{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-many-one-reduction-01",
  "mko_id": "mko-many-one-reduction",
  "title": {
    "zh-Hant": "完整有限reduction與方向保持",
    "en": "A complete finite reduction with preserved direction"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-many-one-reduction-01",
    "source": {
      "mko_id": "mko-many-one-reduction",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c1a3c40ef9e301b3c5bce75380905c6f44c8cfc3bdb5126ad469d6587f9af2c6",
      "english_sha256": "1f0bd01bf78477131defe2c7465aa0fc959339c884a40075d0373878e7a0cb78"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "逐x核iff、組合及decider transfer。",
      "en": "Check every x equivalence, composition and decider transfer."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "source S={0,1,2,3}，A={1,3}；target T={a,b,c}，B={b}；f=(a,b,c,b)。U={u,v}，C={v}；g(a)=u、g(b)=v、g(c)=u。",
          "en": "Source S={0,1,2,3}, A={1,3}; target T={a,b,c}, B={b}; f=(a,b,c,b). U={u,v}, C={v}; g(a)=u,g(b)=v,g(c)=u."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "f=(a,b,c,b),\\quad g=(u,v,u)",
        "after_tex": "f:S\\to T,\\quad g:T\\to U\\text{ are total}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個source entry都有合法target像；g也完整定義。",
          "en": "Every source entry has a legal target image, and g is fully defined too."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "partial table不能先稱many-one reduction。",
            "en": "A partial table cannot be qualified as a many-one reduction."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "x=0,2",
        "after_tex": "x\\notin A,\\quad f(0)=a\\notin B,\\quad f(2)=c\\notin B",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部no instances逐一保持no。",
          "en": "Both no-instances preserve no-membership."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "只核yes會漏掉這兩個責任。",
            "en": "Yes-only checking would omit these two obligations."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "x=1,3",
        "after_tex": "x\\in A,\\quad f(1)=f(3)=b\\in B",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部yes instances也保持yes。",
          "en": "Both yes-instances preserve yes-membership."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "x=1、3不同但可同像，reduction不要求injective。",
            "en": "Distinct1,3 may share an image; reductions need not be injective."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\forall x\\in S:\\ x\\in A\\iff f(x)\\in B",
        "after_tex": "A\\le_m B\\quad\\text{on the declared carriers}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四source元素全部完成雙向membership equivalence。",
          "en": "All four source elements satisfy both directions of membership equivalence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這個有限結論不延伸未列的整數。",
            "en": "This finite conclusion does not extend to unlisted integers."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "a,c\\notin B,\\quad b\\in B",
        "after_tex": "g(a)=g(c)=u\\notin C,\\quad g(b)=v\\in C",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整三target元素給B≤mC。",
          "en": "All three target elements establish B≤mC."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "g定義域是整個T，而不只B。",
            "en": "The domain of g is all T, not only B."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "h=g\\circ f",
        "after_tex": "h=(u,v,u,v),\\quad x\\in A\\iff h(x)\\in C",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐表合成保持方向A到C，四entry完整。",
          "en": "Compose the tables in direction A to C, retaining all four entries."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能用f∘g，型別和方向都不匹配。",
            "en": "f composed after g would mismatch both type and direction."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "D_B(a)=0,D_B(b)=1,D_B(c)=0",
        "after_tex": "D_A(x)=D_B(f(x))=(0,1,0,1)_x",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先算f再查B decider，得到完整A truth table且有限查表halt。",
          "en": "Compute f then consult the B decider, producing the complete A truth table with finite termination."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "decider transfer是從target B到source A。",
            "en": "Decider transfer runs from target B to source A."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "A\\le_mB\\text{ and }B\\text{ decidable}",
        "after_tex": "A\\text{ decidable; no reverse implication supplied}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "演算法composition解釋正向transfer；對一般問題不能由A可決推B可決。",
          "en": "Algorithm composition explains the valid transfer; for general problems A decidable does not imply B decidable."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "有限carriers本來皆可查表，不能用它們偽造無限不可決反例。",
            "en": "Finite carriers are decidable by lookup; they must not be misused as infinite undecidability counterexamples."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "f證A≤mB，g證B≤mC，g∘f證A≤mC；先f後DB決定A。",
      "en": "f gives A≤mB, g gives B≤mC, and g∘f gives A≤mC; applying f then DB decides A."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "只要f在1、3送b即可，0也可送b。",
          "en": "Checking1,3 map to b suffices;0 may also map to b."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "0非A但b屬B會破壞iff反向。",
          "en": "0 is outside A while b lies in B, violating the reverse implication."
