{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-subset",
  "semantic_family_id": "sf-subset",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": { "zh-Hant": "子集合", "en": "Subset" },
  "summary": { "zh-Hant": "A⊆B 表示 A 的每個元素也都是 B 的元素。" },
  "statement": { "zh-Hant": "給定同一論域中的集合 A、B，A⊆B 當且僅當對每個對象 x，若 x∈A，則 x∈B。" },
  "formula": {
    "tex": "A\\subseteq B",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>⊆</mo><mi>B</mi></mrow></math>",
    "semantic_ast": {
      "type": "set_subset",
      "subset": { "type": "symbol", "name": "A", "glyph": "A" },
      "superset": { "type": "symbol", "name": "B", "glyph": "B" }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "efc0cc6ac1b9e1de37605b91a9c892c4176f0a6246a022fec19c70efb269f453"
    }
  },
  "assumptions": [
    { "id": "a1", "zh-Hant": "A 與 B 是同一 ambient type 或同一明示論域中的集合。" }
  ],
  "symbols": [
    { "token": "A", "role": "candidate_subset", "role_zh": "候選子集合", "scope": "definition" },
    { "token": "B", "role": "candidate_superset", "role_zh": "候選母集合", "scope": "definition" },
    { "token": "⊆", "role": "subset_relation", "role_zh": "非嚴格子集合關係", "scope": "definition" }
  ],
  "dependencies": [
    { "id": "mko-set-membership", "title_zh": "集合隸屬", "reason_zh": "子集合以每個元素的隸屬蘊含關係定義。" }
  ],
  "explanation": {
    "paragraphs_zh": [
      "要證明 A⊆B，必須從任意 x∈A 出發，說明同一個 x 也屬於 B；只列出一個共同元素並不充分。",
      "A⊆B 允許 A=B，所以它是非嚴格包含。若要表達 A 是 B 的嚴格子集合，還必須排除 A=B。",
      "x∈A 與 A⊆B 的型別角色不同：前者比較元素與集合，後者比較兩個集合。在 Lean 的 Set α 中，兩個集合也必須共享元素型別 α。"
    ],
    "common_misconception_zh": "常見誤解是把 x∈A 與 {x}⊆A 混為同一語句、認為 A⊆B 必須 A≠B，或只檢查部分元素便斷言包含。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本條目是子集合定義的 AI 候選；有限列舉可重算具體包含關係，但不能證成任意集合上的普遍性質。",
    "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": "綁定 Set.Subset 的精確定義展開、Mathlib 版本與 axiom receipt，並保持非嚴格包含、嚴格包含及集合相等三者分離。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-subset.md 的來源與語義決定整理；內容為 OCME 原創改寫，待人工審查。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-15"
  }
}