x="1552.14" y="-421.04" font-family="Inter" font-size="12.00" fill="#333333">M5</text>
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-<text text-anchor="middle" x="1524.07" y="-502.04" font-family="Inter" font-size="12.00" fill="#333333">M6</text>
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-<!-- exec_step -->
+<polygon fill="white" stroke="transparent" points="-4,4 -4,-926 949.29,-926 949.29,4 -4,4"/>
+<text text-anchor="start" x="236.65" y="-909.4" font-family="Inter" font-weight="bold" font-size="12.00">Milner&#39;s Lambda&#45;Calculus with Partial Substitutions &#45; theorem dependency (M2)</text>
+<text text-anchor="start" x="399.65" y="-899.8" font-family="Inter" font-size="9.00" fill="#38761d">proved</text>
+<text text-anchor="start" x="429.65" y="-899.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="436.65" y="-899.8" font-family="Inter" font-size="9.00" fill="#b45f06">stated</text>
+<text text-anchor="start" x="463.65" y="-899.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="470.65" y="-899.8" font-family="Inter" font-size="9.00" fill="#777777">planned</text>
+<text text-anchor="start" x="504.65" y="-899.8" font-family="Inter" font-size="9.00"> &#160;&#160;</text>
+<text text-anchor="start" x="511.65" y="-899.8" font-family="Inter" font-size="9.00" fill="#cc0000">blocked</text>
+<!-- fvs_lift -->
 <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>
+<title>fvs_lift</title>
+<g id="a_node1"><a xlink:title="fvs_lift theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-563" rx="27" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-560.8" font-family="Inter" font-size="9.00" fill="#222222">fvs_lift</text>
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-<!-- exec_examples -->
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-<title>exec_examples</title>
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-<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>
+<!-- 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 https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-615" rx="41.64" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-612.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_fvs</text>
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-<!-- exec_step&#45;&gt;exec_examples -->
-<g id="edge1" class="edge">
-<title>exec_step&#45;&gt;exec_examples</title>
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-<polygon fill="#333333" stroke="#333333" points="117.18,-230.14 127.18,-226.64 117.18,-223.14 117.18,-230.14"/>
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-<!-- m1_syntax -->
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-<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 -->
+<!-- fvs_lift&#45;&gt;open_rec_fvs -->
 <g id="edge2" class="edge">
-<title>exec_examples&#45;&gt;m1_syntax</title>
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-<polygon fill="#333333" stroke="#333333" points="256.92,-230.14 266.92,-226.64 256.92,-223.14 256.92,-230.14"/>
+<title>fvs_lift&#45;&gt;open_rec_fvs</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M90.24,-570.23C118.42,-578.72 165.55,-592.91 199.19,-603.04"/>
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-<!-- 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>
+<!-- fvs_zplug_lift -->
+<g id="node25" class="node">
+<title>fvs_zplug_lift</title>
+<g id="a_node25"><a xlink:title="fvs_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-563" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-560.8" font-family="Inter" font-size="9.00" fill="#222222">fvs_zplug_lift</text>
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-<!-- m1_syntax&#45;&gt;m1_open_close -->
-<g id="edge3" class="edge">
-<title>m1_syntax&#45;&gt;m1_open_close</title>
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+<!-- fvs_lift&#45;&gt;fvs_zplug_lift -->
+<g id="edge21" class="edge">
+<title>fvs_lift&#45;&gt;fvs_zplug_lift</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M92.46,-563C118.9,-563 160.28,-563 192.31,-563"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="192.58,-565.1 198.58,-563 192.58,-560.9 192.58,-565.1"/>
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-<!-- m1_red1 -->
-<g id="node5" class="node">
-<title>m1_red1</title>
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-<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>
+<!-- lift_lc -->
+<g id="node2" class="node">
+<title>lift_lc</title>
+<g id="a_node2"><a xlink:title="lift_lc theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-719" rx="27" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-716.8" font-family="Inter" font-size="9.00" fill="#222222">lift_lc</text>
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-<!-- 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"/>
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-<!-- 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>
+<!-- lift_0_lc -->
+<g id="node3" class="node">
+<title>lift_0_lc</title>
+<g id="a_node3"><a xlink:title="lift_0_lc theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-719" rx="27.93" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-716.8" font-family="Inter" font-size="9.00" fill="#222222">lift_0_lc</text>
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-<!-- m1_syntax&#45;&gt;m5_mvar -->
-<g id="edge20" class="edge">
-<title>m1_syntax&#45;&gt;m5_mvar</title>
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+<!-- 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="#9a9a9a" stroke-width="0.8" d="M92.46,-719C122.79,-719 172.78,-719 205.77,-719"/>
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-<!-- m1_open_close&#45;&gt;m1_red1 -->
-<g id="edge5" class="edge">
-<title>m1_open_close&#45;&gt;m1_red1</title>
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-<!-- lemma_2_1_fv_preserved -->
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-<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>
+<!-- subst_fvar_self -->
+<g id="node12" class="node">
+<title>subst_fvar_self</title>
+<g id="a_node12"><a xlink:title="subst_fvar_self theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-719" rx="45.72" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-716.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvar_self</text>
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-<!-- m1_red1&#45;&gt;lemma_2_1_fv_preserved -->
+<!-- lift_0_lc&#45;&gt;subst_fvar_self -->
 <g id="edge6" class="edge">
-<title>m1_red1&#45;&gt;lemma_2_1_fv_preserved</title>
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-<!-- lemma_2_2_full_comp -->
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-<title>lemma_2_2_full_comp</title>
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-<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>
+<!-- open_rec_bvar -->
+<g id="node4" class="node">
+<title>open_rec_bvar</title>
+<g id="a_node4"><a xlink:title="open_rec_bvar theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-511" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-508.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_bvar</text>
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-<!-- m1_red1&#45;&gt;lemma_2_2_full_comp -->
+<!-- open_rec_bvar&#45;&gt;open_rec_fvs -->
+<g id="edge3" class="edge">
+<title>open_rec_bvar&#45;&gt;open_rec_fvs</title>
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+<!-- open_close -->
+<g id="node11" class="node">
+<title>open_close</title>
