{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-diophantine-equation",
  "semantic_family_id": "sf-linear-diophantine-equation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性Diophantine方程"
  },
  "summary": {
    "zh-Hant": "非退化整數方程ax+by=c有解恰當gcd(a,b)整除c，全部解由一個整數參數給出。"
  },
  "statement": {
    "zh-Hant": "使用extended Euclidean Bézout證書構造particular solution；若d=gcd(a,b)整除c，全部解為x=x0+(b/d)t、y=y0-(a/d)t。a=b=0時c=0有所有整數對，c非零則無解且不得除以d=0。"
  },
  "formula": {
    "tex": "a(x)+b(y)=c",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mrow><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mrow><mi>b</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>=</mo><mi>c</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "addition",
        "operands": [
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "a",
              "glyph": "a"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "x",
                "glyph": "x"
              }
            ]
          },
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "b",
              "glyph": "b"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "y",
                "glyph": "y"
              }
            ]
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "c",
        "glyph": "c"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "d40d1a0e63d3546af4761ef4c579def7b5380a5bedfccabb77229d97b9cdd157"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "a,b,c為exact integers並保留負號與單一零係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "完整參數化依賴canonical Euclidean演算法而非有限搜尋。"
    }
  ],
  "symbols": [
    {
      "token": "t",
      "role": "integer_solution_parameter",
      "role_zh": "整數參數",
      "scope": "representative_formula"
    },
    {
      "token": "d",
      "role": "normalized_gcd",
      "role_zh": "非負最大公因數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-integer",
      "title_zh": "整數",
      "reason_zh": "解域。"
    },
    {
      "id": "mko-gcd-coprime",
      "title_zh": "最大公因數與互質",
      "reason_zh": "solvability criterion。"
    },
    {
      "id": "mko-linear-equation",
      "title_zh": "一元一次方程式",
      "reason_zh": "線性等式形式。"
    },
    {
      "id": "mko-euclidean-algorithm",
      "title_zh": "Euclidean演算法",
      "reason_zh": "Bézout構造與完整參數化。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "gcd divisibility同時是必要與充分條件。",
      "一組解不是全部解；步長必須除以normalized gcd。",
      "雙零係數是獨立退化分支。"
    ],
    "common_misconception_zh": "找到一組整數解便稱已列完，或在a=b=0時除以gcd零。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理二變數線性整數方程；高維格點與非線性Diophantine另列。",
    "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": "加入多變數整數線性系統與Smith normal form。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-diophantine-equation.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-06 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"
  }
}