initial commit: confluence and normalisation
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_build/
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*.vo
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*.vok
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*.vos
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*.glob
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*.aux
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*.cmi
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*.cmo
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*.cmx
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__pycache__/
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*.pyc
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This was (mostly) me being bored in a Monday evening and in an attempt to formalise in Rocq the confluence and normalisation results that Delia Kesner and Shane Ó Conchúir obtain for the lambda calculus of Milner with partial substitutions which is available as arXiv:2312.13270.
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## Graphs
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The dependency graph covers the six milestones and every edge runs from a dependency to the result that uses it while the node colour encodes the status proved or stated or planned or blocked:
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The rule sketch gives the transition system generated by the rules B and Gc and R:
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The reduction graph gives the bounded reduct set of the term that applies the duplicating identity to a variable with every edge labelled by its rule and with the normal form marked:
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(lang dune 3.21)
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(name lambda_sub)
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(using rocq 0.11)
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(version 0.1.0)
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digraph theorem_deps {
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rankdir=LR;
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bgcolor="white";
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||||
node [fontname=Inter, fontsize=10, fontcolor="#333333"];
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||||
edge [fontname=Inter, fontsize=8, color="#333333"];
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graph [fontname=Inter, fontsize=12, labelloc=t, label="Milner's Lambda-Calculus with Partial Substitutions - theorem dependency (M0)"];
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subgraph cluster_M0 {
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||||
label="M0";
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||||
style="rounded";
|
||||
color="#bbbbbb";
|
||||
fontcolor="#333333";
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||||
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];
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||||
}
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||||
subgraph cluster_M1 {
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label="M1";
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||||
style="rounded";
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||||
color="#bbbbbb";
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||||
fontcolor="#333333";
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||||
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];
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||||
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];
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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];
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}
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subgraph cluster_M2 {
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label="M2";
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style="rounded";
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color="#bbbbbb";
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fontcolor="#333333";
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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];
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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];
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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];
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||||
}
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||||
subgraph cluster_M3 {
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||||
label="M3";
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||||
style="rounded";
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color="#bbbbbb";
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fontcolor="#333333";
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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];
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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];
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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];
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||||
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];
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}
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subgraph cluster_M4 {
|
||||
label="M4";
|
||||
style="rounded";
|
||||
color="#bbbbbb";
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||||
fontcolor="#333333";
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||||
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];
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||||
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];
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||||
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];
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||||
}
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||||
subgraph cluster_M5 {
|
||||
label="M5";
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||||
style="rounded";
|
||||
color="#bbbbbb";
|
||||
fontcolor="#333333";
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||||
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];
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||||
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];
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||||
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];
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||||
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];
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||||
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];
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}
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||||
subgraph cluster_M6 {
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||||
label="M6";
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||||
style="rounded";
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||||
color="#bbbbbb";
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||||
fontcolor="#333333";
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||||
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];
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||||
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];
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||||
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];
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||||
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];
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||||
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];
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||||
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];
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||||
}
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||||
exec_step -> exec_examples;
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exec_examples -> m1_syntax;
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m1_syntax -> m1_open_close;
