{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-counterexample",
  "semantic_family_id": "sf-counterexample",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "反例與論域資格"
  },
  "summary": {
    "zh-Hant": "反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。"
  },
  "statement": {
    "zh-Hant": "對全稱陳述 ∀x∈U P(x)，c 是有效反例當且僅當 c∈U 且 P(c) 為假。一個有效反例即可否定該全稱陳述；沒有找到反例本身不是證明，除非論域有限且已完整驗證所有元素。"
  },
  "formula": {
    "tex": "P(c)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>c</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "c",
          "glyph": "c"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "fe90fe475370b0df5338a167c585b8aaac41e769ff62e1ed9a469661b59fcb2b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "全稱陳述的論域及述詞已明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "最小反例只指本輪有限整數反例集合的數值最小值；集合為空時明示不存在。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "predicate_under_challenge",
      "role_zh": "待檢驗的全稱述詞",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "candidate_counterexample",
      "role_zh": "候選反例",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-quantifier",
      "title_zh": "量詞與明示論域",
      "reason_zh": "反例否定的是綁定論域的全稱陳述。"
    },
    {
      "id": "mko-proof",
      "title_zh": "證明",
      "reason_zh": "需區分合法反駁、有限枚舉及一般證明義務。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先查 c 是否在 U，再查 P(c) 是否失敗。即使同一個代數公式在域外也可計算，域外失敗值仍不是這個全稱命題的反例。",
      "限制到 V⊆U 後，原反例可能離開 V 而失去資格；這只表示該見證不再有效，不等於新全稱陳述必定為真。",
      "有限整數論域可以枚舉全部失敗值、篩選候選集合並選最小值。最小值和『不存在』都以完整失敗集合重算，不從抽樣失敗／成功推論無限論域。"
    ],
    "common_misconception_zh": "以域外值反駁受限全稱，或以一段搜尋沒有找到反例當成普遍證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪有限見證可精確驗證；沒有聲稱所有一般命題的反例搜尋可判定或完備。",
    "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": "綁定見證的集合成員資格、否定述詞和受限全稱反駁；有限最小值另明示存在性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-counterexample.md 整理；2026-10-03 脈衝 R02（依使用者要求提前完成 2026-10-04 的流程），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}