        }
      },
      {
        "claim": {
          "zh-Hant": "f必須injective。",
          "en": "f must be injective."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "1與3同像b仍完整保持iff。",
          "en": "1 and 3 share image b while preserving every iff."
        }
      },
      {
        "claim": {
          "zh-Hant": "composition應是f∘g。",
          "en": "Composition should be f after g."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "g輸出U但f輸入S，不合型別；正確g∘f。",
          "en": "g outputs U while f expects S; the well-typed composition is g after f."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "有限題族明示source carrier、target carrier、decision predicates與完整total mapping table，逐x雙向驗x∈A iff f(x)∈B；只驗yes instances不足。composition須保留方向：A≤mB且B≤mC推出A≤mC。若A≤mB且B decidable，先算f再跑B decider可得A decider；反向transfer不成立。partial map、倒轉方向與Turing reduction混同均須拒絕。有限mapping table只證指定finite domain；無限domain仍需完整可計算算法及termination certificate。",
          "en": "Finite families declare complete source and target carriers, both decision predicates and an aligned mapping table. Qualify totality, then check both directions of membership equivalence for every x; yes-only checks are insufficient. Composition preserves direction: A≤mB and B≤mC give A≤mC. If A≤mB and B is decidable, compute f then run the B decider to decide A; the reverse implication is not supplied. Partial maps, reversed transfer and conflation with Turing reduction must be rejected. A finite table certifies only its declared finite domain; an infinite domain still needs a complete computable algorithm and termination argument."
        },
        "source": {
          "mko_id": "mko-many-one-reduction",
          "sha256": "c1a3c40ef9e301b3c5bce75380905c6f44c8cfc3bdb5126ad469d6587f9af2c6",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "many-one reduction是total computable map保持x∈A iff f(x)∈B；有限完整table可計算且halt；函數合成與iff傳遞。",
          "en": "A many-one reduction is a total computable map preserving x∈A iff f(x)∈B; complete finite tables are computable and halt; function composition and iff transitivity."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "4",
        "rhs": "2+2"
      },
      {
        "id": "N02",
        "lhs": "3",
        "rhs": "2+1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限完整truth tables；不宣稱無限domain算法。",
      "en": "Complete finite truth tables, not an infinite-domain algorithm claim."
    },
    "md_symbols": [
      [
        "f,g,h",
        "total reductions及composition",
        "total reductions and composition"
      ],
      [
        "D_B",
        "target decider",
        "target decider"
      ]
    ],
    "md_error_controls": [
      {
        "title": "no cases / No cases",
        "zh": "0、2的membership否定也逐一核。",
        "en": "Audit nonmembership separately at 0,2."
      },
      {
        "title": "transfer方向 / Transfer direction",
        "zh": "沿f計算方向走，使用target decider來解source。",
        "en": "Follow f forward, using the target decider to decide the source."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "1a1c76bdcb2cbe175d47046cb14542bfd1c9bd765be098a093d56ec92f7bd817",
    "original_source_sha256": "64cb4efb6449a3a14fcb1d7a0a1135fd167442065877da4e143010419d64892f",
    "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": [
      "BR142-S217",
      "BR142-S216"
    ],
    "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."
  }
}