+<g id="a_node11"><a xlink:title="open_close theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-511" rx="36.9" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-508.8" font-family="Inter" font-size="9.00" fill="#222222">open_close</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>
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+<!-- close_rec_fvar -->
+<g id="node5" class="node">
+<title>close_rec_fvar</title>
+<g id="a_node5"><a xlink:title="close_rec_fvar theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-771" rx="45.07" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-768.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_fvar</text>
+</a>
+</g>
+</g>
+<!-- open_rec_occurs_false -->
+<g id="node6" class="node">
+<title>open_rec_occurs_false</title>
+<g id="a_node6"><a xlink:title="open_rec_occurs_false theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-196" rx="65.48" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-193.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_occurs_false</text>
+</a>
+</g>
+</g>
+<!-- open_rec_zplug_lift -->
+<g id="node22" class="node">
+<title>open_rec_zplug_lift</title>
+<g id="a_node22"><a xlink:title="open_rec_zplug_lift theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-122" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-119.8" font-family="Inter" font-size="9.00" fill="#222222">open_rec_zplug_lift</text>
+</a>
+</g>
+</g>
+<!-- open_rec_occurs_false&#45;&gt;open_rec_zplug_lift -->
+<g id="edge13" class="edge">
+<title>open_rec_occurs_false&#45;&gt;open_rec_zplug_lift</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M99.89,-180.5C120.94,-170.91 148.68,-158.48 173.48,-148 181.1,-144.78 189.27,-141.44 197.16,-138.28"/>
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+<!-- full_comp_aux -->
+<g id="node23" class="node">
+<title>full_comp_aux</title>
+<g id="a_node23"><a xlink:title="full_comp_aux theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-139" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-136.8" font-family="Inter" font-size="9.00" fill="#222222">full_comp_aux</text>
+</a>
+</g>
+</g>
+<!-- open_rec_occurs_false&#45;&gt;full_comp_aux -->
+<g id="edge16" class="edge">
+<title>open_rec_occurs_false&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M120.34,-205.79C169.78,-212.67 244.39,-217.82 306.27,-200 327.96,-193.76 329.96,-184.67 349.27,-173 358.26,-167.57 368.1,-161.78 377.11,-156.55"/>
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+<!-- close_rec_notin -->
+<g id="node7" class="node">
+<title>close_rec_notin</title>
+<g id="a_node7"><a xlink:title="close_rec_notin theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-823" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-820.8" font-family="Inter" font-size="9.00" fill="#222222">close_rec_notin</text>
+</a>
+</g>
+</g>
+<!-- close_rec_fvs -->
+<g id="node8" class="node">
+<title>close_rec_fvs</title>
+<g id="a_node8"><a xlink:title="close_rec_fvs theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-667" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-664.8" font-family="Inter" font-size="9.00" fill="#222222">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 https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-667" rx="32.16" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-664.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvs</text>
+</a>
+</g>
+</g>
+<!-- close_rec_fvs&#45;&gt;subst_fvs -->
 <g id="edge7" class="edge">
-<title>m1_red1&#45;&gt;lemma_2_2_full_comp</title>
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+<!-- 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 https://arxiv.org/html/2312.13270v1">
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+<text text-anchor="middle" x="65.24" y="-612.8" font-family="Inter" font-size="9.00" fill="#222222">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>
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+<!-- open_rec_fvs&#45;&gt;subst_fvs -->
+<g id="edge8" class="edge">
+<title>open_rec_fvs&#45;&gt;subst_fvs</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M274.24,-625.4C303.28,-634.45 345.15,-647.49 374.23,-656.56"/>
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+<!-- subst_fvar_other -->
+<g id="node13" class="node">
+<title>subst_fvar_other</title>
+<g id="a_node13"><a xlink:title="subst_fvar_other theory/Binding.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-875" rx="49.8" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-872.8" font-family="Inter" font-size="9.00" fill="#222222">subst_fvar_other</text>
+</a>
+</g>
+</g>
+<!-- Plus_red1_context -->
+<g id="node15" class="node">
+<title>Plus_red1_context</title>
+<g id="a_node15"><a xlink:title="Plus_red1_context theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-191" rx="53.23" ry="18"/>
+<text text-anchor="middle" x="565.21" y="-188.8" font-family="Inter" font-size="9.00" fill="#222222">Plus_red1_context</text>
+</a>
+</g>
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 <!-- lemma_2_3_beta_sim -->
-<g id="node8" class="node">
+<g id="node30" 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">
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+<g id="a_node30"><a xlink:title="lemma_2_3_beta_sim (section 2.3) theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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-<!-- 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>
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+<!-- Plus_red1_context&#45;&gt;lemma_2_3_beta_sim -->
+<g id="edge26" class="edge">
+<title>Plus_red1_context&#45;&gt;lemma_2_3_beta_sim</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M613.13,-183.13C628.84,-180.49 646.55,-177.52 663.02,-174.76"/>
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-<!-- m3_measure -->
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-<title>m3_measure</title>
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-<text text-anchor="middle" x="779.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">sub measure</text>
+<!-- occurs_count_zero -->
+<g id="node16" class="node">
+<title>occurs_count_zero</title>
+<g id="a_node16"><a xlink:title="occurs_count_zero theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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-<!-- m1_red1&#45;&gt;m3_measure -->
-<g id="edge10" class="edge">
-<title>m1_red1&#45;&gt;m3_measure</title>
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-<text text-anchor="middle" x="1120.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">{R,Gc} local confluence</text>
+<!-- occurs_count_pos -->
+<g id="node18" class="node">
+<title>occurs_count_pos</title>
+<g id="a_node18"><a xlink:title="occurs_count_pos theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-18" rx="53.89" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-15.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_count_pos</text>
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 </g>
-<!-- m1_red1&#45;&gt;m3_local_confluence -->
-<g id="edge13" class="edge">
-<title>m1_red1&#45;&gt;m3_local_confluence</title>
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+<!-- occurs_count_zero&#45;&gt;occurs_count_pos -->
+<g id="edge9" class="edge">
+<title>occurs_count_zero&#45;&gt;occurs_count_pos</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M295.05,-18C312.17,-18 331.17,-18 348.62,-18"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="348.97,-20.1 354.97,-18 348.97,-15.9 348.97,-20.1"/>
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-<g id="node13" class="node">
-<title>m4_par</title>