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m1_syntax -> m1_red1;
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m1_open_close -> m1_red1;
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m1_red1 -> lemma_2_1_fv_preserved;
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m1_red1 -> lemma_2_2_full_comp;
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m1_red1 -> lemma_2_3_beta_sim;
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lemma_2_2_full_comp -> lemma_2_3_beta_sim;
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m1_red1 -> m3_measure;
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m3_measure -> m3_termination;
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m3_termination -> m3_local_confluence;
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m1_red1 -> m3_local_confluence;
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m3_local_confluence -> lemma_3_2_sub_nf_unique;
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m1_red1 -> m4_par;
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lemma_3_2_sub_nf_unique -> m4_par;
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m4_par -> m4_par_diamond;
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m4_par_diamond -> theorem_confluence_terms;
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lemma_2_3_beta_sim -> theorem_confluence_terms;
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m1_syntax -> m5_mvar;
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m5_mvar -> m5_rx;
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m1_red1 -> m5_rx;
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m5_mvar -> m5_eq_C;
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m5_rx -> m5_coherence;
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||||
m5_eq_C -> m5_coherence;
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lemma_3_2_sub_nf_unique -> m5_coherence;
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m5_coherence -> theorem_confluence_metaterms;
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m4_par_diamond -> theorem_confluence_metaterms;
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m1_red1 -> m6_translation;
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||||
m6_translation -> m6_sn;
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m6_sn -> m6_intersection;
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m6_intersection -> m6_typable_sn;
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m6_typable_sn -> m6_sn_typable;
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m6_sn_typable -> corollary_6_21_psn;
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theorem_confluence_terms -> corollary_6_21_psn;
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||||
}
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||||
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<text text-anchor="middle" x="996.18" y="-531.04" font-family="Inter" font-size="12.00">Milner's Lambda-Calculus with Partial Substitutions - theorem dependency (M0)</text>
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<g id="clust1" class="cluster">
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<text text-anchor="middle" x="468.37" y="-260.04" font-family="Inter" font-size="12.00" fill="#333333">M1</text>
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<g id="clust3" class="cluster">
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<title>cluster_M2</title>
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<text text-anchor="middle" x="862.78" y="-137.04" font-family="Inter" font-size="12.00" fill="#333333">M2</text>
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</g>
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<g id="clust4" class="cluster">
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<title>cluster_M3</title>
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<g id="clust5" class="cluster">
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<title>cluster_M4</title>
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<text text-anchor="middle" x="1624.32" y="-256.04" font-family="Inter" font-size="12.00" fill="#333333">M4</text>
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<g id="clust6" class="cluster">
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<title>cluster_M5</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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<g id="clust7" class="cluster">
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<title>cluster_M6</title>
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<!-- exec_step -->
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<g id="node1" class="node">
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</g>
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<!-- exec_examples -->
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<!-- exec_step->exec_examples -->
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<g id="edge1" class="edge">
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<title>exec_step->exec_examples</title>
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<!-- m1_syntax -->
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||||
<g id="node3" class="node">
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<title>m1_syntax</title>
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<g id="a_node3"><a xlink:title="m1_syntax (section 2) https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="326.41" y="-224.14" font-family="Inter" font-size="10.00" fill="#333333">M1 syntax/binding</text>
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</a>
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||||
</g>
|
||||
</g>
|
||||
<!-- exec_examples->m1_syntax -->
|
||||
<g id="edge2" class="edge">
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<title>exec_examples->m1_syntax</title>
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<!-- m1_open_close -->
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||||
<g id="node4" class="node">
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||||
<title>m1_open_close</title>
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||||
<g id="a_node4"><a xlink:title="m1_open_close (section 2) https://arxiv.org/html/2312.13270v1">
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<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="477.2" cy="-206.64" rx="55.17" ry="18"/>
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<text text-anchor="middle" x="477.2" y="-204.14" font-family="Inter" font-size="10.00" fill="#333333">open/close laws</text>
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||||
</a>
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</g>
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||||
</g>
|
||||
<!-- m1_syntax->m1_open_close -->
|
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<g id="edge3" class="edge">
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<title>m1_syntax->m1_open_close</title>
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<path fill="none" stroke="#333333" d="M381.12,-219.42C392.21,-217.93 403.97,-216.35 415.29,-214.83"/>
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<polygon fill="#333333" stroke="#333333" points="416.09,-218.25 425.54,-213.45 415.16,-211.31 416.09,-218.25"/>
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</g>
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<!-- m1_red1 -->
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<g id="node5" class="node">
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<title>m1_red1</title>
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<g id="a_node5"><a xlink:title="m1_red1 (section 2) https://arxiv.org/html/2312.13270v1">
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<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="619.16" cy="-206.64" rx="50.75" ry="18"/>
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<text text-anchor="middle" x="619.16" y="-204.14" font-family="Inter" font-size="10.00" fill="#333333">red1 (B, Gc, R)</text>
|
||||
</a>
|
||||
</g>
|
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</g>
|
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<!-- m1_syntax->m1_red1 -->
