{
  "$schema": "../../../schemas/mko-v0.5.schema.json",
  "schema_version": "mko-v0.5",
  "id": "mko-subsequence",
  "semantic_family_id": "sf-subsequence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "子數列"
  },
  "summary": {
    "zh-Hant": "子數列由嚴格遞增的自然數索引映射選取原數列項並保留次序。"
  },
  "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。"
  },
  "formula": {
    "tex": "b_j=a_{k(j)}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>b</mi><mi>j</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>(</mo><mi>j</mi><mo>)</mo></mrow></msub></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "b",
          "glyph": "b"
        },
        "subscript": {
          "type": "symbol",
          "name": "j",
          "glyph": "j"
        }
      },
      "rhs": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        "subscript": {
          "type": "function_call",
          "callee": {
            "type": "symbol",
            "name": "k",
            "glyph": "k"
          },
          "arguments": [
            {
              "type": "symbol",
              "name": "j",
              "glyph": "j"
            }
          ]
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-source13",
      "version": "0.5.0",
      "source_sha256": "a2dad55c5c5a9671f10754f941b73c7d113d3b8a53b250c948d1a0e087e979c1"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "索引起點為非負整數且step為正整數。"
    },
    {
      "id": "a2",
      "zh-Hant": "arithmetic source sequence與index composition使用exact integers。"
    }
  ],
  "symbols": [
    {
      "token": "k",
      "role": "strictly_increasing_index_map",
      "role_zh": "嚴格遞增索引映射",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "subsequence",
      "role_zh": "子數列",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "source_sequence",
      "role_zh": "原數列",
      "scope": "representative_formula"
    },
    {
      "token": "j",
      "role": "sequence_index",
      "role_zh": "數列索引",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "原數列與選取列。"
    },
    {
      "id": "mko-injective",
      "title_zh": "單射",
      "reason_zh": "嚴格遞增蘊含injective。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "索引次序。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "strictly increasing比任意injective更強。",
      "composition of subsequences仍保次序。",
      "finite index prefix不足以證無限索引映射。",
      "記號對齊候選：先將索引映射 k 作用於 j，再取 a 的第 k(j) 項。k 仍須嚴格遞增；單射、重排或有限遞增前綴不能自行取代該前提。"
    ],
    "common_misconception_zh": "把任意重排或有限抽樣稱為子數列。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪symbolic certificate限affine increasing indices；更一般index map需自己的全稱證書。",
    "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Bolzano-Weierstrass theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-subsequence.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-07 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}