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-<text text-anchor="middle" x="1468.07" y="-220.14" font-family="Inter" font-size="10.00" fill="#333333">parallel reduction par</text>
-</a>
+<!-- occurs_count_zero&#45;&gt;full_comp_aux -->
+<g id="edge14" class="edge">
+<title>occurs_count_zero&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M279.4,-30.54C288.54,-34.26 298.02,-38.76 306.27,-44 338.66,-64.56 369.95,-96.22 389.02,-117.26"/>
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-<!-- m1_red1&#45;&gt;m4_par -->
-<g id="edge15" class="edge">
-<title>m1_red1&#45;&gt;m4_par</title>
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-<!-- m5_rx -->
+<!-- occurs_count_false -->
 <g id="node17" class="node">
-<title>m5_rx</title>
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+<title>occurs_count_false</title>
+<g id="a_node17"><a xlink:title="occurs_count_false theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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-<!-- m1_red1&#45;&gt;m5_rx -->
-<g id="edge22" class="edge">
-<title>m1_red1&#45;&gt;m5_rx</title>
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-<!-- m6_translation -->
+<!-- occurs_count_zfill_plug -->
 <g id="node21" class="node">
-<title>m6_translation</title>
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+<title>occurs_count_zfill_plug</title>
+<g id="a_node21"><a xlink:title="occurs_count_zfill_plug theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-70" rx="66.29" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-67.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_count_zfill_plug</text>
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-<!-- m1_red1&#45;&gt;m6_translation -->
-<g id="edge29" class="edge">
-<title>m1_red1&#45;&gt;m6_translation</title>
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+<!-- occurs_count_false&#45;&gt;occurs_count_zfill_plug -->
+<g id="edge10" class="edge">
+<title>occurs_count_false&#45;&gt;occurs_count_zfill_plug</title>
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+<!-- occurs_lift_self -->
+<g id="node19" class="node">
+<title>occurs_lift_self</title>
+<g id="a_node19"><a xlink:title="occurs_lift_self theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="65.24" cy="-122" rx="46.38" ry="18"/>
+<text text-anchor="middle" x="65.24" y="-119.8" font-family="Inter" font-size="9.00" fill="#222222">occurs_lift_self</text>
+</a>
+</g>
+</g>
+<!-- occurs_lift_self&#45;&gt;occurs_count_zfill_plug -->
+<g id="edge11" class="edge">
+<title>occurs_lift_self&#45;&gt;occurs_count_zfill_plug</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M102.43,-111.1C127.54,-103.54 161.3,-93.37 188.96,-85.03"/>
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+<!-- occurs_lift_self&#45;&gt;open_rec_zplug_lift -->
+<g id="edge12" class="edge">
+<title>occurs_lift_self&#45;&gt;open_rec_zplug_lift</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M111.6,-122C131.6,-122 155.41,-122 177.01,-122"/>
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+<!-- zdecs_nonempty -->
+<g id="node20" class="node">
+<title>zdecs_nonempty</title>
+<g id="a_node20"><a xlink:title="zdecs_nonempty theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-174" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-171.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_nonempty</text>
+</a>
+</g>
+</g>
+<!-- zdecs_nonempty&#45;&gt;full_comp_aux -->
+<g id="edge18" class="edge">
+<title>zdecs_nonempty&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M284.28,-164.9C308.41,-159.84 338.38,-153.55 362.75,-148.44"/>
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+<!-- occurs_count_zfill_plug&#45;&gt;full_comp_aux -->
+<g id="edge15" class="edge">
+<title>occurs_count_zfill_plug&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M278.28,-84.71C287.45,-88.36 297.23,-92.29 306.27,-96 328.08,-104.96 352.41,-115.26 371.78,-123.53"/>
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+<!-- open_rec_zplug_lift&#45;&gt;full_comp_aux -->
+<g id="edge17" class="edge">
+<title>open_rec_zplug_lift&#45;&gt;full_comp_aux</title>
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+<!-- lemma_2_2_full_comp -->
+<g id="node24" class="node">
+<title>lemma_2_2_full_comp</title>
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+</a>
+</g>
+</g>
+<!-- full_comp_aux&#45;&gt;lemma_2_2_full_comp -->
+<g id="edge20" class="edge">
+<title>full_comp_aux&#45;&gt;lemma_2_2_full_comp</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M453.59,-139C469.56,-139 487.88,-139 504.96,-139"/>
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-<!-- lemma_2_3_beta_sim&#45;&gt;theorem_confluence_terms -->
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-<!-- m3_measure&#45;&gt;m3_termination -->
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-<title>m3_measure&#45;&gt;m3_termination</title>
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-<title>m3_termination&#45;&gt;m3_local_confluence</title>
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-<!-- m3_local_confluence&#45;&gt;lemma_3_2_sub_nf_unique -->
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-<!-- lemma_3_2_sub_nf_unique&#45;&gt;m4_par -->
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-<title>lemma_3_2_sub_nf_unique&#45;&gt;m4_par</title>
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-<!-- m5_coherence -->
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-<!-- lemma_3_2_sub_nf_unique&#45;&gt;m5_coherence -->
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-</a>
-</g>
-</g>
-<!-- m4_par&#45;&gt;m4_par_diamond -->
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-<title>m4_par&#45;&gt;m4_par_diamond</title>
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-<!-- m4_par_diamond&#45;&gt;theorem_confluence_terms -->
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-<title>m4_par_diamond&#45;&gt;theorem_confluence_terms</title>
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-<title>theorem_confluence_metaterms</title>
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-</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>
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-<!-- corollary_6_21_psn -->
-<g id="node26" class="node">
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-</g>
-</g>
-<!-- theorem_confluence_terms&#45;&gt;corollary_6_21_psn -->
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-<!-- m5_mvar&#45;&gt;m5_rx -->
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-<title>m5_mvar&#45;&gt;m5_rx</title>
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-<!-- m5_eq_C -->
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-<!-- m5_mvar&#45;&gt;m5_eq_C -->
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-<title>m5_mvar&#45;&gt;m5_eq_C</title>
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-<title>m5_rx&#45;&gt;m5_coherence</title>
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-<!-- m5_eq_C&#45;&gt;m5_coherence -->
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-<title>m5_eq_C&#45;&gt;m5_coherence</title>
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-<!-- m5_coherence&#45;&gt;theorem_confluence_metaterms -->
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+<title>lemma_2_2_full_comp&#45;&gt;lemma_2_3_beta_sim</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M619.55,-147.95C633.57,-150.3 648.76,-152.85 663.03,-155.24"/>
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-<!-- m6_sn -->
-<g id="node22" class="node">
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-<text text-anchor="middle" x="1292.78" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">SN for simply typed</text>
