{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-quotient-set",
  "semantic_family_id": "sf-quotient-set",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "商集"
  },
  "summary": {
    "zh-Hant": "完整等價關係的全部互異等價類組成商集，並分割原載體。"
  },
  "statement": {
    "zh-Hant": "X/R是所有distinct equivalence classes的集合，不是挑選代表元的集合。classes覆蓋X且兩兩互斥；canonical projection把每個元素送到其class。任何商運算仍需另證代表元無關。"
  },
  "formula": {
    "tex": "Q(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>Q</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "Q",
        "glyph": "Q"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "170e66d3832f99fffcc323747f54a68f719e2b95c9bf7bbc68d72a21ca886a8f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限載體及已認證等價關係。"
    },
    {
      "id": "a2",
      "zh-Hant": "classes以排序元素陣列正規化並去除重複。"
    }
  ],
  "symbols": [
    {
      "token": "Q",
      "role": "quotient_set",
      "role_zh": "商集",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "equivalence_relation",
      "role_zh": "等價關係",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-equivalence-class",
      "title_zh": "等價類",
      "reason_zh": "商集元素是互異classes。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "商集本身為集合。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "canonical projection為總映射。"
    },
    {
      "id": "mko-equivalence-relation",
      "title_zh": "等價關係",
      "reason_zh": "classes來自同一relation。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "列出全部互異classes。",
      "canonical projection值是class而非代表元。",
      "覆蓋與pairwise disjoint形成partition；商運算另證well-defined。"
    ],
    "common_misconception_zh": "把代表元清單當商集，或未證代表元無關就繼承運算。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只構造有限商集與projection；不自動賦予群／環運算。",
    "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": "加入有限環商環與代表元良定性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-quotient-set.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-10-30 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"
  }
}