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<g id="edge4" class="edge">
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<title>m1_syntax->m1_red1</title>
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<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"/>
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<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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</g>
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<!-- m5_mvar -->
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<g id="node16" class="node">
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<title>m5_mvar</title>
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<g id="a_node16"><a xlink:title="m5_mvar (section 3.3) https://arxiv.org/html/2312.13270v1">
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<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="1292.78" cy="-363.64" rx="48.72" ry="18"/>
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<text text-anchor="middle" x="1292.78" y="-361.14" font-family="Inter" font-size="10.00" fill="#333333">XΔ metaterms</text>
|
||||
</a>
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</g>
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</g>
|
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<!-- m1_syntax->m5_mvar -->
|
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<g id="edge20" class="edge">
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<title>m1_syntax->m5_mvar</title>
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<polygon fill="#333333" stroke="#333333" points="1233.87,-366.03 1243.93,-362.71 1234,-359.03 1233.87,-366.03"/>
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</g>
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<!-- m1_open_close->m1_red1 -->
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<g id="edge5" class="edge">
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<title>m1_open_close->m1_red1</title>
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<path fill="none" stroke="#333333" d="M532.71,-206.64C541,-206.64 549.6,-206.64 558.01,-206.64"/>
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<polygon fill="#333333" stroke="#333333" points="558.13,-210.14 568.13,-206.64 558.13,-203.14 558.13,-210.14"/>
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</g>
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<!-- lemma_2_1_fv_preserved -->
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<g id="node6" class="node">
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<title>lemma_2_1_fv_preserved</title>
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<g id="a_node6"><a xlink:title="lemma_2_1_fv_preserved (section 2) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="844.78,-121.64 714.78,-121.64 714.78,-85.64 844.78,-85.64 844.78,-121.64"/>
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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>
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</a>
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</g>
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</g>
|
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<!-- m1_red1->lemma_2_1_fv_preserved -->
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<g id="edge6" class="edge">
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<title>m1_red1->lemma_2_1_fv_preserved</title>
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<path fill="none" stroke="#333333" d="M643.48,-190.76C658.84,-180.38 679.42,-166.57 697.78,-154.64 711.73,-145.58 727.15,-135.81 740.76,-127.27"/>
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<polygon fill="#333333" stroke="#333333" points="742.79,-130.13 749.41,-121.85 739.08,-124.19 742.79,-130.13"/>
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</g>
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<!-- lemma_2_2_full_comp -->
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<g id="node7" class="node">
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<title>lemma_2_2_full_comp</title>
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<g id="a_node7"><a xlink:title="lemma_2_2_full_comp (section 2) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="853.78,-67.64 705.78,-67.64 705.78,-31.64 853.78,-31.64 853.78,-67.64"/>
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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>
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</a>
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</g>
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</g>
|
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<!-- m1_red1->lemma_2_2_full_comp -->
|
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<g id="edge7" class="edge">
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<title>m1_red1->lemma_2_2_full_comp</title>
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<path fill="none" stroke="#333333" d="M626.18,-188.76C636.48,-160.84 660.09,-106.94 697.78,-76.64 699.34,-75.38 700.96,-74.19 702.63,-73.05"/>
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<polygon fill="#333333" stroke="#333333" points="704.6,-75.95 711.35,-67.78 700.98,-69.95 704.6,-75.95"/>
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</g>
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<!-- lemma_2_3_beta_sim -->
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<g id="node8" class="node">
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<title>lemma_2_3_beta_sim</title>
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<g id="a_node8"><a xlink:title="lemma_2_3_beta_sim (section 2) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1019.78,-67.64 889.78,-67.64 889.78,-31.64 1019.78,-31.64 1019.78,-67.64"/>
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<text text-anchor="middle" x="954.78" y="-47.14" font-family="Inter" font-size="10.00" fill="#333333">Lemma 2.3 β-simulation</text>
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</a>
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</g>
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</g>
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<!-- m1_red1->lemma_2_3_beta_sim -->
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<g id="edge8" class="edge">
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<title>m1_red1->lemma_2_3_beta_sim</title>
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<path fill="none" stroke="#333333" d="M621.5,-188.43C625.43,-149.91 640.58,-60.59 697.78,-22.64 760.49,18.97 851.47,-5.84 906.39,-27.7"/>
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<polygon fill="#333333" stroke="#333333" points="905.3,-31.03 915.88,-31.59 907.95,-24.56 905.3,-31.03"/>
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</g>
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<!-- m3_measure -->
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<g id="node9" class="node">
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<title>m3_measure</title>
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<g id="a_node9"><a xlink:title="m3_measure (section A) https://arxiv.org/html/2312.13270v1">
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<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="779.78" cy="-252.64" rx="45.83" ry="18"/>
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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>
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</a>
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</g>
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</g>
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<!-- m1_red1->m3_measure -->
|
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<g id="edge10" class="edge">
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<title>m1_red1->m3_measure</title>
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<path fill="none" stroke="#333333" d="M658.96,-217.89C681.19,-224.34 709.23,-232.47 732.59,-239.24"/>
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<polygon fill="#333333" stroke="#333333" points="731.7,-242.63 742.28,-242.05 733.65,-235.91 731.7,-242.63"/>