+<!-- red1_root_fv -->
+<g id="node27" class="node">
+<title>red1_root_fv</title>
+<g id="a_node27"><a xlink:title="red1_root_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-563" rx="38.87" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-560.8" font-family="Inter" font-size="9.00" fill="#222222">red1_root_fv</text>
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-<!-- m6_translation&#45;&gt;m6_sn -->
+<!-- fvs_zplug_lift&#45;&gt;red1_root_fv -->
+<g id="edge22" class="edge">
+<title>fvs_zplug_lift&#45;&gt;red1_root_fv</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M281.27,-563C306.27,-563 338.37,-563 363.98,-563"/>
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+<!-- plug_fv_mono -->
+<g id="node26" class="node">
+<title>plug_fv_mono</title>
+<g id="a_node26"><a xlink:title="plug_fv_mono theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-615" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-612.8" font-family="Inter" font-size="9.00" fill="#222222">plug_fv_mono</text>
+</a>
+</g>
+</g>
+<!-- red1r_fv -->
+<g id="node28" class="node">
+<title>red1r_fv</title>
+<g id="a_node28"><a xlink:title="red1r_fv theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-563" rx="28.74" ry="18"/>
+<text text-anchor="middle" x="565.21" y="-560.8" font-family="Inter" font-size="9.00" fill="#222222">red1r_fv</text>
+</a>
+</g>
+</g>
+<!-- plug_fv_mono&#45;&gt;red1r_fv -->
+<g id="edge23" class="edge">
+<title>plug_fv_mono&#45;&gt;red1r_fv</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M442.88,-603.84C469.74,-594.8 507.32,-582.15 533.51,-573.33"/>
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+<!-- red1_root_fv&#45;&gt;red1r_fv -->
+<g id="edge24" class="edge">
+<title>red1_root_fv&#45;&gt;red1r_fv</title>
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+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="530.01,-565.1 536.01,-563 530.01,-560.9 530.01,-565.1"/>
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+<!-- lemma_2_1_fv_preserved -->
+<g id="node29" class="node">
+<title>lemma_2_1_fv_preserved</title>
+<g id="a_node29"><a xlink:title="lemma_2_1_fv_preserved (section 2.1) theory/Metatheory.v https://arxiv.org/html/2312.13270v1">
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+<text text-anchor="middle" x="722.21" y="-534.8" font-family="Inter" font-size="9.00" fill="#222222">lemma_2_1_fv_preserved</text>
+</a>
+</g>
+</g>
+<!-- red1r_fv&#45;&gt;lemma_2_1_fv_preserved -->
+<g id="edge25" class="edge">
+<title>red1r_fv&#45;&gt;lemma_2_1_fv_preserved</title>
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+<!-- corpus_fv_preserved -->
+<g id="node50" class="node">
+<title>corpus_fv_preserved</title>
+<g id="a_node50"><a xlink:title="corpus_fv_preserved theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="885.25" cy="-481" rx="60.09" ry="18"/>
+<text text-anchor="middle" x="885.25" y="-478.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_fv_preserved</text>
+</a>
+</g>
+</g>
+<!-- lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved -->
+<g id="edge46" class="edge">
+<title>lemma_2_1_fv_preserved&#45;&gt;corpus_fv_preserved</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M775.05,-518.97C795.8,-511.76 819.48,-503.52 839.47,-496.57"/>
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+<!-- zdecs_sound -->
+<g id="node31" class="node">
+<title>zdecs_sound</title>
+<g id="a_node31"><a xlink:title="zdecs_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-260" rx="40.98" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-257.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_sound</text>
+</a>
+</g>
+</g>
+<!-- zdecs_sound&#45;&gt;full_comp_aux -->
+<g id="edge19" class="edge">
+<title>zdecs_sound&#45;&gt;full_comp_aux</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M272.62,-249.16C283.94,-244.37 296.34,-237.98 306.27,-230 331,-210.11 325.53,-194.06 349.27,-173 356.62,-166.48 365.41,-160.59 373.91,-155.62"/>
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+<!-- root_steps_sound -->
+<g id="node34" class="node">
+<title>root_steps_sound</title>
+<g id="a_node34"><a xlink:title="root_steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-407" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-404.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_sound</text>
+</a>
+</g>
+</g>
+<!-- zdecs_sound&#45;&gt;root_steps_sound -->
+<g id="edge28" class="edge">
+<title>zdecs_sound&#45;&gt;root_steps_sound</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M276.31,-268.53C286.94,-272.46 297.96,-278.07 306.27,-286 339.8,-318 316.39,-348.34 349.27,-381 353.65,-385.35 358.87,-389.03 364.37,-392.13"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="363.66,-394.13 369.95,-395.05 365.6,-390.41 363.66,-394.13"/>
+</g>
+<!-- zdecs_complete -->
+<g id="node32" class="node">
+<title>zdecs_complete</title>
+<g id="a_node32"><a xlink:title="zdecs_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-459" rx="49.15" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-456.8" font-family="Inter" font-size="9.00" fill="#222222">zdecs_complete</text>
+</a>
+</g>
+</g>
+<!-- root_steps_complete -->
+<g id="node35" class="node">
+<title>root_steps_complete</title>
+<g id="a_node35"><a xlink:title="root_steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-459" rx="59.43" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-456.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_complete</text>
+</a>
+</g>
+</g>
+<!-- zdecs_complete&#45;&gt;root_steps_complete -->
+<g id="edge29" class="edge">
+<title>zdecs_complete&#45;&gt;root_steps_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M289.14,-459C306,-459 325.21,-459 343.19,-459"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="343.23,-461.1 349.23,-459 343.23,-456.9 343.23,-461.1"/>
+</g>
+<!-- positions_at -->
+<g id="node33" class="node">
+<title>positions_at</title>
+<g id="a_node33"><a xlink:title="positions_at theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-511" rx="38.21" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-508.8" font-family="Inter" font-size="9.00" fill="#222222">positions_at</text>
+</a>
+</g>
+</g>
+<!-- steps_sound -->
+<g id="node36" class="node">
+<title>steps_sound</title>
+<g id="a_node36"><a xlink:title="steps_sound theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-428" rx="39.02" ry="18"/>
+<text text-anchor="middle" x="565.21" y="-425.8" font-family="Inter" font-size="9.00" fill="#222222">steps_sound</text>
+</a>
+</g>
+</g>
+<!-- positions_at&#45;&gt;steps_sound -->
 <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"/>
+<title>positions_at&#45;&gt;steps_sound</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M438.01,-499.23C447.7,-494.98 458.53,-490 468.21,-485 491.66,-472.88 517.33,-457.52 536.22,-445.8"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="537.38,-447.55 541.36,-442.6 535.16,-443.99 537.38,-447.55"/>
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-<!-- m6_intersection -->
-<g id="node23" class="node">
-<title>m6_intersection</title>
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-<text text-anchor="middle" x="1468.07" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">intersection typing</text>
-</a>
-</g>
-</g>
-<!-- m6_sn&#45;&gt;m6_intersection -->
+<!-- root_steps_sound&#45;&gt;steps_sound -->
 <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"/>
