{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-subsequence-01",
  "mko_id": "mko-subsequence",
  "title": {
    "zh-Hant": "仿射索引的全稱子數列證書",
    "en": "An all-index subsequence certificate for affine indices"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-subsequence-01",
    "source": {
      "mko_id": "mko-subsequence",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "f4791c70590c6e099961027b14a591894bd691565f6e1bc8cd191792d0e2a44f",
      "english_sha256": "e54379f1fd9f6ccb9ef4be2d69a165c97d4e793ec8b2ecf051c61d8e2163b155"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "sets",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "由 a_n=3n-2、k(j)=1+2j 構造並再取子數列。",
      "en": "Construct a subsequence of a_n=3n-2 using k(j)=1+2j, then take a further subsequence."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "a_n=3n-2（n≥0）；k(j)=1+2j；另一索引 ell(i)=2+3i（i≥0）。",
          "en": "a_n=3n-2 for n≥0; k(j)=1+2j; a further index is ell(i)=2+3i for i≥0."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "j\\in\\mathbb N_0",
        "after_tex": "k(j)=1+2j\\in\\mathbb N_0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "k 把全部非負整數送到合法來源索引。",
          "en": "k maps every nonnegative integer to a legal source index."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不是只列前幾個索引。",
            "en": "The definition is not just a finite list of indices."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "j_2>j_1\\ge0",
        "after_tex": "k(j_2)-k(j_1)=2(j_2-j_1)>0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任意兩個有序索引證嚴格遞增。",
          "en": "Prove strict increase for any ordered pair of indices."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "單射雖由此推出，反向卻不足。",
            "en": "Strict increase implies injectivity, but injectivity alone is not sufficient for an index map to certify a subsequence."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "b_j=a_{k(j)}",
        "after_tex": "b_j=3(1+2j)-2=1+6j",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先算 k(j)，再代入來源數列，不能把 k 當值。",
          "en": "Evaluate k(j) before substituting into the source sequence; k is an index map, not a term value."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "代回 a 的下標確為 k(j)。",
            "en": "The substituted subscript of a is exactly k(j)."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S02",
          "S03"
        ],
        "before_tex": "b_j=a_{k(j)},\\quad k\\text{ strictly increasing on }\\mathbb N_0",
        "after_tex": "(b_j)\\text{ is an infinite subsequence of }(a_n)",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整無限索引映射和取項公式共同構成子數列定義。",
          "en": "The complete infinite increasing map and term formula together establish the subsequence definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "有限前綴本身不提供這個全函數。",
            "en": "A finite prefix alone would not supply this total function."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "i_2>i_1\\ge0",
        "after_tex": "\\ell(i_2)-\\ell(i_1)=3(i_2-i_1)>0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二索引亦合法非負且嚴格遞增。",
          "en": "The second index map is likewise nonnegative and strictly increasing."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "起點 2 非負，步長 3 正。",
            "en": "Its start 2 is nonnegative and step 3 positive."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "m(i)=k(\\ell(i))",
        "after_tex": "m(i)=1+2(2+3i)=5+6i",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "組合索引仍是非負仿射函數。",
          "en": "The composed index remains a nonnegative affine map."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "定義的組合次序是先 ell 後 k；這兩條特定仿射律數值恰好可交換，但一般索引映射不能如此互換。",
            "en": "The defining composition applies ell then k. These particular affine laws accidentally commute numerically, but arbitrary index maps cannot be interchanged."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S03",
          "S06"
        ],
        "before_tex": "c_i=b_{\\ell(i)}=a_{m(i)}",
        "after_tex": "c_i=13+18i",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩種取項路徑都給 13+18i，且 m 的正步長 6 證仍為來源子數列。",
          "en": "Both selection routes give 13+18i; the positive step 6 of m certifies a subsequence of the original sequence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "等式核值，正步長另核次序。",
            "en": "The equality checks values, while the positive step separately checks order."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S04"
        ],
        "before_tex": "r(0)=1,\\ r(1)=0,\\ r(j)=j\\ (j\\ge2)",
        "after_tex": "r\\text{ injective but not increasing};\\quad(a_{r(j)})\\text{ is not certified as a subsequence}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "交換前兩索引給無限單射反例，顯示單射不等於保次序。",
          "en": "Swapping the first two indices gives an infinite injective counterexample, distinguishing injectivity from order preservation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "r(1)<r(0) 是明確失敗，而非有限樣本無法判定。",
            "en": "r(1)<r(0) is an explicit failure, not merely an inconclusive finite sample."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "b_j=1+6j、c_i=13+18i；兩者均由完整嚴格遞增索引證書構成。",
      "en": "b_j=1+6j and c_i=13+18i, both certified by complete strictly increasing index maps."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "單射索引就必為子數列。",
          "en": "An injective index map always gives a subsequence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S08 的交換單射不遞增。",
          "en": "The swapping injection in S08 is not increasing."
