{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-congruence",
  "semantic_family_id": "sf-congruence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "明示對應的非退化三角形全等"
  },
  "summary": {
    "zh-Hant": "限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。"
  },
  "statement": {
    "zh-Hant": "在同一等單位正交歐氏平面，A=(A1,A2,A3)、B=(B1,B2,B3) 都為非退化三角形，且對應明示為 Ai↔Bi。本輪以三組對應邊長全部相等的 SSS 切片判全等；可用平方長度精確比較，鏡射改變方向但不破壞全等。"
  },
  "formula": {
    "tex": "G(A,B)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>G</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "G",
        "glyph": "G"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        },
        {
          "type": "symbol",
          "name": "B",
          "glyph": "B"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "713ee05f6d1ff8e8a7a3cdf8686a166c7a492d0cde6b471b00eb544b3f06d1d2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "兩組點各自互異且不共線；除三角形資格題外，先驗非退化，不把退化點組當本條目的三角形。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫座標為安全整數，平方邊長依12、13、23順序；不猜標籤置換，不靠圖形外觀。"
    },
    {
      "id": "a3",
      "zh-Hant": "資格題以 (A2−A1,A3−A1) 的二階行列式非零判不共線；這不是繼承舊面積或畢氏 proof。"
    }
  ],
  "symbols": [
    {
      "token": "G",
      "role": "labeled_triangle_congruence",
      "role_zh": "本輪標記三角形全等關係",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "first_ordered_triangle",
      "role_zh": "第一標記非退化三角形",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "second_ordered_triangle",
      "role_zh": "明示對應的第二三角形",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-triangle",
      "title_zh": "三角形",
      "reason_zh": "先驗三個互異、不共線的點。"
    },
    {
      "id": "mko-distance-formula",
      "title_zh": "等單位正交座標的歐氏距離",
      "reason_zh": "以平方邊長比較對應長度。"
    },
    {
      "id": "mko-determinant",
      "title_zh": "實二階方陣的行列式",
      "reason_zh": "非共線的精確座標判別。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "本條目先採有明示標籤的 SSS 判定，不把無標記圖形的可能重排當成給定對應已成立。方向可以相反，不能以有向量號不同否定鏡射全等。",
      "任一對應邊平方長度不等即可否定本對應；首個不等見證按12、13、23順序選，不按任意遍歷。若都相等答無不等邊。否定這個對應不等於否定所有其他頂點排列。",
      "對應邊長題先確認全等再取指定合法邊；另一點及另一三角形的無關平移不產生新長度計算。資格題另外容許共線或重複點作不合格候選。"
    ],
    "common_misconception_zh": "忽略非退化和標記對應、猜圖形全等，或把鏡射造成的方向反轉當成不全等。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "標記整數三角形 SSS 切片可重算；一般等距映射、全等與 SSS 的形式橋未新增 Evidence。",
    "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": "橋接 EuclideanSpace 的非共線、對應距離相等與三角形 SSS，保留鏡射及標記對應。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-congruence.md 整理；2026-10-06 脈衝 R05（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}