+<title>root_steps_sound&#45;&gt;steps_sound</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M456.92,-413.41C477.35,-416.19 501.09,-419.42 521.04,-422.13"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="520.87,-424.23 527.1,-422.96 521.44,-420.07 520.87,-424.23"/>
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-<!-- 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"/>
-<text text-anchor="middle" x="1628.35" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">typable → SN</text>
+<!-- steps_complete -->
+<g id="node43" class="node">
+<title>steps_complete</title>
+<g id="a_node43"><a xlink:title="steps_complete theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-316" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="565.21" y="-313.8" font-family="Inter" font-size="9.00" fill="#222222">steps_complete</text>
 </a>
 </g>
 </g>
-<!-- m6_intersection&#45;&gt;m6_typable_sn -->
+<!-- root_steps_complete&#45;&gt;steps_complete -->
+<g id="edge37" class="edge">
+<title>root_steps_complete&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M449.47,-445.8C456.24,-442.36 462.81,-438.13 468.21,-433 498.88,-403.82 482.41,-379.04 511.21,-348 516.35,-342.46 522.66,-337.55 529.12,-333.34"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="530.56,-334.92 534.55,-329.97 528.35,-331.35 530.56,-334.92"/>
+</g>
+<!-- steps_sound_red1 -->
+<g id="node37" class="node">
+<title>steps_sound_red1</title>
+<g id="a_node37"><a xlink:title="steps_sound_red1 theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="722.21" cy="-480" rx="51.92" ry="18"/>
+<text text-anchor="middle" x="722.21" y="-477.8" font-family="Inter" font-size="9.00" fill="#222222">steps_sound_red1</text>
+</a>
+</g>
+</g>
+<!-- steps_sound&#45;&gt;steps_sound_red1 -->
 <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"/>
+<title>steps_sound&#45;&gt;steps_sound_red1</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M597.57,-438.52C620.85,-446.33 652.85,-457.07 678.46,-465.66"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="677.8,-467.65 684.16,-467.57 679.14,-463.67 677.8,-467.65"/>
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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"/>
-<text text-anchor="middle" x="1790.35" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">SN → typable</text>
+<!-- has_red_spec -->
+<g id="node45" class="node">
+<title>has_red_spec</title>
+<g id="a_node45"><a xlink:title="has_red_spec theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="722.21" cy="-374" rx="42.95" ry="18"/>
+<text text-anchor="middle" x="722.21" y="-371.8" font-family="Inter" font-size="9.00" fill="#222222">has_red_spec</text>
 </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"/>
+<!-- steps_sound&#45;&gt;has_red_spec -->
+<g id="edge41" class="edge">
+<title>steps_sound&#45;&gt;has_red_spec</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M596.83,-417.33C621.52,-408.73 656.37,-396.59 682.91,-387.34"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="683.8,-389.26 688.78,-385.3 682.42,-385.29 683.8,-389.26"/>
 </g>
-<!-- m6_sn_typable&#45;&gt;corollary_6_21_psn -->
+<!-- corpus_sound -->
+<g id="node47" class="node">
+<title>corpus_sound</title>
+<g id="a_node47"><a xlink:title="corpus_sound theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="722.21" cy="-428" rx="43.1" ry="18"/>
+<text text-anchor="middle" x="722.21" y="-425.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_sound</text>
+</a>
+</g>
+</g>
+<!-- steps_sound&#45;&gt;corpus_sound -->
+<g id="edge43" class="edge">
+<title>steps_sound&#45;&gt;corpus_sound</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M604.52,-428C625.09,-428 650.65,-428 672.61,-428"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="672.7,-430.1 678.7,-428 672.7,-425.9 672.7,-430.1"/>
+</g>
+<!-- steps_sound_red1&#45;&gt;corpus_fv_preserved -->
+<g id="edge47" class="edge">
+<title>steps_sound_red1&#45;&gt;corpus_fv_preserved</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M774.16,-480.32C788.35,-480.4 803.97,-480.5 818.86,-480.59"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="819.18,-482.7 825.2,-480.63 819.21,-478.5 819.18,-482.7"/>
+</g>
+<!-- positions_hole -->
+<g id="node38" class="node">
+<title>positions_hole</title>
+<g id="a_node38"><a xlink:title="positions_hole theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="239.88" cy="-312" rx="44.41" ry="18"/>
+<text text-anchor="middle" x="239.88" y="-309.8" font-family="Inter" font-size="9.00" fill="#222222">positions_hole</text>
+</a>
+</g>
+</g>
+<!-- root_steps_in_steps -->
+<g id="node39" class="node">
+<title>root_steps_in_steps</title>
+<g id="a_node39"><a xlink:title="root_steps_in_steps theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-303" rx="56.66" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-300.8" font-family="Inter" font-size="9.00" fill="#222222">root_steps_in_steps</text>
+</a>
+</g>
+</g>
+<!-- positions_hole&#45;&gt;root_steps_in_steps -->
+<g id="edge33" class="edge">
+<title>positions_hole&#45;&gt;root_steps_in_steps</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M284.28,-309.66C303.35,-308.63 326.08,-307.4 346.81,-306.29"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="347,-308.38 352.88,-305.96 346.78,-304.19 347,-308.38"/>
+</g>
+<!-- root_steps_in_steps&#45;&gt;steps_complete -->
+<g id="edge38" class="edge">
+<title>root_steps_in_steps&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M463.76,-307.55C479.43,-308.87 496.5,-310.3 512.06,-311.61"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="512.01,-313.71 518.16,-312.12 512.36,-309.53 512.01,-313.71"/>
+</g>
+<!-- plug_comp -->
+<g id="node40" class="node">
+<title>plug_comp</title>
+<g id="a_node40"><a xlink:title="plug_comp theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-199" rx="35.59" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-196.8" font-family="Inter" font-size="9.00" fill="#222222">plug_comp</text>
+</a>
+</g>
+</g>
+<!-- plug_comp&#45;&gt;steps_complete -->
+<g id="edge35" class="edge">
+<title>plug_comp&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M438.12,-209.33C448.02,-213.52 458.97,-218.83 468.21,-225 497.91,-244.83 526.96,-274.15 545.18,-294.12"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="543.77,-295.68 549.35,-298.73 546.88,-292.87 543.77,-295.68"/>
+</g>
+<!-- positions_comp -->
+<g id="node41" class="node">
+<title>positions_comp</title>
+<g id="a_node41"><a xlink:title="positions_comp theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-355" rx="47.19" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-352.8" font-family="Inter" font-size="9.00" fill="#222222">positions_comp</text>
+</a>
+</g>
+</g>
+<!-- positions_comp&#45;&gt;steps_complete -->
+<g id="edge36" class="edge">
+<title>positions_comp&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M448.72,-345.16C470.13,-339.75 496.79,-333.02 519.1,-327.39"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="519.67,-329.41 524.97,-325.91 518.64,-325.34 519.67,-329.41"/>
+</g>
+<!-- at_ctx_comp -->
+<g id="node42" class="node">
+<title>at_ctx_comp</title>
+<g id="a_node42"><a xlink:title="at_ctx_comp theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="408.74" cy="-251" rx="40.33" ry="18"/>
+<text text-anchor="middle" x="408.74" y="-248.8" font-family="Inter" font-size="9.00" fill="#222222">at_ctx_comp</text>
+</a>
+</g>
+</g>
+<!-- at_ctx_comp&#45;&gt;steps_complete -->
 <g id="edge34" class="edge">
-<title>m6_sn_typable&#45;&gt;corollary_6_21_psn</title>
-<path fill="none" stroke="#333333" d="M1831.08,-468.64C1848.14,-468.64 1868.25,-468.64 1886.23,-468.64"/>
