{ "id": "rp.belnap4-embed-image", "wikidata_q": null, "item_type": "law", "name": { "ja": "D-FUMT₈の値dがBelnap-Dunn FOURの埋め込みの像に入るのは、d ∈ {TRUE, FALSE, BOTH, NEITHER}のときに限る", "en": null }, "summary": { "ja": "標準的な埋め込みembedDfumt8 : Belnap4 → Dfumt8の像（= {embedDfumt8 b | b ∈ Belnap4}）は、D-FUMT₈の8値のうちTRUE / FALSE / BOTH / NEITHERの4値ちょうどに一致する。Lean 4で両方向の同値（`↔`）として形式化。依存する公理はpropext, Classical.choice, Quot.sound（mathlib base）。", "en": null }, "formula": "∀ d : Dfumt8, (∃ b : Belnap4, embedDfumt8 b = d) ↔ (d = TRUE ∨ d = FALSE ∨ d = BOTH ∨ d = NEITHER)", "expression_latex": "\\forall d \\in \\mathrm{Dfumt8}.\\quad \\bigl(\\exists b \\in \\mathrm{Belnap4}.\\ \\mathrm{embedDfumt8}(b) = d\\bigr) \\;\\Longleftrightarrow\\; \\bigl(d = \\mathrm{TRUE} \\lor d = \\mathrm{FALSE} \\lor d = \\mathrm{BOTH} \\lor d = \\mathrm{NEITHER}\\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": "embedDfumt8", "meaning": "canonical mapping T↦TRUE, F↦FALSE, B↦BOTH, N↦NEITHER", "type": "Belnap4 → Dfumt8" }, { "name": "d", "meaning": "D-FUMT₈の任意の値", "type": "Dfumt8" }, { "name": "b", "meaning": "Belnap4の存在量化変数", "type": "Belnap4" } ], "example": { "description": { "ja": "Lean 4で `embedDfumt8_image_characterization` 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_image_characterization\n#print axioms embedDfumt8_image_characterization", "expected_output": "embedDfumt8_image_characterization : ∀ (d : CollatzRei.Dfumt8Category.Dfumt8),\n (∃ b, embedDfumt8 b = d) ↔\n d = CollatzRei.Dfumt8Category.Dfumt8.TRUE ∨\n d = CollatzRei.Dfumt8Category.Dfumt8.FALSE ∨\n d = CollatzRei.Dfumt8Category.Dfumt8.BOTH ∨ d = CollatzRei.Dfumt8Category.Dfumt8.NEITHER\n'CollatzRei.Belnap4.embedDfumt8_image_characterization' depends on axioms: [propext, Classical.choice, Quot.sound]" }, "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": "単射性（関連項目）と像の特定（本項目）を合わせると、embedDfumt8はBelnap4とD-FUMT₈の4値｛TRUE, FALSE, BOTH, NEITHER｝の間の全単射になる" }, { "kind": "related_to", "target_id": "rp.belnap4-embed-preserves-meet", "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（本項目は像の特定、関連項目はmeet保存）" } ], "verify_source_theorem": "CollatzRei.Belnap4.embedDfumt8_image_characterization", "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": "human", "name": "藤本 伸樹", "contribution": "Specified 5-item draft scope (2026-10-15)", "at": "2026-10-15T00:00:00+09:00" } ], "corrections": [ { "id": "C-image-corr-001", "date": "2026-10-28", "found_by": "fujimoto-nobuki", "found_by_stage": "post-publication-audit", "target": "走る最小例の期待出力・作成者欄", "what_was_wrong": "(1) 走る最小例の期待出力を、Leanを実際に走らせずに書いていた。実際にLeanで実行すると、`open CollatzRei.Belnap4` の影響で短い名前（embedDfumt8_image_characterization）で表示され、定理の署名が複数行に折り返される。以前の期待出力では完全な名前（@CollatzRei.Belnap4.embedDfumt8_image_characterization）で、かつ単一行で書いていた。