{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-minimum-vertex-cover-01",
  "mko_id": "mko-minimum-vertex-cover",
  "title": {
    "zh-Hant": "P3全部八子集求minimum cover",
    "en": "All eight subsets for minimum cover on P3"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-minimum-vertex-cover-01",
    "source": {
      "mko_id": "mko-minimum-vertex-cover",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "4f8d510264312335a0f27c32a69f66f31350aede08398538e94a015583d8250b",
      "english_sha256": "0ec533e9f581c06215d447562ddc54c5249c8bf96d6f057f82089a97614d035c"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整分類所有covers與全部minimizers。",
      "en": "Classify every cover and all minimizers."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2}，E={01,12}。",
          "en": "V={0,1,2}, E={01,12}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "|V|=3",
        "after_tex": "2^3=8\\text{ vertex subsets}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每頂點選或不選，完整候選八個。",
          "en": "Each vertex is selected or not, giving eight complete candidates."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是只測幾個可能最優候選。",
            "en": "This is not a sample of plausible optimizers."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "C=\\varnothing",
        "after_tex": "01\\cap C=\\varnothing\\implies\\text{invalid}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空集合沒有覆蓋原邊01。",
          "en": "The empty set fails to cover original edge 01."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "圖有邊，所以空圖convention不適用。",
            "en": "The graph has edges, so the edgeless convention does not apply."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "C=\\{0\\},\\quad C=\\{2\\}",
        "after_tex": "12\\text{ uncovered},\\quad01\\text{ uncovered}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個端點singleton各給明確失敗邊。",
          "en": "Each endpoint singleton has an explicit uncovered edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩案都保留。",
            "en": "Both cases are retained."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "C=\\{1\\}",
        "after_tex": "01\\cap C=12\\cap C=\\{1\\}\\implies\\text{valid, size }1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中心singleton覆蓋兩邊。",
          "en": "The center singleton covers both edges."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "提供大小一上界但仍完成其餘分類。",
            "en": "It gives an upper bound of one while the remaining cases are still classified."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "C=\\{0,1\\},\\quad C=\\{1,2\\}",
        "after_tex": "\\text{both cover }01,12\\text{ via vertex }1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "含中心的兩個二點子集皆cover。",
          "en": "Both two-vertex sets containing the center are covers."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "它們刪掉端點仍cover，所以非minimal。",
            "en": "Deleting their endpoint still leaves a cover, so they are not minimal."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "C=\\{0,2\\}",
        "after_tex": "01\\text{ covered by }0,\\quad12\\text{ by }2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "不含中心的二點集也cover，且刪任一點就失敗。",
          "en": "The two-point set without the center also covers, and deleting either point fails."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "故此集合minimal但不是minimum，因{1}更小。",
            "en": "It is minimal but not minimum because{1} is smaller."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "C=\\{0,1,2\\}",
        "after_tex": "\\text{cover, size }3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "最後triple覆蓋兩邊，八候選全部完成。",
          "en": "The final triple covers both edges, completing all eight candidates."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "有效covers為{1}、{0,1}、{0,2}、{1,2}、V。",
            "en": "The valid covers are{1},{0,1},{0,2},{1,2},V."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\text{valid sizes }1,2,2,2,3",
        "after_tex": "\\tau(G)=1,\\quad\\operatorname{Argmin}=\\{\\{1\\}\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "取所有合法covers最小基數並列全部minimizers。",
          "en": "Take the minimum cardinality over all covers and list all minimizers."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "empty-edge graph則唯一size0 minimizer為空集；一般圖可能多minima。",
            "en": "For an edgeless graph the size-zero minimizer is empty; general graphs may have multiple minima."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "minimum size1，唯一minimizer{1}；{0,2}為minimal但非minimum。",
      "en": "Minimum size is 1 with unique minimizer{1};{0,2} is minimal but not minimum."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "{0,2} minimum因刪任何點都失敗。",
          "en": "{0,2} is minimum because every deletion fails."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{1}是更小cover。",
          "en": "{1} is a smaller cover."
        }
      },
      {
        "claim": {
          "zh-Hant": "{0}是cover。",
          "en": "{0} is a cover."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "邊12未被覆蓋。",
          "en": "Edge 12 is uncovered."
        }
      },
      {
        "claim": {
          "zh-Hant": "minimum cover一定唯一。",
          "en": "Minimum covers are always unique."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "單邊圖的任一端點singleton都是minimum。",
          "en": "Either endpoint singleton is minimum on a one-edge graph."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "對bounded finite simple undirected graph完整枚舉全部vertex subsets，逐一做all-edge cover audit，得到minimum cardinality與全部minimizers。給定nonminimum cover時須明示更小cover witness；inclusion-minimal只表示刪任一點即失去cover，不等於minimum。empty graph、multiple minima與minimal-but-not-minimum均保留。不得混同maximal、edge cover、independent set，也不使用König theorem或weighted/directed variants。",
          "en": "For a fully specified bounded finite simple undirected graph, enumerate every vertex subset and audit every edge to obtain minimum cardinality and all minimizing covers. A nonminimum cover needs an explicit smaller-cover witness. Inclusion-minimal means each single-vertex deletion loses coverage; it does not imply minimum cardinality. Retain edgeless graphs, multiple minimum solutions and minimal-but-nonminimum examples. Do not substitute maximality, edge covers or independent sets, or invoke Konig theorem or weighted/directed variants."
        },
        "source": {
          "mko_id": "mko-minimum-vertex-cover",
          "sha256": "4f8d510264312335a0f27c32a69f66f31350aede08398538e94a015583d8250b",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限無向簡單圖的邊為無loop、無重複unordered pairs；圖性質須對完整明示V、E逐一檢查。 cover必與每條邊有共同端點；minimum基數在全部合法covers中最小，argmin列出全部達此值的covers；minimal只禁止刪選點。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A cover intersects every edge in an endpoint; minimum cardinality is least over all valid covers and argmin lists all attaining covers; minimal only forbids deleting a selected vertex."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1+3+3+1",
        "rhs": "2**3"
      },
      {
        "id": "N02",
        "lhs": "min(1,2,2,2,3)",
        "rhs": "1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "bounded finite exact enumeration，不使用König theorem。",
      "en": "Bounded exact finite enumeration, with no Konig theorem."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "τ(G)",
        "minimum cover基數",
        "minimum cover cardinality"
      ]
    ],
    "md_error_controls": [
      {
        "title": "八案 / Eight cases",
        "zh": "0點一案、1點三案、2點三案、3點一案。",
        "en": "Count one empty, three singleton, three pair and one triple case."
      },
      {
        "title": "全部argmin / Full argmin",
        "zh": "不只報大小1，還報唯一達標集合{1}。",
        "en": "Report not only size 1 but the sole attaining set{1}."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "0c5dcd41a728ed5ea6c8d3a9073f8c93e7136545d0106b2b4387bb0d79708b0c",
    "original_source_sha256": "62c1c8fdcda2e9d32b26f78d86b566895ca1e88ad41eac54b7a4454148d6bafb",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S215",
      "BR142-S214",
      "BR142-S213",
      "BR142-P074",
      "BR142-S212"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}