{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-program-standard-form",
  "semantic_family_id": "sf-linear-program-standard-form",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性規劃標準形"
  },
  "summary": {
    "zh-Hant": "本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。"
  },
  "statement": {
    "zh-Hant": "A為m×n exact rational matrix，b∈R^m、c∈R^n，decision x∈R^n。資格先核dimensions與sign convention。Ax≤b以nonnegative slack s=b-Ax轉為Ax+s=b，並在原點與slack點間雙向保存feasibility及objective c^Tx。unrestricted variable以x=x^+-x^-、x^±≥0 split；≥ constraint乘-1，equality可拆成兩個≤，minimize可negate objective。轉換不證原問題feasible、bounded或存在optimal solution，也不混用另一套standard-form convention。"
  },
  "formula": {
    "tex": "L(A,b,c)=z",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>L</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "A",
            "glyph": "A"
          },
          {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          },
          {
            "type": "symbol",
            "name": "c",
            "glyph": "c"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "z",
        "glyph": "z"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f811e3e9ba85e4e6793958c5e946fa95cbbc5393749c1c5c1a27ab4b90f98adf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有matrix/vector/points為bounded exact rationals。"
    },
    {
      "id": "a2",
      "zh-Hant": "題幹與輸出始終固定maximize/≤/nonnegative convention。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "linear_program_standard_form",
      "role_zh": "線性規劃標準形",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "constraint_matrix",
      "role_zh": "限制矩陣",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "right_hand_side",
      "role_zh": "右端向量",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "objective_vector",
      "role_zh": "目標向量",
      "scope": "representative_formula"
    },
    {
      "token": "z",
      "role": "objective_value",
      "role_zh": "目標值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-linear-programming",
      "title_zh": "線性規劃",
      "reason_zh": "objective與constraints。"
    },
    {
      "id": "mko-matrix",
      "title_zh": "矩陣",
      "reason_zh": "A與matrix-vector product。"
    },
    {
      "id": "mko-system-linear-equations",
      "title_zh": "聯立一次方程組",
      "reason_zh": "slack equality form。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "componentwise inequalities。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "唯一convention避免standard-form歧義。",
      "slack conversion附雙向point mapping。",
      "unrestricted、≥、equality與minimize分別明示conversion。"
    ],
    "common_misconception_zh": "混用maximize/minimize或≤/≥ convention，或把conversion誤當feasibility/optimality證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只證表示轉換與point correspondence；不宣稱可行、有界、最優或dual。",
    "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": "加入線性規劃對偶。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-program-standard-form.md 整理；2026-10-15 脈衝 R06（使用者手動接續 2026-11-14 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}