また、必要な公理の並び順も実際の出力と違っていた（公理の種類は同じ：propext・Classical.choice・Quot.sound の3つ）。定理の内容と必要な公理の集合は実機の出力と一致している。(2) 作成者欄の bot と AI の記述に、公開前の状態を表す文言（「draft, not yet published」「awaiting independent review」）が残っていた。", "fix": "走る最小例の期待出力を、実際にLeanで実行した結果に合わせて直しました。以前の期待出力は実行せずに書いたもので、定理名の表示の仕方（完全な名前か短い名前か）や改行の位置が実際の出力と違っていました。定理の内容、必要な公理、検査状態の表示は変わっていません。必要な公理の並び順も実際の出力に合わせました（公理の種類は同じです）。作成者欄の、公開前の状態を表す記述を直しました。", "severity": "typo", "error_type": "citation-mismatch", "commit": "2dfb14ea15c1bab822e4cee7badb2bac00925564", "recheck_status": "pending", "found_by_stage_revision": [ { "to": "post-publish", "by": "fujimoto-nobuki", "date": "2026-10-29", "reason": "STEP 2451 で 書いた 'post-publication-audit' は addendum-3 §2.1 の 3 値 enum (machine | independent-review | post-publish) 外、 正しくは 'post-publish' (= 公開後に発見)。 内容 (= 発見経緯 + 訂正内容) は 変わらない、 記録 の 値 の 文字列 の 直し。", "approval_commit": "b41db9e6b36a3fe66d5e2af8da76482c1fbda57a", "addendum_ref": "addendum-7-v1.0 §2" } ], "recheck_status_revision": [ { "to": "not_applicable", "by": "fujimoto-nobuki", "date": "2026-10-30", "reason": "STEP 2451 で severity=typo に 変更 した 際、 g-4 rule (= typo なら recheck_status=not_applicable を 要求) を 確認 せず recheck_status=pending の まま 残した。 内容 (= 訂正 事実 + learning_log C-image-corr-001-recheck-001 の recheck_pass 事実) は 変わらない、 記録 の 値 の 文字列 の 直し。 本 件 は learning_log 側 で 再確認 済 + 藤本さん approve 済。", "approval_commit": "29b353aa59dd0adbea4e0e2b5baf7609fe199387", "addendum_ref": "addendum-8-v1.0 §2" } ] } ], "learning_log": [ { "id": "C-image-corr-001-recheck-001", "date": "2026-10-28", "type": "recheck_pass", "correction_ref": "C-image-corr-001", "recheck_result": "passed", "evidence_source": "本訂正の後に、別の確認者がLean 4で例を実際に実行し、訂正後の期待出力と一字一句照合して一致を確認。", "change": "別の確認者がLean 4で例を実行し、訂正後の期待出力と実機の出力が一字一句一致することを確認しました。", "found_by": "claude-code-subagent", "session": "claude-code-subagent:rei-aios-1b:preserves_meet_siblings_lean_audit:2026-10-28", "commit": "8b9f1bf84a2f2b33b4226054d953749d1bac8ada", "human_approver": "fujimoto-nobuki", "approval_date": "2026-10-28" } ], "authorship": { "created_by": [ { "agent": "rei-aios-session", "model": "claude-opus-4-7", "session": "rei-aios-77", "role": "draft", "date": "2026-10-15" } ], "reviewed_by": [ { "agent": "claude-code-subagent", "kind": "independent", "session": "claude-code-subagent:rei-aios-9c:belnap4-embed-image-review:2026-10-27", "date": "2026-10-27" } ], "human_approver": "fujimoto-nobuki", "approval_date": "2026-10-27" }, "inspected_at": null, "reinspect_due": null, "inspector_error_rate": null, "inspection": { "state": "S2", "state_label": "形式証明済み (mathlib base)", "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "is_zero_axiom": false, "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_image_characterization", "verify_source_file": "CollatzRei/Belnap4EmbeddingAxiomCheck.lean" } }