{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-flow-network-01",
  "mko_id": "mko-finite-flow-network",
  "title": {
    "zh-Hant": "逐邊與逐內點核可行流",
    "en": "Checking every edge and internal vertex of a feasible flow"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-flow-network-01",
    "source": {
      "mko_id": "mko-finite-flow-network",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c6cc774f53f5114e2b238ff77e24ed61a2adc5f2d6d7ffa5a969b25f287ad5d0",
      "english_sha256": "eaeb9873e61c9b144bb18d9037f62d7125afdee46da133467f5a106dbddc04b7"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "sets",
      "flow_def"
    ],
    "goal": {
      "zh-Hant": "核一個含反平行、零容量邊與零流量點的完整網路流值。",
      "en": "Audit a complete flow network with antiparallel edges, a zero-capacity edge and a zero-flow vertex."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={s,a,b,t,z}，來源 s≠t 終點；邊依序 sa,sb,ab,at,bt,ba,za，容量 (3,2,1,2,3,1,0)，流 (2,1,1,1,2,0,0)。z 除零容量 za 外沒有邊。",
          "en": "V={s,a,b,t,z}, source s and distinct sink t; ordered edges sa,sb,ab,at,bt,ba,za have capacities (3,2,1,2,3,1,0) and flows (2,1,1,1,2,0,0). z has no edge except zero-capacity za."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "E=\\{(s,a),(s,b),(a,b),(a,t),(b,t),(b,a),(z,a)\\}",
        "after_tex": "|E|=7,\\quad u\\ne v\\text{ and each ordered pair occurs once}",
        "relation": "implies",
        "rule": "flow_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐邊核兩端在 V、無 self-loop 且不重複；ab 與 ba 是合法不同邊。",
          "en": "Each edge has endpoints in V, no loop and no duplicate ordered pair; ab and ba are legal distinct edges."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "反平行不是 duplicate；流與容量鍵恰對應全部七邊。",
            "en": "Antiparallel is not duplicate; both capacity and flow keys match all seven edges."
          }
        },
        "leaf_basis": "flow_def"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(f_e,c_e)= (2,3),(1,2),(1,1),(1,2),(2,3),(0,1),(0,0)",
        "after_tex": "0\\le f_e\\le c_e\\quad\\forall e\\in E",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整七對逐個核非負及容量界，零容量邊流恰零。",
          "en": "Check all seven pairs for nonnegativity and capacity; the zero-capacity edge carries zero flow."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "流值小不代表每條邊合法，此處每邊另核。",
            "en": "A small total value does not imply edge legality; each edge is separately checked."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "G1"
        ],
        "before_tex": "\\operatorname{in}(a)=f_{sa}+f_{ba}+f_{za}=2+0+0",
        "after_tex": "\\operatorname{in}(a)=2=f_{ab}+f_{at}=\\operatorname{out}(a)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "a 的全部三條流入與兩條流出均列出。",
          "en": "List all three incoming and both outgoing edges of a."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不能漏反向 ba 或零容量 za，即使它們流為零。",
            "en": "Do not omit ba or za merely because their flows are zero."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "G1"
        ],
        "before_tex": "\\operatorname{in}(b)=f_{sb}+f_{ab}=1+1",
        "after_tex": "\\operatorname{in}(b)=2=f_{bt}+f_{ba}=\\operatorname{out}(b)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "b 的完整流入流出相等。",
          "en": "The complete inflow and outflow of b agree."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "ab 在 b 是流入，在 a 才是流出。",
            "en": "ab is incoming at b and outgoing at a."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "\\operatorname{in}(z)=0,\\quad\\operatorname{out}(z)=f_{za}=0",
        "after_tex": "\\operatorname{in}(z)=\\operatorname{out}(z)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "z 沒有流入且唯一流出容量為零，守恆成立；這是零流點，不是圖論孤立點。",
          "en": "z has no inflow and its sole outgoing edge has zero capacity, so conservation holds; it is a zero-flow vertex, not an isolated graph vertex."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "零容量邊仍是圖邊；z→a→t 是圖路徑，但含容量零邊，不能運送正流，z 不是孤立點。",
            "en": "A zero-capacity edge remains a graph edge: z→a→t is a graph path but carries no positive flow because of the zero capacity; z is not isolated."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "|f|=\\operatorname{out}(s)-\\operatorname{in}(s)=2+1-0",
        "after_tex": "|f|=3=\\operatorname{in}(t)-\\operatorname{out}(t)=1+2-0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "來源與終點兩種淨流值獨立計算一致。",
          "en": "Source and sink net values independently evaluate to the same result."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "淨值要扣反向流；本圖沒有來源入邊及終點出邊所以扣零。",
            "en": "Net values subtract reverse-direction flow; this graph has no source in-edge or sink out-edge, so these are zero."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S01",
          "S02",
          "S03",
          "S04",
          "S05",
          "S06"
        ],
        "before_tex": "0\\le f\\le c,\\quad\\operatorname{in}=\\operatorname{out}\\text{ at }a,b,z",
        "after_tex": "f\\text{ is feasible with value }3",
        "relation": "definition",
        "rule": "flow_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部內點與邊均已核，才可作可行性結論。",
          "en": "Only after all edges and internal vertices are checked is feasibility concluded."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S03",
            "S04",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "尚未給 optimality 證書，所以不稱最大。",
            "en": "No optimality certificate has been supplied, so it is not called maximum."