-<polygon fill="#333333" stroke="#333333" points="1886.25,-472.14 1896.25,-468.64 1886.25,-465.14 1886.25,-472.14"/>
+<title>at_ctx_comp&#45;&gt;steps_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M438.79,-263.23C463.76,-273.73 499.91,-288.94 526.99,-300.34"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="526.28,-302.32 532.62,-302.71 527.91,-298.45 526.28,-302.32"/>
+</g>
+<!-- steps_complete&#45;&gt;has_red_spec -->
+<g id="edge40" class="edge">
+<title>steps_complete&#45;&gt;has_red_spec</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M599.84,-328.59C624.67,-337.88 658.46,-350.52 684.09,-360.11"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="683.4,-362.09 689.75,-362.23 684.87,-358.16 683.4,-362.09"/>
+</g>
+<!-- corpus_complete -->
+<g id="node48" class="node">
+<title>corpus_complete</title>
+<g id="a_node48"><a xlink:title="corpus_complete theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="722.21" cy="-316" rx="51.27" ry="18"/>
+<text text-anchor="middle" x="722.21" y="-313.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_complete</text>
+</a>
+</g>
+</g>
+<!-- steps_complete&#45;&gt;corpus_complete -->
+<g id="edge44" class="edge">
+<title>steps_complete&#45;&gt;corpus_complete</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M613.13,-316C629.32,-316 647.63,-316 664.51,-316"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="664.66,-318.1 670.66,-316 664.66,-313.9 664.66,-318.1"/>
+</g>
+<!-- has_red_f -->
+<g id="node44" class="node">
+<title>has_red_f</title>
+<g id="a_node44"><a xlink:title="has_red_f theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="565.21" cy="-374" rx="32.16" ry="18"/>
+<text text-anchor="middle" x="565.21" y="-371.8" font-family="Inter" font-size="9.00" fill="#222222">has_red_f</text>
+</a>
+</g>
+</g>
+<!-- has_red_f&#45;&gt;has_red_spec -->
+<g id="edge39" class="edge">
+<title>has_red_f&#45;&gt;has_red_spec</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M597.57,-374C619.26,-374 648.54,-374 673.15,-374"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="673.31,-376.1 679.31,-374 673.31,-371.9 673.31,-376.1"/>
+</g>
+<!-- red1_dec -->
+<g id="node46" class="node">
+<title>red1_dec</title>
+<g id="a_node46"><a xlink:title="red1_dec theory/Reduction.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="885.25" cy="-374" rx="30.7" ry="18"/>
+<text text-anchor="middle" x="885.25" y="-371.8" font-family="Inter" font-size="9.00" fill="#222222">red1_dec</text>
+</a>
+</g>
+</g>
+<!-- has_red_spec&#45;&gt;red1_dec -->
+<g id="edge42" class="edge">
+<title>has_red_spec&#45;&gt;red1_dec</title>
+<path fill="none" stroke="#9a9a9a" stroke-width="0.8" d="M765.1,-374C790.96,-374 823.79,-374 848.43,-374"/>
+<polygon fill="#9a9a9a" stroke="#9a9a9a" stroke-width="0.8" points="848.51,-376.1 854.51,-374 848.51,-371.9 848.51,-376.1"/>
+</g>
+<!-- corpus_decides -->
+<g id="node49" class="node">
+<title>corpus_decides</title>
+<g id="a_node49"><a xlink:title="corpus_decides theory/Tests.v https://arxiv.org/html/2312.13270v1">
+<ellipse fill="#d9ead3" stroke="#38761d" cx="885.25" cy="-426" rx="47.84" ry="18"/>
+<text text-anchor="middle" x="885.25" y="-423.8" font-family="Inter" font-size="9.00" fill="#222222">corpus_decides</text>
+</a>
+</g>
+</g>
+<!-- has_red_spec&#45;&gt;corpus_decides -->
+<g id="edge45" class="edge">
+<title>has_red_spec&#45;&gt;corpus_decides</title>
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diff --git a/graphs/theorems.json b/graphs/theorems.json
index f36a280..5f664a4 100644
--- a/graphs/theorems.json
+++ b/graphs/theorems.json
@@ -1,272 +1,592 @@
 {
   "paper": {
     "title": "Milner's Lambda-Calculus with Partial Substitutions",
-    "authors": ["Delia Kesner", "Shane Ó Conchúir"],
+    "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",
+  "milestones": [
+    "M1",
+    "M2"
+  ],
+  "current": "M2",
   "nodes": [
     {
-      "id": "exec_step",
-      "label": "exec_step",
+      "id": "fvs_lift",
+      "label": "fvs_lift",
       "kind": "infra",
       "status": "proved",
-      "milestone": "M0",
+      "milestone": "M1",
       "section": "",
-      "file": "theory/ExecReducer.v",
+      "file": "theory/Binding.v",
       "depends": []
     },
     {
-      "id": "exec_examples",
-      "label": "exec_examples",
+      "id": "lift_lc",
+      "label": "lift_lc",
       "kind": "infra",
       "status": "proved",
-      "milestone": "M0",
+      "milestone": "M1",
       "section": "",
-      "file": "theory/ExecReducer.v",
-      "depends": ["exec_step"]
+      "file": "theory/Binding.v",
+      "depends": []
     },
     {
-      "id": "m1_syntax",
-      "label": "M1 syntax/binding",
+      "id": "lift_0_lc",
+      "label": "lift_0_lc",
       "kind": "infra",
-      "status": "planned",
+      "status": "proved",
       "milestone": "M1",
-      "section": "2",
-      "file": "",
-      "depends": ["exec_examples"]
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": [
+        "lift_lc"
+      ]
     },
     {
-      "id": "m1_open_close",
-      "label": "open/close laws",
+      "id": "open_rec_bvar",
+      "label": "open_rec_bvar",
       "kind": "infra",
-      "status": "planned",
+      "status": "proved",
       "milestone": "M1",
-      "section": "2",
-      "file": "",
-      "depends": ["m1_syntax"]
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
     },
     {
-      "id": "m1_red1",
-      "label": "red1 (B, Gc, R)",
+      "id": "close_rec_fvar",
+      "label": "close_rec_fvar",
       "kind": "infra",
-      "status": "planned",
+      "status": "proved",
       "milestone": "M1",
-      "section": "2",
-      "file": "",
-      "depends": ["m1_syntax", "m1_open_close"]
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
     },
     {
-      "id": "lemma_2_1_fv_preserved",
-      "label": "Lemma 2.1 fv preserved",
-      "kind": "paper",
-      "status": "planned",
+      "id": "open_rec_occurs_false",
+      "label": "open_rec_occurs_false",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
+    },
+    {
+      "id": "close_rec_notin",
+      "label": "close_rec_notin",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
+    },
+    {
+      "id": "close_rec_fvs",
+      "label": "close_rec_fvs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
+    },
+    {
+      "id": "open_rec_bvar_neq",
+      "label": "open_rec_bvar_neq",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
+    },
+    {
+      "id": "open_rec_fvs",
+      "label": "open_rec_fvs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": [
+        "fvs_lift",
+        "open_rec_bvar",
+        "open_rec_bvar_neq"
+      ]
+    },
+    {
+      "id": "open_close",
+      "label": "open_close",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": [
+        "open_rec_bvar"
+      ]
+    },
+    {
+      "id": "subst_fvar_self",
+      "label": "subst_fvar_self",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": [
+        "lift_0_lc"
+      ]
+    },
+    {
+      "id": "subst_fvar_other",
+      "label": "subst_fvar_other",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": []
+    },
+    {
+      "id": "subst_fvs",
+      "label": "subst_fvs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Binding.v",
+      "depends": [
+        "close_rec_fvs",
+        "open_rec_fvs"
+      ]
+    },
+    {
+      "id": "Plus_red1_context",
+      "label": "Plus_red1_context",
+      "kind": "infra",
+      "status": "proved",
       "milestone": "M2",
-      "section": "2",
-      "file": "",
-      "depends": ["m1_red1"]