        }
      },
      {
        "claim": {
          "zh-Hant": "c_i 的來源索引是 k(i)。",
          "en": "The original index of c_i is k(i)."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "正確索引 k(ell(i))=5+6i；k(i)=1+2i，在 i=0 已是 5 與 1。",
          "en": "The correct index is k(ell(i))=5+6i, whereas k(i)=1+2i; already at i=0 they are five and one."
        }
      },
      {
        "claim": {
          "zh-Hant": "前十索引遞增已證無限子數列。",
          "en": "Ten increasing indices prove an infinite subsequence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "仍需整個 N_0 上的映射及遞增證書。",
          "en": "A map and increasing certificate on all of N_0 are still required."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "索引k:N→N必須strictly increasing，不只是injective；b_j=a_{k(j)}。本輪以affine k(j)=s+dj、d>=1給通用symbolic certificate，並區分有限遞增prefix與真正無限subsequence。重排、任意子集合或finite sample都不是自動的subsequence證書。",
          "en": "The map k:N→N must be strictly increasing, not merely injective, and b_j=a_{k(j)}. This slice uses affine k(j)=s+d*j with d≥1 as a symbolic all-j certificate. A finite increasing prefix is not an already-supplied infinite map; arbitrary rearrangements and finite samples are not automatic certificates."
        },
        "source": {
          "mko_id": "mko-subsequence",
          "sha256": "f4791c70590c6e099961027b14a591894bd691565f6e1bc8cd191792d0e2a44f",
          "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": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "b_j=a_{k(j)} 是子數列須 k 在整個非負整數域嚴格遞增；指標合成先算內層。",
          "en": "A subsequence b_j=a_{k(j)} requires k strictly increasing on all nonnegative integers; index composition evaluates the inner map first."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "composed-value",
        "lhs": "3*(1+2*(2+3*i))-2",
        "rhs": "13+18*i",
        "step": "S07",
        "vars": [
          "i"
        ],
        "degree": 1
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "n,j≥0 整數；完整仿射索引，不以有限前綴或任意重排代替。",
      "en": "Nonnegative integer indices and complete affine maps, not finite prefixes or arbitrary rearrangements."
    },
    "md_symbols": [
      [
        "k,\\ell,m",
        "索引映射",
        "index maps"
      ],
      [
        "b_j,c_i",
        "選出的項",
        "selected terms"
      ]
    ],
    "md_error_controls": [
      {
        "title": "索引先行 / Index first",
        "zh": "先代 k(j) 再算 a，保留括號。",
        "en": "Evaluate k(j) before a and retain the parentheses."
      },
      {
        "title": "無限證書 / Infinite certificate",
        "zh": "用正步長差值證全部索引，而非有限表。",
        "en": "Use the positive-step difference to certify all indices, not a finite table."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "723f86ede56e20fd17954f55eaf9ac1bb183cabb027071619eab37816d8ea520",
    "original_source_sha256": "b3f9513bca7f0ae4d56cfc0a6777cfda19771c065785d1014d30696c2f912b93",
    "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-P046"
    ],
    "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."
  }
}