{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-equivalence-relation",
  "semantic_family_id": "sf-equivalence-relation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "等價關係"
  },
  "summary": {
    "zh-Hant": "固定載體上的關係若完整滿足反身、對稱與傳遞，則為等價關係。"
  },
  "statement": {
    "zh-Hant": "關係R在集合X上為等價關係，若每個x都有xRx；xRy推出yRx；xRy且yRz推出xRz。本輪在完整有限載體與完整關係對集合上逐項核對。"
  },
  "formula": {
    "tex": "E(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>E</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "E",
        "glyph": "E"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e8969e0b6be8a590333fd1d3cd6213198075146de9e07013a04fe0c1920fef25"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "載體為0到n-1，關係由互異有序對完整列出。"
    },
    {
      "id": "a2",
      "zh-Hant": "只有三項全通過後才輸出等價類；單一反例足以否證對應公理。"
    }
  ],
  "symbols": [
    {
      "token": "E",
      "role": "equivalence_record",
      "role_zh": "等價關係紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "binary_relation",
      "role_zh": "二元關係",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-relation",
      "title_zh": "二元關係",
      "reason_zh": "等價關係是固定載體上的關係。"
    },
    {
      "id": "mko-logical-connectives",
      "title_zh": "邏輯連接詞",
      "reason_zh": "公理含蘊涵與合取。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "三個公理量化全部載體元素。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "反身核全部對角對。",
      "對稱核每個關係對的反向。",
      "傳遞核所有可串接關係對。"
    ],
    "common_misconception_zh": "只看關係圖像分群，或只核反身與對稱就叫等價關係。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "等價類只在完整資格後導出。",
    "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": "加入等價類、商集與基數比較。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-equivalence-relation.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-26 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}