{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-integer","semantic_family_id":"sf-integer","type":"concept","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"整數","en":"Integer"},
  "summary":{"zh-Hant":"整數由零、正整數與負整數組成，擴充自然數以表示相反方向與差額。"},
  "statement":{"zh-Hant":"整數集合 Z 包含 0、每個正自然數及其負數；符號 n∈Z 表示 n 是整數。"},
  "formula":{"tex":"n\\in Z","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>n</mi><mo>∈</mo><mi>Z</mi></mrow></math>","semantic_ast":{"type":"membership","element":{"type":"symbol","name":"n","glyph":"n"},"set":{"type":"symbol","name":"Z","glyph":"Z"}},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"d4b65f67ed8018e006b189012c175423ec108689e648e4b447599cd4a8e72c66"}},
  "assumptions":[{"id":"a1","zh-Hant":"本條目沿用 OCME 自然數包含 0 的慣例。"}],
  "symbols":[{"token":"n","role":"integer_candidate","role_zh":"候選整數","scope":"definition"},{"token":"Z","role":"integer_set","role_zh":"整數集合","scope":"definition"},{"token":"∈","role":"membership_relation","role_zh":"隸屬關係","scope":"definition"}],
  "dependencies":[{"id":"mko-natural-number","title_zh":"自然數","reason_zh":"正整數與零由自然數語言進入，負整數再加入相反數方向。"}],
  "explanation":{"paragraphs_zh":["自然數可用來計數；整數再加入負方向，使溫度差、海拔差、債務與位移等情境可用同一數系表示。","負號不是數字品質的評價，而是相對於零與加法相反數的結構記號。","本條目建立成員與符號邊界；加法、乘法、次序與整除的普遍性質需要各自的聲明與證據。"],"common_misconception_zh":"常見誤解是把負號當成減法操作本身、把所有帶小數點的數視為整數，或由幾個例子推得全部整數代數律。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"有限分類或算例不能取代整數構造與代數性質的普遍證明。","evidence_refs":[],"producers":[{"id":"lean-mathlib","adapter":"formal_system","config_path":"formal/lean/lakefile.toml","status":"configured"}]},
  "formalization":{"status":"not_formalized","target_systems":["Lean 4","Mathlib"],"next_obligation_zh":"綁定 Lean Int 的構造、自然數嵌入、負數與零的精確語義，再分離後續序與環性質。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-integer.md 整理；待人工審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-19"}
}