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Sources were drawn from model memory of SST and GeoCoq; no network access and no Lean toolchain were available, so all proofs were verified by hand-tracing (and, for SST 7.13, by an additional coordinate check) rather than by compilation." } ], "@type": [ "http://www.w3.org/ns/prov#Activity" ], "http://www.w3.org/ns/prov#endedAtTime": [ { "@value": "2026-07-19", "@type": "http://www.w3.org/2001/XMLSchema#date" } ], "http://www.w3.org/ns/prov#startedAtTime": [ { "@value": "2026-07-18", "@type": "http://www.w3.org/2001/XMLSchema#date" } ], "http://www.w3.org/ns/prov#used": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/src-conversation" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/src-geocoq" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/src-sst" } ], "http://www.w3.org/ns/prov#wasAssociatedWith": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-claude" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-user" } ], "https://schema.org/name": [ { "@value": "Interactive formalization dialogue" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/src-conversation", "http://purl.org/dc/terms/description": [ { "@value": "Full transcript export (36 messages, participants 'Claude (Anthropic PBC)' and the human ORCID holder, per the export's own metadata) of the sandboxed formalization dialogue recorded as sub:activity; the source-of-record for this nanopublication's account of that session, ISCC-identified for tamper-evident reference." } ], "http://purl.org/iscc/terms/datahash": [ { "@value": "1e20ba75fff4f8f4c444367ab1a8b2a6dc174d8594a5e0a608629c783c3672ed9d13" } ], "http://purl.org/iscc/terms/generator": [ { "@value": "iscc-sdk - v0.9.4" } ], "http://purl.org/iscc/terms/iscc": [ { "@value": "ISCC:KACVLHAC56XUUSQOQMH3UGAUADDVMRS5VJT3THHCNW5HL77U7D2MIRA" } ], "http://purl.org/iscc/terms/metahash": [ { "@value": "1e203058b6a3e88931d06f05aaf71f1baef07909f330d8d6fa48952fa3a8184e823f" } ], "@type": [ "http://www.w3.org/ns/prov#Entity", "https://schema.org/CreativeWork" ], "https://schema.org/dateCreated": [ { "@value": "2026-07-18T08:47:56Z", "@type": "http://www.w3.org/2001/XMLSchema#dateTime" } ], "https://schema.org/dateModified": [ { "@value": "2026-07-19T07:51:20Z", "@type": "http://www.w3.org/2001/XMLSchema#dateTime" } ], "https://schema.org/encodingFormat": [ { "@value": "application/json" } ], "https://schema.org/name": [ { "@value": "Pythagorean theorem in Lean4 inner product spaces" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/src-geocoq", "http://purl.org/dc/terms/description": [ { "@value": "Used from model memory as the formalization map; no code consulted or ported in-session." } ], "@type": [ "http://www.w3.org/ns/prov#Entity" ], "https://schema.org/name": [ { "@value": "GeoCoq repository and papers (Narboux et al., 2006–)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/src-sst", "http://purl.org/dc/terms/description": [ { "@value": "Used from model memory as the architectural source; not consulted directly in-session." } ], "@type": [ "http://www.w3.org/ns/prov#Entity" ], "https://schema.org/name": [ { "@value": "Schwabhäuser, Szmielew, Tarski — Metamathematische Methoden in der Geometrie (Springer, 1983)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/activity-iscc", "http://purl.org/dc/terms/description": [ { "@value": "Generated the ISCC-CODE (ISO 24138) for Tarski.lean using iscc-sdk v0.9.4 via a standalone content-identification workflow (iscc-workflow/generate_iscc.py). Fetched the commit-pinned GitHub blob (johnmaxton/lean4-starter @ 90018aec70b2bb8950c1045cac443731825d79ef) and confirmed it is byte-identical (same SHA-256) to the local working copy before generating the code. Because the .lean extension is unrecognized by iscc-sdk's mediatype map, content-based text auto-detection (added to the workflow for this case) was used to process the file as text/plain rather than a raw-bytes fallback." } ], "@type": [ "http://www.w3.org/ns/prov#Activity" ], "http://www.w3.org/ns/prov#startedAtTime": [ { "@value": "2026-07-21", "@type": "http://www.w3.org/2001/XMLSchema#date" } ], "http://www.w3.org/ns/prov#used": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/artifact" } ], "http://www.w3.org/ns/prov#wasAssociatedWith": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-claude" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-user" } ], "https://schema.org/name": [ { "@value": "ISCC-CODE generation" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/activity-verify", "http://purl.org/dc/terms/description": [ { "@value": "A follow-up session with a working environment (network access, elan/Lean toolchains). Compiled `lean Tarski.lean` standalone and, separately, built it as a `lake` library target in two projects (pythagoras4, using leanprover/lean4-nightly:nightly-2023-05-16; and lean4-starter, using leanprover/lean4:v4.32.0). Found and fixed one error: the closing `example` used Mathlib's `∃!` notation, unavailable in core Lean 4, rewritten as its explicit unfolding. Both toolchains now build the file with zero errors and zero `sorry`s." } ], "@type": [ "http://www.w3.org/ns/prov#Activity" ], "http://www.w3.org/ns/prov#startedAtTime": [ { "@value": "2026-07-18", "@type": "http://www.w3.org/2001/XMLSchema#date" } ], "http://www.w3.org/ns/prov#used": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/artifact" } ], "http://www.w3.org/ns/prov#wasAssociatedWith": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-claude" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-user" } ], "https://schema.org/name": [ { "@value": "Build, fix, and compilation verification" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/assertion", "http://www.w3.org/ns/prov#wasGeneratedBy": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/activity" } ], "http://www.w3.org/ns/prov#wasInfluencedBy": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/activity-iscc" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/activity-verify" } ] } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/assertion", "@graph": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-claude", "http://purl.org/dc/terms/description": [ { "@value": "Drafted all Lean code and documentation; reconstructed and in places re-derived the proofs (see sub:faia). In a follow-up session, diagnosed and fixed the sole compile error and verified compilation under two toolchains (see sub:activity-verify). In a later session, generated the file's ISCC-CODE (see sub:activity-iscc). Not an author under prevailing scholarly norms; correctly credited by acknowledgment. All errors in the draft are attributable here, not to the mathematical tradition." } ], "@type": [ "http://www.w3.org/ns/prov#SoftwareAgent" ], "http://www.w3.org/ns/prov#hadRole": [ { "@id": "https://credit.niso.org/contributor-roles/software" }, { "@id": "https://credit.niso.org/contributor-roles/validation" }, { "@id": "https://credit.niso.org/contributor-roles/writing-original-draft" } ], "https://schema.org/name": [ { "@value": "Claude (Anthropic)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-euclid", "http://purl.org/dc/terms/description": [ { "@value": "The axiomatic ideal the artifact rebuilds; the original tower (Elements, Book I), including target propositions I.47–I.48 (Pythagoras and converse) and Definition I.10, echoed in the planned `Per` (SST ch. 8)." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Intellectual ancestor" } ], "https://schema.org/name": [ { "@value": "Euclid of Alexandria (fl. c. 300 BCE)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-geocoq", "http://purl.org/dc/terms/description": [ { "@value": "First machine-checked ascent of the SST development (Coq, from Narboux 2006 onward). The artifact reconstructs the GeoCoq architecture and navigated by its lemma map (l4_2 → inner_five_segment, l4_3 → cong_sub, l4_5 → cong3_construction, l4_6 → betw_transfer, l7_13 → reflect_cong, l7_15 → reflect_betw) but ports no GeoCoq code." } ], "@type": [ "http://www.w3.org/ns/prov#Organization" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Formal ancestor" } ], "https://schema.org/name": [ { "@value": "GeoCoq project (J. Narboux, M. Beeson, P. Boutry, G. Braun, C. Gries, P. Schreck, et al.)" } ], "https://schema.org/url": [ { "@id": "https://github.com/GeoCoq/GeoCoq" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-gupta", "http://purl.org/dc/terms/description": [ { "@value": "PhD dissertation, UC Berkeley 1965, under Tarski: axiom independence and simplification (several axioms are as lean as A1–A8 because of him), and the perpendicular and midpoint constructions without any continuity axiom — SST 8.18 and 8.22, the declared next stage of this artifact, not yet contained in it." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Axiom simplifier; author of the continuity-free constructions ahead" } ], "https://schema.org/name": [ { "@value": "Haragauri Narayan Gupta (1925–2016)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-hilbert", "http://purl.org/dc/terms/description": [ { "@value": "Grundlagen der Geometrie (1899): the modern axiomatic program, and the segment arithmetic (Streckenrechnung, cf. SST ch. 14–15) that is the declared future route from this artifact to Pythagoras." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Program architect" } ], "https://schema.org/name": [ { "@value": "David Hilbert (1862–1943)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-pasch", "http://purl.org/dc/terms/description": [ { "@value": "Discovered that order is an assumption, not a triviality (Vorlesungen über neuere Geometrie, 1882). Namesake of axiom A7 (`inner_pasch`), on which the whole of Stage 2 rests: SST 3.1 (betw_trivial, indirectly), 3.2 (betw_symm), 3.5 (betw_inner_trans)." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Axiom source" } ], "https://schema.org/name": [ { "@value": "Moritz Pasch (1843–1930)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-schwabhaeuser", "http://purl.org/dc/terms/description": [ { "@value": "Completed and published the treatise: Schwabhäuser, Szmielew, Tarski, 'Metamathematische Methoden in der Geometrie', Springer 1983. Every 'SST n.m' tag in the artifact cites all three authors through his numbering." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Systematizer and publisher" } ], "https://schema.org/name": [ { "@value": "Wolfram Schwabhäuser (1931–1985)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-szmielew", "http://purl.org/dc/terms/description": [ { "@value": "Part I of SST grew from her lectures; the artifact follows her development lemma-for-lemma: SST 2.1–2.5, 2.8, 2.11, 2.12 (congruence calculus, segment addition, construction uniqueness); 3.1–3.3, 3.5, 3.6(1), 3.6(2), 3.7(1), 3.7(2) (betweenness calculus); 4.2, 4.3, 4.5, 4.6 (congruence–betweenness bridge); 7.4, 7.5, 7.13, 7.15 (point reflection and its isometry). She died before publication; her share of the credit is frequently understated." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Development author (followed lemma-for-lemma)" } ], "https://schema.org/name": [ { "@value": "Wanda Szmielew (1918–1976)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-tarski", "http://purl.org/dc/terms/description": [ { "@value": "The axiom system formalized as the `Tarski` typeclass, axioms A1–A8 (cong_pseudo_refl, cong_inner_trans, cong_identity, segment_construction, five_segment, betw_identity, inner_pasch, lower_dim), developed 1926–27; completeness and decidability of elementary geometry, guaranteeing the synthetic development agrees with EuclideanSpace ℝ (Fin n) on all first-order sentences. Survey citation: Tarski & Givant, 'Tarski's System of Geometry', Bulletin of Symbolic Logic 5(2), 1999." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@value": "Axiom-system author; metatheorist" } ], "https://schema.org/name": [ { "@value": "Alfred Tarski (1901–1983)" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-user", "http://purl.org/dc/terms/description": [ { "@value": "Set the organizing question and metaphor (being 'trapped' in EuclideanSpace ℝ (Fin n) with only straightedge and compass), chose each descent (perpendiculars → compiling file → SST ch. 4 block → SST 7.13), and supplied the persistence that drove the artifact to zero `sorry`s. Directed the follow-up session that found a working Lean toolchain, compiled the artifact, and requested the fix for the sole compile error, and later directed the ISCC-CODE generation session. Holds the director's and verifier's credit." } ], "@type": [ "http://www.w3.org/ns/prov#Person" ], "http://www.w3.org/ns/prov#hadRole": [ { "@id": "https://credit.niso.org/contributor-roles/conceptualization" }, { "@id": "https://credit.niso.org/contributor-roles/supervision" }, { "@id": "https://credit.niso.org/contributor-roles/validation" } ], "https://schema.org/identifier": [ { "@id": "https://orcid.org/0000-0002-8042-4131" } ], "https://schema.org/name": [ { "@value": "Myles Axton" } ], "https://schema.org/sameAs": [ { "@id": "https://orcid.org/0000-0002-8042-4131" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/artifact", "http://purl.org/dc/terms/contributor": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-claude" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-euclid" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-geocoq" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-gupta" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-hilbert" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-pasch" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-schwabhaeuser" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-szmielew" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-tarski" }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/agent-user" } ], "http://purl.org/dc/terms/relation": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/dedication" } ], "http://purl.org/dc/terms/tableOfContents": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/sst-map" } ], "http://purl.org/faia/terms/declaration": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/faia" } ], "http://purl.org/iscc/terms/datahash": [ { "@value": "1e20eb61415a5383e8037259c33ad1bc4fe912516562a125c862e03648571b86d938" } ], "http://purl.org/iscc/terms/generator": [ { "@value": "iscc-sdk - v0.9.4" } ], "http://purl.org/iscc/terms/iscc": [ { "@value": "ISCC:KAC67GWDV372567ANMUX3VYVTO7YIYHV2AILIDWJV3VWCQK2KOB6QAY" } ], "http://purl.org/iscc/terms/metahash": [ { "@value": "1e20bde6645eb61ec39e972274a1bd94924c754662a55ed8a84999eca3bd6eacdfc4" } ], "@type": [ "http://www.w3.org/ns/prov#Entity", "https://schema.org/SoftwareSourceCode" ], "https://schema.org/codeRepository": [ { "@id": "https://github.com/johnmaxton/lean4-starter/blob/90018aec70b2bb8950c1045cac443731825d79ef/Tarski.lean" } ], "https://schema.org/contentSize": [ { "@value": "28435 bytes (648 lines)" } ], "https://schema.org/description": [ { "@value": "A synthetic development of Tarski's Euclidean geometry following Schwabhäuser–Szmielew–Tarski (SST) chapters 2–7: eleven axioms over two primitives (betweenness, segment congruence), developed through SST 7.13/7.15 (point reflection is an isometry). Zero `sorry`s. Compiles cleanly (`lean Tarski.lean` and `lake build`, exit 0) under both leanprover/lean4-nightly:nightly-2023-05-16 and leanprover/lean4:v4.32.0, confirming the dependency-free (core-Lean-4-only) claim across toolchain versions." } ], "https://schema.org/encodingFormat": [ { "@value": "text/plain" } ], "https://schema.org/name": [ { "@value": "Tarski.lean" } ], "https://schema.org/programmingLanguage": [ { "@value": "Lean 4 (core language only; no Mathlib dependency)" } ], "https://schema.org/sha256": [ { "@value": "1372181e2284465648cf15fdd9b7a761ceef9288224feccfd71fb63a5417bfd1" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/dedication", "http://purl.org/dc/terms/description": [ { "@value": "Myles Axton thanks his father, Richard Axton (1941–2021), for his introduction to Euclidean geometry and the inspiration to do more with less (straight edge and compass only)." } ], "@type": [ "https://schema.org/Comment" ], "https://schema.org/about": [ { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/person-richard-axton" } ], "https://schema.org/name": [ { "@value": "Dedication" } ] }, { "@id": "https://w3id.org/np/RAZnSuo6CBwVYALiaZ45kg_9T9Oq_EBAKwbSb5SW8iduc/sst-map", "@type": [ "https://schema.org/Dataset" ], "https://schema.org/description": [ { "@value": "Lean name = SST number [= GeoCoq name where applicable]: cong_refl = 2.1; cong_symm = 2.2; cong_trans = 2.3; cong_left_comm = 2.4; cong_right_comm = 2.5; cong_trivial = 2.8; cong_add = 2.11 [l2_11]; construction_uniqueness = 2.12; betw_trivial = 3.1; betw_symm = 3.2; betw_left_trivial = 3.3; betw_inner_trans = 3.5; betw_exchange_left = 3.6(1); betw_exchange2 = 3.6(2) [between_exchange2]; betw_outer_trans = 3.7(1); betw_outer_trans' = 3.7(2); Cong3 = Def. 4.1 [Cong_3]; inner_five_segment = 4.2 [l4_2]; cong_sub = 4.3 [l4_3]; cong3_construction = 4.5 [l4_5]; betw_transfer = 4.6 [l4_6]; symmetric_point_exists = 7.4; symmetric_point_uniqueness = 7.5; reflect_cong = 7.13 [l7_13]; reflect_betw = 7.15 [l7_15]. 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