          }
        },
        "leaf_basis": "flow_def"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07"
        ],
        "before_tex": "f'=(3,1,1,2,2,0,0)\\text{ in the same edge order}",
        "after_tex": "f'\\text{ is feasible with }|f'|=4>3",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "替代流每邊合法，a 入3出1+2、b 入1+1出2、z 為零，證原流其實不是最大。",
          "en": "The alternative satisfies every capacity; a has 3 in and 1+2 out, b has 1+1 in and 2 out, and z stays zero, proving the original is not maximum."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "只以更大可行流否定最大性，未偷用 min-cut 定理。",
            "en": "A larger feasible flow refutes maximum-value optimality without invoking the min-cut theorem."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "原流可行、值 3；另有值 4 可行流，故原流非最大。",
      "en": "The original flow is feasible with value 3; a value-4 feasible flow proves it is not maximum."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "ab 和 ba 同時存在違法。",
          "en": "Having both ab and ba is illegal."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "簡單有向圖只禁相同 ordered pair 重複，不禁反平行。",
          "en": "A simple directed graph forbids duplicate ordered pairs, not antiparallel edges."
        }
      },
      {
        "claim": {
          "zh-Hant": "只算來源值就完成可行性。",
          "en": "The source value alone proves feasibility."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "每邊與每內點仍須核。",
          "en": "Every edge and internal vertex still requires auditing."
        }
      },
      {
        "claim": {
          "zh-Hant": "可行必最大。",
          "en": "Feasible means maximum."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S08 給更大可行流。",
          "en": "S08 gives a larger feasible flow."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "network使用finite loop-free simple directed graph，source s與sink t不同；antiparallel edges可同時存在但每個ordered pair最多一條。capacity與flow為exact nonnegative rationals。feasible flow逐邊滿足0≤f(e)≤c(e)，且每個internal vertex的完整inflow等於outflow。flow value同時由source net outflow與sink net inflow計算並須相等。允許zero capacity與disconnected vertices。feasible不等於maximum，不得提前宣稱min-cut equality或augmenting-path optimality。",
          "en": "Use a finite loop-free simple directed graph with distinct source s and sink t. Antiparallel edges are allowed, but each ordered pair appears at most once. Valid capacities/flows are exact nonnegative rationals. Feasibility requires0≤f(e)≤c(e) and complete inflow=outflow at every internal vertex. Cross-check source net outflow against sink net inflow. Zero capacities and disconnected vertices are legal. Feasibility is not maximum flow; no min-cut equality or augmenting-path optimality is claimed ahead of its theorem."
        },
        "source": {
          "mko_id": "mko-finite-flow-network",
          "sha256": "c6cc774f53f5114e2b238ff77e24ed61a2adc5f2d6d7ffa5a969b25f287ad5d0",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "sets",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合的聯、交、補集、包含與函數逆像以逐元素定義計算。",
          "en": "Compute unions, intersections, complements, inclusion and preimages element by element."
        }
      },
      {
        "id": "flow_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限簡單有向網路無 self-loop、無重複 ordered pair，但容許反平行邊；可行流滿足每邊 0≤f≤c 及每內點流入等於流出，值為來源淨流出及終點淨流入。",
          "en": "A finite simple directed network has no self-loop or duplicate ordered pair, but permits antiparallel edges; feasible flows satisfy every capacity bound and internal conservation, with value equal to source net outflow and sink net inflow."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "source-value",
        "lhs": "2+1",
        "rhs": "3",
        "step": "S06"
      },
      {
        "id": "sink-value",
        "lhs": "1+2",
        "rhs": "3",
        "step": "S06"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只核資格、capacity、conservation 與流值，不聲稱任意可行流最大。",
      "en": "Only network qualification, capacity, conservation and flow value; feasibility alone does not establish maximum flow."
    },
    "md_symbols": [
      [
        "f_e,c_e",
        "每條有向邊的流及容量",
        "flow and capacity per directed edge"
      ],
      [
        "|f|",
        "淨流值",
        "net flow value"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全鍵對應 / Complete key correspondence",
        "zh": "容量與流的鍵不能漏邊或多邊。",
        "en": "Capacity and flow keys may neither omit nor add edges."
      },
      {
        "title": "淨值交核 / Net-value cross-check",
        "zh": "獨立算 s 與 t，不能只保留一側。",
        "en": "Compute both s and t independently."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "original_source_sha256": "9ddde2b86d73fa9084b762788ec009ac727c1ea81b32343655d29131ebaa1df4",
    "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": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    ],
    "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."
  }
}