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</g>
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<!-- m3_local_confluence -->
|
||||
<g id="node11" class="node">
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<title>m3_local_confluence</title>
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<g id="a_node11"><a xlink:title="m3_local_confluence (section 3.1) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1185.78,-270.64 1055.78,-270.64 1055.78,-234.64 1185.78,-234.64 1185.78,-270.64"/>
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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>
|
||||
</a>
|
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</g>
|
||||
</g>
|
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<!-- m1_red1->m3_local_confluence -->
|
||||
<g id="edge13" class="edge">
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<title>m1_red1->m3_local_confluence</title>
|
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<path fill="none" stroke="#333333" d="M669.82,-205.16C746.98,-203.71 900.09,-204.24 1027.78,-225.64 1037.16,-227.21 1046.94,-229.43 1056.47,-231.92"/>
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<polygon fill="#333333" stroke="#333333" points="1055.72,-235.34 1066.29,-234.6 1057.56,-228.59 1055.72,-235.34"/>
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</g>
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<!-- m4_par -->
|
||||
<g id="node13" class="node">
|
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<title>m4_par</title>
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<g id="a_node13"><a xlink:title="m4_par (section 3.2) https://arxiv.org/html/2312.13270v1">
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<ellipse fill="none" stroke="#777777" stroke-dasharray="5,2" cx="1468.07" cy="-222.64" rx="68.07" ry="18"/>
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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>
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</g>
|
||||
</g>
|
||||
<!-- m1_red1->m4_par -->
|
||||
<g id="edge15" class="edge">
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<title>m1_red1->m4_par</title>
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<path fill="none" stroke="#333333" d="M664.23,-198.39C727.9,-187.22 849.43,-168.64 953.78,-168.64 953.78,-168.64 953.78,-168.64 1121.78,-168.64 1223.5,-168.64 1340.07,-191.8 1408.42,-207.78"/>
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<polygon fill="#333333" stroke="#333333" points="1408.01,-211.28 1418.55,-210.18 1409.63,-204.47 1408.01,-211.28"/>
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</g>
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<!-- m5_rx -->
|
||||
<g id="node17" class="node">
|
||||
<title>m5_rx</title>
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<g id="a_node17"><a xlink:title="m5_rx (section 3.3) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1495.07,-351.64 1441.07,-351.64 1441.07,-315.64 1495.07,-315.64 1495.07,-351.64"/>
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||||
<text text-anchor="middle" x="1468.07" y="-331.14" font-family="Inter" font-size="10.00" fill="#333333">RX rule</text>
|
||||
</a>
|
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</g>
|
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</g>
|
||||
<!-- m1_red1->m5_rx -->
|
||||
<g id="edge22" class="edge">
|
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<title>m1_red1->m5_rx</title>
|
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<path fill="none" stroke="#333333" d="M628.47,-224.44C640.27,-247.38 664.22,-286.37 697.78,-303.64 799.1,-355.78 839.83,-317.64 953.78,-317.64 953.78,-317.64 953.78,-317.64 1121.78,-317.64 1234.39,-317.64 1367.43,-326.23 1430.45,-330.83"/>
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<polygon fill="#333333" stroke="#333333" points="1430.43,-334.33 1440.66,-331.58 1430.95,-327.35 1430.43,-334.33"/>
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</g>
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<!-- m6_translation -->
|
||||
<g id="node21" class="node">
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<title>m6_translation</title>
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<g id="a_node21"><a xlink:title="m6_translation (section 4) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1169.78,-486.64 1071.78,-486.64 1071.78,-450.64 1169.78,-450.64 1169.78,-486.64"/>
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<text text-anchor="middle" x="1120.78" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">translation to λβp</text>
|
||||
</a>
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</g>
|
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</g>
|
||||
<!-- m1_red1->m6_translation -->
|
||||
<g id="edge29" class="edge">
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<title>m1_red1->m6_translation</title>
|
||||
<path fill="none" stroke="#333333" d="M626.69,-224.75C637.39,-251.88 661.18,-302.97 697.78,-331.64 808.02,-417.99 974.51,-450.79 1061.76,-462.55"/>
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<polygon fill="#333333" stroke="#333333" points="1061.38,-466.03 1071.75,-463.85 1062.28,-459.08 1061.38,-466.03"/>
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</g>
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<!-- lemma_2_2_full_comp->lemma_2_3_beta_sim -->
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<g id="edge9" class="edge">
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<title>lemma_2_2_full_comp->lemma_2_3_beta_sim</title>
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<path fill="none" stroke="#333333" d="M853.9,-49.64C862.36,-49.64 870.99,-49.64 879.46,-49.64"/>
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<polygon fill="#333333" stroke="#333333" points="879.7,-53.14 889.7,-49.64 879.7,-46.14 879.7,-53.14"/>
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</g>
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<!-- theorem_confluence_terms -->
|
||||
<g id="node15" class="node">
|
||||
<title>theorem_confluence_terms</title>
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<g id="a_node15"><a xlink:title="theorem_confluence_terms (section 3.2) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1848.85,-240.64 1731.85,-240.64 1731.85,-204.64 1848.85,-204.64 1848.85,-240.64"/>
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<text text-anchor="middle" x="1790.35" y="-220.14" font-family="Inter" font-size="10.00" fill="#333333">Confluence on terms</text>
|
||||
</a>
|
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</g>
|
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</g>
|
||||
<!-- lemma_2_3_beta_sim->theorem_confluence_terms -->
|
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<g id="edge19" class="edge">
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<title>lemma_2_3_beta_sim->theorem_confluence_terms</title>
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<path fill="none" stroke="#333333" d="M1019.91,-62.16C1087.32,-74.22 1196.58,-90.64 1291.78,-90.64 1291.78,-90.64 1291.78,-90.64 1469.07,-90.64 1581.25,-90.64 1699.98,-160.71 1756.17,-198.75"/>
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<polygon fill="#333333" stroke="#333333" points="1754.42,-201.79 1764.64,-204.56 1758.38,-196.02 1754.42,-201.79"/>
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</g>
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<!-- m3_termination -->
|
||||
<g id="node10" class="node">
|
||||
<title>m3_termination</title>
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<g id="a_node10"><a xlink:title="m3_termination (section 3.1) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1006.78,-270.64 902.78,-270.64 902.78,-234.64 1006.78,-234.64 1006.78,-270.64"/>
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<text text-anchor="middle" x="954.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">{R,Gc} termination</text>
|
||||
</a>
|
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</g>
|
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</g>
|
||||
<!-- m3_measure->m3_termination -->
|
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<g id="edge11" class="edge">