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": []
+    },
+    {
+      "id": "occurs_count_zero",
+      "label": "occurs_count_zero",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": []
+    },
+    {
+      "id": "occurs_count_false",
+      "label": "occurs_count_false",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": []
+    },
+    {
+      "id": "occurs_count_pos",
+      "label": "occurs_count_pos",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "occurs_count_zero"
+      ]
+    },
+    {
+      "id": "occurs_lift_self",
+      "label": "occurs_lift_self",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": []
+    },
+    {
+      "id": "zdecs_nonempty",
+      "label": "zdecs_nonempty",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": []
+    },
+    {
+      "id": "occurs_count_zfill_plug",
+      "label": "occurs_count_zfill_plug",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "occurs_count_false",
+        "occurs_lift_self"
+      ]
+    },
+    {
+      "id": "open_rec_zplug_lift",
+      "label": "open_rec_zplug_lift",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "occurs_lift_self",
+        "open_rec_occurs_false"
+      ]
+    },
+    {
+      "id": "full_comp_aux",
+      "label": "full_comp_aux",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "occurs_count_zero",
+        "occurs_count_zfill_plug",
+        "open_rec_occurs_false",
+        "open_rec_zplug_lift",
+        "zdecs_nonempty",
+        "zdecs_sound"
+      ]
     },
     {
       "id": "lemma_2_2_full_comp",
-      "label": "Lemma 2.2 full composition",
+      "label": "lemma_2_2_full_comp",
       "kind": "paper",
-      "status": "planned",
+      "status": "proved",
       "milestone": "M2",
-      "section": "2",
-      "file": "",
-      "depends": ["m1_red1"]
+      "section": "2.2",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "full_comp_aux"
+      ]
+    },
+    {
+      "id": "fvs_zplug_lift",
+      "label": "fvs_zplug_lift",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "fvs_lift"
+      ]
+    },
+    {
+      "id": "plug_fv_mono",
+      "label": "plug_fv_mono",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": []
+    },
+    {
+      "id": "red1_root_fv",
+      "label": "red1_root_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "fvs_zplug_lift"
+      ]
+    },
+    {
+      "id": "red1r_fv",
+      "label": "red1r_fv",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "plug_fv_mono",
+        "red1_root_fv"
+      ]
+    },
+    {
+      "id": "lemma_2_1_fv_preserved",
+      "label": "lemma_2_1_fv_preserved",
+      "kind": "paper",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "2.1",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "red1r_fv"
+      ]
     },
     {
       "id": "lemma_2_3_beta_sim",
-      "label": "Lemma 2.3 β-simulation",
+      "label": "lemma_2_3_beta_sim",
       "kind": "paper",
-      "status": "planned",
+      "status": "proved",
       "milestone": "M2",
-      "section": "2",
-      "file": "",
-      "depends": ["m1_red1", "lemma_2_2_full_comp"]
+      "section": "2.3",
+      "file": "theory/Metatheory.v",
+      "depends": [
+        "Plus_red1_context",
+        "lemma_2_2_full_comp"
+      ]
     },
     {
-      "id": "m3_measure",
-      "label": "sub measure",
+      "id": "zdecs_sound",
+      "label": "zdecs_sound",
       "kind": "infra",
-      "status": "planned",
-      "milestone": "M3",
-      "section": "A",
-      "file": "",
-      "depends": ["m1_red1"]
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "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",
+      "id": "zdecs_complete",
+      "label": "zdecs_complete",
       "kind": "infra",
-      "status": "planned",
-      "milestone": "M4",
-      "section": "3.2",
-      "file": "",
-      "depends": ["m1_red1", "lemma_3_2_sub_nf_unique"]
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "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",
+      "id": "positions_at",
+      "label": "positions_at",
       "kind": "infra",
-      "status": "planned",
-      "milestone": "M5",
-      "section": "3.3",
-      "file": "",
-      "depends": ["m1_syntax"]
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "id": "m5_rx",
-      "label": "RX rule",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M5",
-      "section": "3.3",
-      "file": "",
-      "depends": ["m5_mvar", "m1_red1"]
+      "id": "root_steps_sound",
+      "label": "root_steps_sound",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "zdecs_sound"
+      ]
     },
     {
-      "id": "m5_eq_C",
-      "label": "=C setoid",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M5",
-      "section": "3.3",
-      "file": "",
-      "depends": ["m5_mvar"]
+      "id": "root_steps_complete",
+      "label": "root_steps_complete",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "zdecs_complete"
+      ]
     },
     {
-      "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": "steps_sound",
+      "label": "steps_sound",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "positions_at",
+        "root_steps_sound"
+      ]
     },
     {
-      "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": "steps_sound_red1",
+      "label": "steps_sound_red1",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "steps_sound"
+      ]
     },
     {
-      "id": "m6_translation",
-      "label": "translation to λβp",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M6",
-      "section": "4",
-      "file": "",
-      "depends": ["m1_red1"]
+      "id": "positions_hole",
+      "label": "positions_hole",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "id": "m6_sn",
-      "label": "SN for simply typed",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M6",
-      "section": "6",
-      "file": "",
-      "depends": ["m6_translation"]
+      "id": "root_steps_in_steps",
+      "label": "root_steps_in_steps",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "positions_hole"
+      ]
     },
     {
-      "id": "m6_intersection",
-      "label": "intersection typing",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M6",
-      "section": "6",
-      "file": "",
-      "depends": ["m6_sn"]
+      "id": "plug_comp",
+      "label": "plug_comp",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "id": "m6_typable_sn",
-      "label": "typable → SN",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M6",
-      "section": "6",
-      "file": "",
-      "depends": ["m6_intersection"]
+      "id": "positions_comp",
+      "label": "positions_comp",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "id": "m6_sn_typable",
-      "label": "SN → typable",
-      "kind": "paper",
-      "status": "planned",
-      "milestone": "M6",
-      "section": "6",
-      "file": "",
-      "depends": ["m6_typable_sn"]
+      "id": "at_ctx_comp",
+      "label": "at_ctx_comp",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
     },
     {
-      "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"]
+      "id": "steps_complete",
+      "label": "steps_complete",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "at_ctx_comp",
+        "plug_comp",
+        "positions_comp",
+        "root_steps_complete",
+        "root_steps_in_steps"
+      ]
+    },
+    {
+      "id": "has_red_f",
+      "label": "has_red_f",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": []
+    },
+    {
+      "id": "has_red_spec",
+      "label": "has_red_spec",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "has_red_f",
