{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-equivalence-class",
  "semantic_family_id": "sf-equivalence-class",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "等價類"
  },
  "summary": {
    "zh-Hant": "固定完整等價關係下，元素a的等價類收集全部與a相關的載體元素。"
  },
  "statement": {
    "zh-Hant": "給完整有限載體X及已認證等價關係R，[a]={x屬X:aRx}。若aRb則[a]=[b]；任兩等價類相等或不交。不得從抽樣關係或未通過等價公理的relation形成canonical classes。"
  },
  "formula": {
    "tex": "K(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>K</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "K",
        "glyph": "K"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3580686c4878ce4be7e188b0d4b7e5f3cdb309d5e68a63777c6272f4c10d541e"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "載體0到n-1及完整relation pair集合明示，且先完整通過反身、對稱、傳遞。"
    },
    {
      "id": "a2",
      "zh-Hant": "class使用同一relation與carrier，不以圖像分群猜測。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "equivalence_class",
      "role_zh": "等價類",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "equivalence_relation",
      "role_zh": "完整等價關係",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-equivalence-relation",
      "title_zh": "等價關係",
      "reason_zh": "class只由同一已認證relation形成。"
    },
    {
      "id": "mko-set-membership",
      "title_zh": "集合隸屬",
      "reason_zh": "class membership等價於aRx。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "每個class是carrier子集。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "[a]精確列出所有aRx元素。",
      "相關代表元產生同一class。",
      "classes相等或互斥並覆蓋carrier；商集另立節點。"
    ],
    "common_misconception_zh": "從未認證relation直接分群，或把有交集但不相等的兩個等價類保留下來。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理指定完整有限等價關係；不建立商集運算或代表元良定性。",
    "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": "加入商集與有限環商環。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-equivalence-class.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-10-29 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}