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<title>m3_measure->m3_termination</title>
|
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<path fill="none" stroke="#333333" d="M825.78,-252.64C846.09,-252.64 870.41,-252.64 892.36,-252.64"/>
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<polygon fill="#333333" stroke="#333333" points="892.59,-256.14 902.58,-252.64 892.58,-249.14 892.59,-256.14"/>
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</g>
|
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<!-- m3_termination->m3_local_confluence -->
|
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<g id="edge12" class="edge">
|
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<title>m3_termination->m3_local_confluence</title>
|
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<path fill="none" stroke="#333333" d="M1007.22,-252.64C1019.3,-252.64 1032.42,-252.64 1045.29,-252.64"/>
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<polygon fill="#333333" stroke="#333333" points="1045.53,-256.14 1055.53,-252.64 1045.53,-249.14 1045.53,-256.14"/>
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</g>
|
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<!-- lemma_3_2_sub_nf_unique -->
|
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<g id="node12" class="node">
|
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<title>lemma_3_2_sub_nf_unique</title>
|
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<g id="a_node12"><a xlink:title="lemma_3_2_sub_nf_unique (section 3.1) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1363.78,-270.64 1221.78,-270.64 1221.78,-234.64 1363.78,-234.64 1363.78,-270.64"/>
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<text text-anchor="middle" x="1292.78" y="-250.14" font-family="Inter" font-size="10.00" fill="#333333">Lemma 3.2 sub NF unique</text>
|
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</a>
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</g>
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</g>
|
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<!-- m3_local_confluence->lemma_3_2_sub_nf_unique -->
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<g id="edge14" class="edge">
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<title>m3_local_confluence->lemma_3_2_sub_nf_unique</title>
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<path fill="none" stroke="#333333" d="M1186.05,-252.64C1194.43,-252.64 1203.1,-252.64 1211.7,-252.64"/>
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<polygon fill="#333333" stroke="#333333" points="1211.74,-256.14 1221.74,-252.64 1211.74,-249.14 1211.74,-256.14"/>
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</g>
|
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<!-- lemma_3_2_sub_nf_unique->m4_par -->
|
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<g id="edge16" class="edge">
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<title>lemma_3_2_sub_nf_unique->m4_par</title>
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<path fill="none" stroke="#333333" d="M1364.26,-240.44C1376.15,-238.38 1388.46,-236.25 1400.24,-234.21"/>
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<polygon fill="#333333" stroke="#333333" points="1401.19,-237.6 1410.45,-232.44 1400,-230.7 1401.19,-237.6"/>
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</g>
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<!-- m5_coherence -->
|
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<g id="node19" class="node">
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<title>m5_coherence</title>
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<g id="a_node19"><a xlink:title="m5_coherence (section 3.3) https://arxiv.org/html/2312.13270v1">
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<polygon fill="none" stroke="#777777" stroke-dasharray="5,2" points="1684.35,-351.64 1572.35,-351.64 1572.35,-315.64 1684.35,-315.64 1684.35,-351.64"/>
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<text text-anchor="middle" x="1628.35" y="-331.14" font-family="Inter" font-size="10.00" fill="#333333">RX + =C coherence</text>
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</a>
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</g>
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</g>
|
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<!-- lemma_3_2_sub_nf_unique->m5_coherence -->
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<g id="edge26" class="edge">
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<title>lemma_3_2_sub_nf_unique->m5_coherence</title>
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<path fill="none" stroke="#333333" d="M1363.89,-267.68C1423.47,-280.57 1504.3,-298.33 1536.35,-306.64 1544.81,-308.83 1553.67,-311.28 1562.41,-313.77"/>
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<polygon fill="#333333" stroke="#333333" points="1561.6,-317.18 1572.18,-316.6 1563.55,-310.46 1561.6,-317.18"/>
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</g>
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<!-- m4_par_diamond -->
|
||||
<g id="node14" class="node">
|
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<title>m4_par_diamond</title>
|
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<g id="a_node14"><a xlink:title="m4_par_diamond (section 3.2) https://arxiv.org/html/2312.13270v1">
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<text text-anchor="middle" x="1790.35" y="-331.14" font-family="Inter" font-size="10.00" fill="#333333">Confluence on metaterms</text>
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<text text-anchor="middle" x="1468.07" y="-385.14" font-family="Inter" font-size="10.00" fill="#333333">=C setoid</text>
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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>
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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>
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<text text-anchor="middle" x="1628.35" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">typable → SN</text>
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<!-- m6_intersection->m6_typable_sn -->
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<g id="edge32" class="edge">
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<title>m6_intersection->m6_typable_sn</title>
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<text text-anchor="middle" x="1790.35" y="-466.14" font-family="Inter" font-size="10.00" fill="#333333">SN → typable</text>
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<g id="edge34" class="edge">
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<title>m6_sn_typable->corollary_6_21_psn</title>
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nf [label="y y (normal form)", shape=doublecircle, style=filled, fillcolor="#d9ead3", color="#38761d", fontcolor="#333333"];
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n0 -> n1 [label="B", color="#990000"];
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n1 -> n2b [label="R", color="#674ea7"];
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n2 -> n3 [label="R", color="#674ea7"];
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n2b -> n3 [label="R", color="#674ea7"];
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||||
{
|
||||
"id": "m6_typable_sn",
|
||||
"label": "typable → SN",
|
||||
"kind": "paper",
|
||||
"status": "planned",
|
||||
"milestone": "M6",
|
||||
"section": "6",
|
||||
"file": "",
|
||||
"depends": ["m6_intersection"]
|
||||
},
|
||||
{
|
||||
"id": "m6_sn_typable",
|
||||
"label": "SN → typable",
|
||||
"kind": "paper",
|
||||
"status": "planned",
|
||||
"milestone": "M6",
|
||||
"section": "6",
|
||||
"file": "",
|
||||
"depends": ["m6_typable_sn"]
|
||||
},
|
||||
{
|
||||
"id": "corollary_6_21_psn",
|
||||
"label": "Cor 6.21 PSN",
|
||||
"kind": "paper",
|
||||
"status": "planned",
|
||||
"milestone": "M6",
|
||||
"section": "6",
|
||||
"file": "",
|
||||
"depends": ["m6_sn_typable", "theorem_confluence_terms"]
|
||||
}
|
||||
]
|
||||
}
|
||||
Executable
+52
@@ -0,0 +1,52 @@
|
||||
#!/usr/bin/env bash
|
||||
set -euo pipefail
|
||||
ROOT="$(cd "$(dirname "$0")/.." && pwd)"
|
