{ "id": "rp.belnap4-embed-preserves-meet", "wikidata_q": null, "item_type": "law", "name": { "ja": "埋め込みembedDfumt8は、Belnap–Dunnのmeet（AND）をD-FUMT₈のand8に移す", "en": null }, "summary": { "ja": "Belnap–Dunnの四値論理FOUR（Belnap 1977 / Dunn 1976）の真理順序のmeetは、標準的な埋め込みembedDfumt8 : Belnap4 → Dfumt8を通してD-FUMT₈の8値AND演算（and8）に移る。Lean 4で証明し、`#print axioms`は依存する公理がないこと（`does not depend on any axioms`）を示す。", "en": null }, "formula": "∀ x y : Belnap4, embedDfumt8 (Belnap4.meet x y) = and8 (embedDfumt8 x) (embedDfumt8 y)", "expression_latex": "\\forall x, y \\in \\mathrm{Belnap4}.\\quad \\mathrm{embedDfumt8}(x \\sqcap y) \\;=\\; \\mathrm{and8}\\bigl(\\mathrm{embedDfumt8}(x),\\, \\mathrm{embedDfumt8}(y)\\bigr)", "symbols": [ { "name": "Belnap4", "meaning": "Belnap–Dunn四値論理の値型（T / F / B / N）", "type": "Type" }, { "name": "Dfumt8", "meaning": "D-FUMT₈八値論理の値型（TRUE / FALSE / BOTH / NEITHER / INFINITY / ZERO / FLOWING / SELF）", "type": "Type" }, { "name": "Belnap4.meet", "meaning": "Belnap–Dunnの真理順序におけるmeet（Belnap 1977 / Dunn 1976）", "type": "Belnap4 → Belnap4 → Belnap4" }, { "name": "embedDfumt8", "meaning": "canonical mapping T↦TRUE, F↦FALSE, B↦BOTH, N↦NEITHER", "type": "Belnap4 → Dfumt8" }, { "name": "and8", "meaning": "D-FUMT₈の8値AND演算（Rei-AIOS CollatzRei.Dfumt8Category）", "type": "Dfumt8 → Dfumt8 → Dfumt8" }, { "name": "x, y", "meaning": "Belnap4の任意の値", "type": "Belnap4" } ], "example": { "description": { "ja": "Lean 4で `embedDfumt8_preserves_meet` を証明し、`#print axioms` でaxiom profileを出力する（= lean-verify.ymlがGitHub Actionsで実行、本項目の検査状態はその出力から生成）。", "en": null }, "code": "import CollatzRei.Belnap4Embedding\nopen CollatzRei.Belnap4\n#check @embedDfumt8_preserves_meet\n#print axioms embedDfumt8_preserves_meet", "expected_output": "embedDfumt8_preserves_meet : ∀ (x y : Belnap4),\n embedDfumt8 (x.meet y) = CollatzRei.Dfumt8Category.and8 (embedDfumt8 x) (embedDfumt8 y)\n'CollatzRei.Belnap4.embedDfumt8_preserves_meet' does not depend on any axioms" }, "sources": [ { "title": "A useful four-valued logic", "author": "Belnap, N. D.", "year": 1977, "url": "https://doi.org/10.1007/978-94-010-1161-7_2", "wayback_url": null, "retrieved_at": null, "license": null, "license_note": "未確認（出版社Springer）" }, { "title": "Intuitive semantics for first-degree entailments and coupled trees", "author": "Dunn, J. M.", "year": 1976, "url": "https://doi.org/10.1007/BF00373152", "wayback_url": null, "retrieved_at": null, "license": null, "license_note": "未確認（出版社Springer）" }, { "title": "Rei-AIOS Lean 4 CollatzRei/Belnap4Embedding.lean", "author": "Fujimoto, N.", "year": 2026, "url": null, "wayback_url": null, "retrieved_at": null, "license": "AGPL-3.0 + Commercial" } ], "relations": [ { "kind": "related_to", "target_id": "rp.belnap4-embed-injective", "note": "両者ともLean 4 CollatzReiプロジェクト内でaxiom