+        "steps_complete",
+        "steps_sound"
+      ]
+    },
+    {
+      "id": "red1_dec",
+      "label": "red1_dec",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M1",
+      "section": "",
+      "file": "theory/Reduction.v",
+      "depends": [
+        "has_red_spec"
+      ]
+    },
+    {
+      "id": "corpus_sound",
+      "label": "corpus_sound",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Tests.v",
+      "depends": [
+        "steps_sound"
+      ]
+    },
+    {
+      "id": "corpus_complete",
+      "label": "corpus_complete",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Tests.v",
+      "depends": [
+        "steps_complete"
+      ]
+    },
+    {
+      "id": "corpus_decides",
+      "label": "corpus_decides",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Tests.v",
+      "depends": [
+        "has_red_spec"
+      ]
+    },
+    {
+      "id": "corpus_fv_preserved",
+      "label": "corpus_fv_preserved",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M2",
+      "section": "",
+      "file": "theory/Tests.v",
+      "depends": [
+        "lemma_2_1_fv_preserved",
+        "steps_sound_red1"
+      ]
     }
   ]
 }
diff --git a/scripts/audit.sh b/scripts/audit.sh
index a7f6d35..7a4166a 100755
--- a/scripts/audit.sh
+++ b/scripts/audit.sh
@@ -33,9 +33,11 @@ while IFS= read -r -d '' f; do
 done < <(find theory -name '*.v' -print0 2>/dev/null)
 
 echo "== Print Assumptions =="
-for f in ExecReducer Binding Reduction; do
+WORK="$(mktemp -d ./.audit-work.XXXXXX)"
+cp theory/*.v "$WORK"/
+for f in ExecReducer Binding Reduction Metatheory Tests; do
   echo "-- $f"
-  out="$(rocq compile -Q theory LambdaSub "theory/$f.v" 2>&1 || true)"
+  out="$(rocq compile -Q "$WORK" LambdaSub "$WORK/$f.v" 2>&1 || true)"
   echo "$out"
   if echo "$out" | grep -q "Axioms:"; then
     echo "unexpected axioms reported in $f" >&2
@@ -46,6 +48,7 @@ for f in ExecReducer Binding Reduction; do
     fail=1
   fi
 done
+rm -rf "$WORK"
 
 if [ "$fail" -ne 0 ]; then
   echo "AUDIT FAIL"
diff --git a/scripts/difftest.sh b/scripts/difftest.sh
new file mode 100755
index 0000000..3fc69fa
--- /dev/null
+++ b/scripts/difftest.sh
@@ -0,0 +1,20 @@
+#!/usr/bin/env bash
+set -euo pipefail
+ROOT="$(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
+
+dune build
+./scripts/audit.sh
+echo "CONFORMANCE OK: 7 term corpus checked against the relational semantics by theory/Tests.v"
+echo "DIFFTEST OK"
diff --git a/scripts/extract_metadata.py b/scripts/extract_metadata.py
index 425950f..211d60c 100755
--- a/scripts/extract_metadata.py
+++ b/scripts/extract_metadata.py
@@ -85,7 +85,7 @@ def extract(theory_dir):
         with open(path, "r", encoding="utf-8") as fh:
             text = fh.read()
         for decl in split_decls(text):
-            if decl["kind"] not in ("Lemma", "Theorem", "Corollary", "Proposition", "Example"):
+            if decl["kind"] not in ("Lemma", "Theorem", "Corollary", "Proposition"):
                 continue
             name = decl["name"]
             nodes.append(
diff --git a/scripts/gen_graphs.py b/scripts/gen_graphs.py
index 7dd2502..9e73456 100755
--- a/scripts/gen_graphs.py
+++ b/scripts/gen_graphs.py
@@ -42,48 +42,59 @@ def load(path):
         return json.load(f)
 
 
+def esc_html(s):
+    return str(s).replace("&", "&amp;").replace("<", "&lt;").replace(">", "&gt;")
+
+
 def dependency_dot(data):
     nodes = data["nodes"]
     by_id = {n["id"]: n for n in nodes}
     title = data["paper"]["title"] + " - theorem dependency (" + data["current"] + ")"
+    legend = (
+        "<B>" + esc_html(title) + "</B>"
+        "<BR/><FONT POINT-SIZE=\"9\">"
+        "<FONT COLOR=\"#38761d\">proved</FONT>   "
+        "<FONT COLOR=\"#b45f06\">stated</FONT>   "
+        "<FONT COLOR=\"#777777\">planned</FONT>   "
+        "<FONT COLOR=\"#cc0000\">blocked</FONT>"
+        "</FONT>"
+    )
     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),
+        '  bgcolor="white";',
+        "  splines=spline;",
+        "  nodesep=0.22;",
+        "  ranksep=0.6;",
+        '  node [fontname=Inter, fontsize=9, fontcolor="#222222"];',
+        '  edge [fontname=Inter, fontsize=8, color="#9a9a9a", arrowsize=0.6, penwidth=0.8];',
+        "  graph [fontname=Inter, fontsize=12, labelloc=t, label=<%s>];" % legend,
     ]
-    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:
+        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"]
+        if n["kind"] == "paper":
+            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:
diff --git a/theory/Tests.v b/theory/Tests.v
new file mode 100644
index 0000000..1383a25
--- /dev/null
+++ b/theory/Tests.v
@@ -0,0 +1,77 @@
+From Stdlib Require Import List Bool Arith Lia PeanoNat.
+Import ListNotations.
+From LambdaSub Require Import ExecReducer Binding Reduction Metatheory.
+
+Definition t_id : trm := App (Lam (BVar 0)) (FVar 1).
+Definition t_dup : trm := App (Lam (App (BVar 0) (BVar 0))) (FVar 1).
+Definition t_dup_es : trm := ESub (App (BVar 0) (BVar 0)) (FVar 1).
+Definition t_gc : trm := ESub (FVar 2) (FVar 1).
+Definition t_nested : trm := ESub (ESub (BVar 0) (FVar 1)) (FVar 2).
+Definition t_capture : trm := App (Lam (Lam (BVar 1))) (FVar 1).
+Definition t_under : trm := ESub (Lam (BVar 1)) (FVar 2).
+
+Definition corpus : list trm :=
+  [t_id; t_dup; t_dup_es; t_gc; t_nested; t_capture; t_under].
+
+Example g_id_beta : has_rule RB (steps t_id) = true.
+Proof. reflexivity. Qed.
+
+Example g_dup_beta : has_rule RB (steps t_dup) = true.
+Proof. reflexivity. Qed.
+
+Example g_dup_es_two_R : count_rule RR (steps t_dup_es) = 2.
+Proof. reflexivity. Qed.
+
+Example g_dup_es_no_Gc : has_rule RGc (steps t_dup_es) = false.
+Proof. reflexivity. Qed.
+
+Example g_gc_gc : has_rule RGc (steps t_gc) = true.
+Proof. reflexivity. Qed.
+
+Example g_gc_no_R : has_rule RR (steps t_gc) = false.
+Proof. reflexivity. Qed.
+
+Example g_nested_gc : has_rule RGc (steps t_nested) = true.
+Proof. reflexivity. Qed.
+
+Example g_under_R : has_rule RR (steps t_under) = true.
+Proof. reflexivity. Qed.
+
+Example g_capture_beta : has_rule RB (steps t_capture) = true.
+Proof. reflexivity. Qed.
+
+Example g_normal_var : normal_form (FVar 5) = true.
+Proof. reflexivity. Qed.
+
+Example g_normal_lam : normal_form (Lam (BVar 0)) = true.
+Proof. reflexivity. Qed.
+
+Example g_capture_avoid_lam : subst 0 (FVar 1) (Lam (Lam (BVar 1))) = Lam (Lam (BVar 1)).
+Proof. reflexivity. Qed.
+
+Example g_capture_avoid_es : subst 0 (FVar 1) (ESub (FVar 0) (FVar 2)) = ESub (FVar 1) (FVar 2).
+Proof. reflexivity. Qed.
+
+Example g_occurs0_lam : occurs0 (Lam (BVar 0)) = false.
+Proof. reflexivity. Qed.
+
+Example g_occurs0_bvar : occurs0 (BVar 0) = true.
+Proof. reflexivity. Qed.
+
+Lemma corpus_sound : forall t p, In t corpus -> In p (steps t) -> red1r (fst p) t (snd p).
+Proof. intros t [r s] _ H. simpl. apply steps_sound. exact H. Qed.
+
+Lemma corpus_complete : forall t r t', In t corpus -> red1r r t t' -> In (r,t') (steps t).
+Proof. intros t r t' _ H. apply steps_complete. exact H. Qed.
+
+Lemma corpus_decides : forall t t', In t corpus -> has_red t t' = true <-> red1 t t'.
+Proof. intros t t' _. apply has_red_spec. Qed.
+
+Lemma corpus_fv_preserved : forall t p, In t corpus -> In p (steps t) ->
+  incl (fvs (snd p)) (fvs t).
+Proof. intros t [r s] _ H. simpl. apply (lemma_2_1_fv_preserved t s). exact (steps_sound_red1 t r s H). Qed.
+
+Print Assumptions corpus_sound.
+Print Assumptions corpus_complete.
+Print Assumptions corpus_decides.
+Print Assumptions corpus_fv_preserved.