||||
cd "$ROOT"
|
||||
|
||||
if ! command -v rocq >/dev/null 2>&1; then
|
||||
if command -v opam >/dev/null 2>&1; then
|
||||
eval "$(opam env --switch=rocq-system --set-switch 2>/dev/null || true)"
|
||||
fi
|
||||
fi
|
||||
|
||||
if ! command -v rocq >/dev/null 2>&1; then
|
||||
echo "rocq not on PATH; run: eval \$(opam env --switch=rocq-system --set-switch)" >&2
|
||||
exit 1
|
||||
fi
|
||||
|
||||
fail=0
|
||||
|
||||
echo "== forbidden tokens =="
|
||||
while IFS= read -r -d '' f; do
|
||||
if grep -nE '\b(Admitted|admit|Axiom|Conjecture|native_compute)\b|TODO|FIXME|XXX' "$f"; then
|
||||
echo "forbidden token in $f" >&2
|
||||
fail=1
|
||||
fi
|
||||
done < <(find theory -name '*.v' -print0 2>/dev/null)
|
||||
|
||||
echo "== forbidden Parameter =="
|
||||
while IFS= read -r -d '' f; do
|
||||
if grep -nE '^[[:space:]]*Parameter[[:space:]]' "$f"; then
|
||||
echo "forbidden Parameter in $f" >&2
|
||||
fail=1
|
||||
fi
|
||||
done < <(find theory -name '*.v' -print0 2>/dev/null)
|
||||
|
||||
echo "== Print Assumptions =="
|
||||
mkdir -p /tmp/opencode/audit
|
||||
out="$(rocq compile -o /tmp/opencode/audit/ExecReducer.vo theory/ExecReducer.v 2>&1 || true)"
|
||||
echo "$out"
|
||||
if echo "$out" | grep -q "Axioms:"; then
|
||||
echo "unexpected axioms reported" >&2
|
||||
fail=1
|
||||
fi
|
||||
if ! echo "$out" | grep -q "Closed under the global context"; then
|
||||
echo "no Closed-under-global-context report found" >&2
|
||||
fail=1
|
||||
fi
|
||||
|
||||
if [ "$fail" -ne 0 ]; then
|
||||
echo "AUDIT FAIL"
|
||||
exit 1
|
||||
fi
|
||||
echo "AUDIT OK"
|
||||
Executable
+180
@@ -0,0 +1,180 @@
|
||||
#!/usr/bin/env python3
|
||||
import json
|
||||
import os
|
||||
import subprocess
|
||||
import sys
|
||||
|
||||
ROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
|
||||
GRAPH_DIR = os.path.join(ROOT, "graphs")
|
||||
META = os.path.join(GRAPH_DIR, "theorems.json")
|
||||
|
||||
STATUS_FILL = {
|
||||
"proved": "#d9ead3",
|
||||
"stated": "#fff2cc",
|
||||
"planned": "#f3f3f3",
|
||||
"blocked": "#f4cccc",
|
||||
}
|
||||
STATUS_PEN = {
|
||||
"proved": "#38761d",
|
||||
"stated": "#b45f06",
|
||||
"planned": "#777777",
|
||||
"blocked": "#cc0000",
|
||||
}
|
||||
STATUS_STYLE = {
|
||||
"proved": "filled",
|
||||
"stated": "filled,dashed",
|
||||
"planned": "dashed",
|
||||
"blocked": "filled,bold",
|
||||
}
|
||||
INK = "#333333"
|
||||
RULE_B = "#990000"
|
||||
RULE_R = "#674ea7"
|
||||
RULE_G = "#38761d"
|
||||
|
||||
|
||||
def q(s):
|
||||
return '"' + str(s).replace("\\", "\\\\").replace('"', '\\"') + '"'
|
||||
|
||||
|
||||
def load():
|
||||
with open(META, "r", encoding="utf-8") as f:
|
||||
return json.load(f)
|
||||
|
||||
|
||||
def dependency_dot(data):
|
||||
nodes = data["nodes"]
|
||||
by_id = {n["id"]: n for n in nodes}
|
||||
title = data["paper"]["title"] + " - theorem dependency (" + data["current"] + ")"
|
||||
lines = [
|
||||
"digraph theorem_deps {",
|
||||
" rankdir=LR;",
|
||||
" bgcolor=\"white\";",
|
||||
" node [fontname=Inter, fontsize=10, fontcolor=\"%s\"];" % INK,
|
||||
" edge [fontname=Inter, fontsize=8, color=\"%s\"];" % INK,
|
||||
" graph [fontname=Inter, fontsize=12, labelloc=t, label=%s];" % q(title),
|
||||
]
|
||||
for ms in data["milestones"]:
|
||||
members = [n for n in nodes if n["milestone"] == ms]
|
||||
if not members:
|
||||
continue
|
||||
lines.append(" subgraph cluster_%s {" % ms)
|
||||
lines.append(" label=%s;" % q(ms))
|
||||
lines.append(" style=\"rounded\";")
|
||||
lines.append(" color=\"#bbbbbb\";")
|
||||
lines.append(" fontcolor=\"%s\";" % INK)
|
||||
for n in members:
|
||||
fill = STATUS_FILL.get(n["status"], "#f3f3f3")
|
||||
pen = STATUS_PEN.get(n["status"], "#777777")
|
||||
style = STATUS_STYLE.get(n["status"], "dashed")
|
||||
tip = n["id"]
|
||||
if n.get("section"):
|
||||
tip += " (section " + n["section"] + ")"
|
||||
if n.get("file"):
|
||||
tip += " " + n["file"]
|
||||
tip += " " + data["paper"]["html"]
|
||||
attrs = [
|
||||
"label=%s" % q(n["label"]),
|
||||
"color=%s" % q(pen),
|
||||
"fillcolor=%s" % q(fill),
|
||||
"style=%s" % q(style),
|
||||
"fontcolor=%s" % q(INK),
|
||||
"tooltip=%s" % q(tip),
|
||||
"shape=%s" % ("box" if n["kind"] == "paper" else "ellipse"),
|
||||
]
|
||||
lines.append(" %s [%s];" % (n["id"], ", ".join(attrs)))
|
||||
lines.append(" }")
|
||||
for n in nodes:
|
||||
for d in n.get("depends", []):
|
||||
if d in by_id:
|
||||
lines.append(" %s -> %s;" % (d, n["id"]))
|
||||
lines.append("}")
|
||||
return "\n".join(lines) + "\n"
|
||||
|
||||
|
||||
def rules_dot():
|
||||
box = 'shape=box, style="rounded,filled", fillcolor="#f5f5f5", color="%s", fontcolor="%s"' % (INK, INK)
|
||||
lines = [
|
||||
"digraph lambda_sub_cfg {",
|
||||
" rankdir=TB;",
|
||||
" bgcolor=\"white\";",
|
||||
" node [fontname=Inter, fontsize=10];",
|
||||
" edge [fontname=Inter, fontsize=9, color=\"%s\"];" % INK,
|
||||
" start [label=\"term\", shape=oval, style=filled, fillcolor=\"#f3f3f3\", color=\"%s\"];" % INK,
|
||||
" beta [label=\"(lambda x. t) u\", %s];" % box,
|
||||
" es [label=\"t[x/u]\", %s];" % box,
|
||||
" pure [label=\"pure term\", shape=oval, style=filled, fillcolor=\"#f3f3f3\", color=\"%s\"];" % INK,
|
||||
" nf [label=\"normal form\", shape=doublecircle, style=filled, fillcolor=\"#d9ead3\", color=\"%s\"];" % RULE_G,
|
||||
" start -> beta [label=\"App(Lam,_)\"];",
|
||||
" start -> es [label=\"ESub\"];",
|
||||
" start -> pure [label=\"BVar/FVar/Lam\"];",
|
||||
" beta -> es [label=\"B\", color=\"%s\"];" % RULE_B,
|
||||
" es -> es [label=\"R (one occurrence)\", color=\"%s\"];" % RULE_R,
|
||||
" es -> pure [label=\"Gc (x not in fv)\", color=\"%s\"];" % RULE_G,
|
||||
" pure -> pure [label=\"B under context\", color=\"%s\"];" % RULE_B,
|
||||
" es -> nf [label=\"R/Gc normalisation\", color=\"%s\"];" % RULE_R,
|
||||
"}",
|
||||
]
|
||||
return "\n".join(lines) + "\n"
|
||||
|
||||
|
||||
def reduction_dot():
|
||||
box = 'shape=box, style="filled", fillcolor="#f5f5f5", color="%s", fontcolor="%s"' % (INK, INK)
|
||||
lines = [
|
||||
"digraph reduction_dup {",
|
||||
" rankdir=LR;",
|
||||
" bgcolor=\"white\";",
|
||||
" node [fontname=Inter, fontsize=10];",
|
||||
" edge [fontname=Inter, fontsize=9, color=\"%s\"];" % INK,
|
||||
" graph [labelloc=t, fontcolor=\"%s\", label=\"(lambda x. x x) y reduction graph\"];" % INK,
|
||||
" n0 [label=\"(lambda x. x x) y\", %s];" % box,
|
||||
" n1 [label=\"(x x)[x/y]\", %s];" % box,
|
||||
" n2 [label=\"(y x)[x/y]\", %s];" % box,
|
||||
" n2b [label=\"(x y)[x/y]\", %s];" % box,
|
||||
" n3 [label=\"(y y)[x/y]\", %s];" % box,
|
||||
" nf [label=\"y y (normal form)\", shape=doublecircle, style=filled, fillcolor=\"#d9ead3\", color=\"%s\", fontcolor=\"%s\"];" % (RULE_G, INK),
|
||||
" n0 -> n1 [label=\"B\", color=\"%s\"];" % RULE_B,
|
||||
" n1 -> n2 [label=\"R\", color=\"%s\"];" % RULE_R,
|
||||
" n1 -> n2b [label=\"R\", color=\"%s\"];" % RULE_R,
|
||||
" n2 -> n3 [label=\"R\", color=\"%s\"];" % RULE_R,
|
||||
" n2b -> n3 [label=\"R\", color=\"%s\"];" % RULE_R,
|
||||
" n3 -> nf [label=\"Gc\", color=\"%s\"];" % RULE_G,
|
||||
" {rank=same; n2; n2b;}",
|
||||
"}",
|
||||
]
|
||||
return "\n".join(lines) + "\n"
|
||||
|
||||
|
||||
def run_dot(path):
|
||||
svg = path[:-4] + ".svg"
|
||||
png = path[:-4] + ".png"
|
||||
ok = True
|
||||
for fmt, out in (("svg", svg), ("png", png)):
|
||||
r = subprocess.run(["dot", "-T" + fmt, path, "-o", out], capture_output=True, text=True)
|
||||
if r.returncode != 0:
|
||||
print(r.stderr, file=sys.stderr)
|
||||
ok = False
|
||||
else:
|
||||
print("wrote", os.path.relpath(out, ROOT))
|
||||
return ok
|
||||
|
||||
|
||||
def main():
|
||||
data = load()
|
||||
outputs = {
|
||||
"dependency.dot": dependency_dot(data),
|
||||
"rules.dot": rules_dot(),
|
||||
"reduction.dot": reduction_dot(),
|
||||
}
|
||||
ok = True
|
||||
for name, text in outputs.items():
|
||||
p = os.path.join(GRAPH_DIR, name)
|
||||
with open(p, "w", encoding="utf-8") as f:
|
||||
f.write(text)
|
||||
print("wrote", os.path.relpath(p, ROOT))
|
||||
if not run_dot(p):
|
||||
ok = False
|
||||
return 0 if ok else 1
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
sys.exit(main())
|
||||
@@ -0,0 +1,272 @@
|
||||
From Stdlib Require Import List Bool PeanoNat Lia.