profileを検査した、同じ埋め込みembedDfumt8の性質に関する定理（本項目はmeet保存、関連項目は単射性）" }, { "kind": "related_to", "target_id": "rp.belnap4-embed-image", "note": "両者ともLean 4 CollatzReiプロジェクト内でaxiom profileを検査した、同じ埋め込みembedDfumt8の性質に関する定理（本項目はmeet保存、関連項目は像の特定）" } ], "verify_source_theorem": "CollatzRei.Belnap4.embedDfumt8_preserves_meet", "verify_source_file": "CollatzRei/Belnap4EmbeddingAxiomCheck.lean", "authors": [ { "kind": "bot", "name": "reipedia-renderer", "model": "deterministic-template-step2343", "contribution": "Generated HTML + JSON from source item JSON via Astro static build", "at": "2026-10-15T00:00:00+09:00" }, { "kind": "AI", "name": "Claude （Rei-AIOS）", "model": "claude-opus-4-7", "contribution": "項目本文（一行説明・式・記号表・走る最小例・出典・関係リンク）を起草", "at": "2026-10-15T00:00:00+09:00" }, { "kind": "AI", "name": "Claude （Rei-AIOS）", "model": "claude-opus-4-7", "contribution": "下書きの改訂（表記規則の適用、英語用語の日本語化、項目名の文言修正、and8の名前空間の明記、走る最小例の出力をLeanの実行結果に合わせる、関係項目の追加、作成者とレビューア記録の追加）", "at": "2026-10-27T00:00:00+09:00" }, { "kind": "human", "name": "藤本 伸樹", "contribution": "5項目の下書き範囲を指定", "at": "2026-10-15T00:00:00+09:00" } ], "corrections": [ { "id": "C-preserves-meet-corr-001", "date": "2026-10-31", "found_by": "claude-code", "found_by_stage": "post-publish", "target": "出典のauthor表記", "what_was_wrong": "出典「Rei-AIOS Lean 4 CollatzRei/Belnap4Embedding.lean」のauthorが '藤本 伸樹' と記録されていたが、本人（藤本さん）確認済の正統表記は 'Fujimoto, N.'。他6項目（rp.belnap4-embed-image / rp.belnap4-embed-injective / rp.dvir-bound-3-2 / rp.dvir-bound-3-4 / rp.dvir-bound-5-4 / rp.homogeneous-vanishing-fn）で同人のself-referenceは全て 'Fujimoto, N.' で統一されており、本項目のみ日本語表記のoutlierだった。", "fix": "出典のauthorを 'Fujimoto, N.' に修正。他6項目のauthor表記と一致。", "severity": "typo", "error_type": "citation-mismatch", "commit": "d862a6fb0890ab4dc586289814a7198a4f202e49", "recheck_status": "not_applicable" } ], "authorship": { "created_by": [ { "agent": "rei-aios-session", "model": "claude-opus-4-7", "session": "rei-aios-77", "role": "draft", "date": "2026-10-15" }, { "agent": "rei-aios-session", "model": "claude-opus-4-7", "session": "rei-aios-1b", "role": "revise", "date": "2026-10-27" } ], "reviewed_by": [ { "agent": "claude-code-subagent", "kind": "independent", "session": "claude-code-subagent:rei-aios-1b:preserves-meet-review:2026-10-27", "date": "2026-10-27" } ], "human_approver": "fujimoto-nobuki", "approval_date": "2026-10-28" }, "inspected_at": null, "reinspect_due": null, "inspector_error_rate": null, "inspection": { "state": "S2", "state_label": "形式証明済み (axiom-free)", "axioms": [], "is_zero_axiom": true, "is_mathlib_base": true, "has_sorry_ax": false, "has_native_decide": false, "has_user_axiom": false, "source": "gha", "generated_at": "2026-10-18T23:15:15.522Z", "gha_run_id": "39175017999", "lean_toolchain": "leanprover/lean4:v4.27.0", "verify_source_theorem": "CollatzRei.Belnap4.embedDfumt8_preserves_meet", "verify_source_file": "CollatzRei/Belnap4EmbeddingAxiomCheck.lean" } }