|
||||
Import ListNotations.
|
||||
|
||||
Definition atom := nat.
|
||||
|
||||
Inductive trm : Type :=
|
||||
| BVar : nat -> trm
|
||||
| FVar : atom -> trm
|
||||
| App : trm -> trm -> trm
|
||||
| Lam : trm -> trm
|
||||
| ESub : trm -> trm -> trm.
|
||||
|
||||
Fixpoint fvs (t : trm) : list atom :=
|
||||
match t with
|
||||
| BVar _ => []
|
||||
| FVar x => [x]
|
||||
| App t1 t2 => fvs t1 ++ fvs t2
|
||||
| Lam t1 => fvs t1
|
||||
| ESub t1 t2 => fvs t1 ++ fvs t2
|
||||
end.
|
||||
|
||||
Definition in_fvs (x : atom) (t : trm) : bool :=
|
||||
existsb (Nat.eqb x) (fvs t).
|
||||
|
||||
Fixpoint lift (k : nat) (t : trm) : trm :=
|
||||
match t with
|
||||
| BVar n => if Nat.ltb n k then BVar n else BVar (S n)
|
||||
| FVar x => FVar x
|
||||
| App t1 t2 => App (lift k t1) (lift k t2)
|
||||
| Lam t1 => Lam (lift (S k) t1)
|
||||
| ESub t1 t2 => ESub (lift (S k) t1) (lift k t2)
|
||||
end.
|
||||
|
||||
Fixpoint open_rec (k : nat) (u : trm) (t : trm) : trm :=
|
||||
match t with
|
||||
| BVar n => if Nat.eqb n k then lift k u else BVar n
|
||||
| FVar x => FVar x
|
||||
| App t1 t2 => App (open_rec k u t1) (open_rec k u t2)
|
||||
| Lam t1 => Lam (open_rec (S k) u t1)
|
||||
| ESub t1 t2 => ESub (open_rec (S k) u t1) (open_rec k u t2)
|
||||
end.
|
||||
|
||||
Fixpoint close_rec (x : atom) (k : nat) (t : trm) : trm :=
|
||||
match t with
|
||||
| BVar n => BVar n
|
||||
| FVar y => if Nat.eqb y x then BVar k else FVar y
|
||||
| App t1 t2 => App (close_rec x k t1) (close_rec x k t2)
|
||||
| Lam t1 => Lam (close_rec x (S k) t1)
|
||||
| ESub t1 t2 => ESub (close_rec x (S k) t1) (close_rec x k t2)
|
||||
end.
|
||||
|
||||
Definition subst (x : atom) (u t : trm) : trm :=
|
||||
open_rec 0 u (close_rec x 0 t).
|
||||
|
||||
Fixpoint occurs (n : nat) (t : trm) : bool :=
|
||||
match t with
|
||||
| BVar m => Nat.eqb m n
|
||||
| FVar _ => false
|
||||
| App t1 t2 => occurs n t1 || occurs n t2
|
||||
| Lam t1 => occurs (S n) t1
|
||||
| ESub t1 t2 => occurs (S n) t1 || occurs n t2
|
||||
end.
|
||||
|
||||
Definition occurs0 (t : trm) : bool := occurs 0 t.
|
||||
|
||||
Inductive rule : Type := RB | RGc | RR.
|
||||
|
||||
Definition rule_eqb (r s : rule) : bool :=
|
||||
match r, s with
|
||||
| RB, RB => true
|
||||
| RGc, RGc => true
|
||||
| RR, RR => true
|
||||
| _, _ => false
|
||||
end.
|
||||
|
||||
Inductive zctx : Type :=
|
||||
| ZTop : zctx
|
||||
| ZAppL : zctx -> trm -> zctx
|
||||
| ZAppR : trm -> zctx -> zctx
|
||||
| ZLam : zctx -> zctx
|
||||
| ZESubL : zctx -> trm -> zctx
|
||||
| ZESubR : trm -> zctx -> zctx.
|
||||
|
||||
Fixpoint zplug (C : zctx) (k : nat) (t : trm) : trm :=
|
||||
match C with
|
||||
| ZTop => t
|
||||
| ZAppL C1 u => App (zplug C1 k t) u
|
||||
| ZAppR u C1 => App u (zplug C1 k t)
|
||||
| ZLam C1 => Lam (zplug C1 (S k) t)
|
||||
| ZESubL C1 u => ESub (zplug C1 (S k) t) u
|
||||
| ZESubR t1 C1 => ESub t1 (zplug C1 k t)
|
||||
end.
|
||||
|
||||
Fixpoint zplug_lift (C : zctx) (k : nat) (t : trm) : trm :=
|
||||
match C with
|
||||
| ZTop => lift k t
|
||||
| ZAppL C1 u => App (zplug_lift C1 k t) u
|
||||
| ZAppR u C1 => App u (zplug_lift C1 k t)
|
||||
| ZLam C1 => Lam (zplug_lift C1 (S k) t)
|
||||
| ZESubL C1 u => ESub (zplug_lift C1 (S k) t) u
|
||||
| ZESubR t1 C1 => ESub t1 (zplug_lift C1 k t)
|
||||
end.
|
||||
|
||||
Fixpoint zdecs (t : trm) (k : nat) : list zctx :=
|
||||
match t with
|
||||
| BVar n => if Nat.eqb n k then [ZTop] else []
|
||||
| FVar _ => []
|
||||
| App t1 t2 =>
|
||||
map (fun C => ZAppL C t2) (zdecs t1 k)
|
||||
++ map (fun C => ZAppR t1 C) (zdecs t2 k)
|
||||
| Lam t1 => map ZLam (zdecs t1 (S k))
|
||||
| ESub t1 t2 =>
|
||||
map (fun C => ZESubL C t2) (zdecs t1 (S k))
|
||||
++ map (fun C => ZESubR t1 C) (zdecs t2 k)
|
||||
end.
|
||||
|
||||
Definition root_steps (t : trm) : list (rule * trm) :=
|
||||
match t with
|
||||
| App (Lam body) u => [(RB, ESub body u)]
|
||||
| ESub body u =>
|
||||
(if occurs0 body then [] else [(RGc, body)])
|
||||
++ map (fun C => (RR, ESub (zplug_lift C 0 u) u)) (zdecs body 0)
|
||||
| _ => []
|
||||
end.
|
||||
|
||||
Inductive ctx : Type :=
|
||||
| GHole : ctx
|
||||
| GAppL : ctx -> trm -> ctx
|
||||
| GAppR : trm -> ctx -> ctx
|
||||
| GLam : ctx -> ctx
|
||||
| GESubL : ctx -> trm -> ctx
|
||||
| GESubR : trm -> ctx -> ctx.
|
||||
|
||||
Fixpoint plug (C : ctx) (t : trm) : trm :=
|
||||
match C with
|
||||
| GHole => t
|
||||
| GAppL C1 u => App (plug C1 t) u
|
||||
| GAppR u C1 => App u (plug C1 t)
|
||||
| GLam C1 => Lam (plug C1 t)
|
||||
| GESubL C1 u => ESub (plug C1 t) u
|
||||
| GESubR t1 C1 => ESub t1 (plug C1 t)
|
||||
end.
|
||||
|
||||
Fixpoint positions (t : trm) : list ctx :=
|
||||
GHole ::
|
||||
match t with
|
||||
| BVar _ => []
|
||||
| FVar _ => []
|
||||
| App t1 t2 =>
|
||||
map (fun C => GAppL C t2) (positions t1)
|
||||
++ map (fun C => GAppR t1 C) (positions t2)
|
||||
| Lam t1 => map GLam (positions t1)
|
||||
| ESub t1 t2 =>
|
||||
map (fun C => GESubL C t2) (positions t1)
|
||||
++ map (fun C => GESubR t1 C) (positions t2)
|
||||
end.
|
||||
|
||||
Fixpoint at_ctx (C : ctx) (t : trm) : option trm :=
|
||||
match C, t with
|
||||
| GHole, t => Some t
|
||||
| GAppL C1 _, App t1 _ => at_ctx C1 t1
|
||||
| GAppR _ C2, App _ t2 => at_ctx C2 t2
|
||||
| GLam C1, Lam t1 => at_ctx C1 t1
|
||||
| GESubL C1 _, ESub t1 _ => at_ctx C1 t1
|
||||
| GESubR _ C2, ESub _ t2 => at_ctx C2 t2
|
||||
| _, _ => None
|
||||
end.
|
||||
|
||||
Definition step_at (C : ctx) (t : trm) : list (rule * trm) :=
|
||||
match at_ctx C t with
|
||||
| None => []
|
||||
| Some s => map (fun p => (fst p, plug C (snd p))) (root_steps s)
|
||||
end.
|
||||
|
||||
Definition steps (t : trm) : list (rule * trm) :=
|
||||
flat_map (fun C => step_at C t) (positions t).
|
||||
|
||||
Definition has_rule (r : rule) (l : list (rule * trm)) : bool :=
|
||||
existsb (fun p => rule_eqb r (fst p)) l.
|
||||
|
||||
Definition count_rule (r : rule) (l : list (rule * trm)) : nat :=
|
||||
length (filter (fun p => rule_eqb r (fst p)) l).
|
||||
|
||||
Definition normal_form (t : trm) : bool :=
|
||||
match steps t with [] => true | _ => false end.
|
||||
|
||||
Definition ex_id : trm := App (Lam (BVar 0)) (FVar 1).
|
||||
Definition ex_dup : trm := App (Lam (App (BVar 0) (BVar 0))) (FVar 1).
|
||||
Definition ex_dup_es : trm := ESub (App (BVar 0) (BVar 0)) (FVar 1).
|
||||
Definition ex_es : trm := ESub (BVar 0) (FVar 1).
|
||||
Definition ex_gc : trm := ESub (FVar 2) (FVar 1).
|
||||
Definition ex_nested : trm := ESub (ESub (BVar 0) (FVar 1)) (FVar 2).
|
||||
|
||||
Example ex_id_beta : has_rule RB (steps ex_id) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_id_one_step : length (steps ex_id) = 1.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_dup_beta : has_rule RB (steps ex_dup) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_dup_es_two_R : count_rule RR (steps ex_dup_es) = 2.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_dup_es_no_Gc : has_rule RGc (steps ex_dup_es) = false.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_es_R : has_rule RR (steps ex_es) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_es_no_Gc : has_rule RGc (steps ex_es) = false.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_gc_gc : has_rule RGc (steps ex_gc) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_gc_no_R : has_rule RR (steps ex_gc) = false.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_nested_has_gc : has_rule RGc (steps ex_nested) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example ex_id_terminates : normal_form (plug GHole (FVar 5)) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example fvs_lam : fvs (Lam (App (BVar 0) (FVar 1))) = [1].
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example fvs_es : fvs (ESub (FVar 0) (FVar 1)) = [0; 1].
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example occurs0_lam : occurs0 (Lam (BVar 0)) = false.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example occurs0_bvar : occurs0 (BVar 0) = true.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example open_basic : open_rec 0 (FVar 1) (BVar 0) = FVar 1.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example open_under_lam : open_rec 0 (FVar 1) (Lam (BVar 1)) = Lam (FVar 1).
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example subst_hit : subst 0 (FVar 1) (FVar 0) = FVar 1.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example subst_miss : subst 0 (FVar 1) (FVar 2) = FVar 2.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example capture_avoid_1 :
|
||||
subst 0 (FVar 1) (Lam (BVar 0)) = Lam (BVar 0).
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example capture_avoid_2 :
|
||||
subst 0 (FVar 1) (Lam (App (BVar 0) (FVar 0)))
|
||||
= Lam (App (BVar 0) (FVar 1)).
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Example capture_avoid_es :
|
||||
subst 0 (FVar 1) (ESub (FVar 0) (FVar 2))
|
||||
= ESub (FVar 1) (FVar 2).
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Print Assumptions ex_id_beta.
|
||||
Print Assumptions ex_dup_es_two_R.
|
||||
Print Assumptions ex_gc_gc.
|
||||
Print Assumptions ex_nested_has_gc.
|
||||
Print Assumptions capture_avoid_1.
|
||||
Print Assumptions capture_avoid_2.
|
||||
Print Assumptions capture_avoid_es.
|
||||
Print Assumptions open_under_lam.
|
||||
@@ -0,0 +1,3 @@
|
||||
(rocq.theory
|
||||
(name LambdaSub)
|
||||
(theories Stdlib))
|
||||
Reference in new issue
Block a user