{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-cardinality-comparison-01",
  "mko_id": "mko-cardinality-comparison",
  "title": {
    "zh-Hant": "完整映射表的基數證書",
    "en": "Cardinality certificates from full maps"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-cardinality-comparison-01",
    "source": {
      "mko_id": "mko-cardinality-comparison",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "50efb74b57a576c8cec0b8230fc37bf7cd9c7ecd6303efe09810085d9a570227",
      "english_sha256": "d5d927d20ab09e9e7df6d3a33e4ed265b08e1db823fee7f5cb2a61a667172b4a"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "有單射A→B且|A|=2<3=|B|；反向滿射非單射。",
      "en": "There is an injection A to B with |A|=2<3=|B|; a reverse surjection is not injective."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "A={a,b}，B={1,2,3}；f(a)=1,f(b)=2；g(1)=a,g(2)=b,g(3)=a。",
          "en": "A={a,b}, B={1,2,3}; f(a)=1,f(b)=2; g(1)=a,g(2)=b,g(3)=a."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "A=\\{a,b\\},\\ B=\\{1,2,3\\}",
        "after_tex": "|A|=2,\\quad |B|=3",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "來源與目標均完整且內部元素互異，計數無重複。",
          "en": "Both carriers are complete with distinct entries, so their counts contain no duplicates."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "先固定集合；同一映射換codomain可能改變滿射性。",
            "en": "Fix the sets first; changing a codomain may change surjectivity."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "f:A\\to B",
        "after_tex": "\\operatorname{graph}(f)=\\{(a,1),(b,2)\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個來源元素恰各有一個像，像都在B中，因此f是總映射。",
          "en": "Each source element has exactly one image in B, so f is total."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "不可只檢查已填表格而遺漏一個來源元素。",
            "en": "Checking filled entries alone would miss an omitted source element."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "f(a)=1,\\ f(b)=2",
        "after_tex": "f(a)\\ne f(b)\\Rightarrow f\\text{ injective}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "A只有兩個元素，唯一不同元素對的像不同，已核全部碰撞。",
          "en": "A has only two elements, and the only distinct pair has distinct images, exhausting collision checks."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "1≠2是在B內的相異性；單射並不要求命中3。",
            "en": "1 and 2 are distinct in B; injectivity does not require hitting 3."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S03"
        ],
        "before_tex": "f(A)=\\{1,2\\}",
        "after_tex": "3\\notin f(A),\\quad f\\text{ not surjective},\\quad 2\\le3",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "缺少3否證滿射；兩個不同像仍給|A|≤|B|證書。",
          "en": "Missing 3 refutes surjectivity, while the two distinct images certify |A|≤|B|."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "不可把單射錯寫成雙射；缺像已由完整來源表證實。",
            "en": "Do not upgrade injection to bijection; the missing image is certified by the full source table."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "g:B\\to A",
        "after_tex": "g^{-1}(\\{a\\})=\\{1,3\\},\\quad g^{-1}(\\{b\\})=\\{2\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整反向表給兩個非空fiber，全部B都已列入。",
          "en": "The full reverse map gives two nonempty fibers accounting for all of B."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "fiber是來源元素集合，不能把輸入1與3合併為一個元素。",
            "en": "A fiber is a source subset; inputs 1 and 3 cannot be counted as one element."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "g(B)=A,\\quad g(1)=g(3)",
        "after_tex": "g\\text{ surjective but not injective};\\quad3\\ge2",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "a、b均有前像，所以g滿射；1≠3但像同，故非單射。",
          "en": "Both a and b have preimages, so g is onto; distinct 1 and 3 have the same image, so g is not injective."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "滿射與單射分開判斷；反向滿射只給有限基數反向不等式。",
            "en": "Audit onto and one-to-one separately; the reverse surjection gives the finite reverse inequality only."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "h:B\\to A\\text{ assumed injective}",
        "after_tex": "h(1),h(2),h(3)\\text{ distinct in a two-element set: impossible}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "單射會要求A內三個互異元素；A明列只有兩個，所以任何反向單射都不存在。",
          "en": "An injection would require three distinct elements in explicitly two-element A, which is impossible."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "這否證所有h，不只是g；理由依完整載體數目。",
            "en": "This excludes every h, not merely g, using the complete carrier sizes."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "C=\\{u,v\\},\\quad b(a)=u,\\ b(b)=v",
        "after_tex": "b:A\\to C\\text{ bijective};\\quad |A|=|C|=2",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另列完整雙射：不同來源像不同，且u、v皆命中，給相等基數證書。",
          "en": "A separate full map is both injective and onto u,v, certifying equality of cardinalities."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "等式有具體雙射支撐，並未把原來的f誤稱滿射。",
            "en": "The equality has an actual bijection witness and does not mislabel the original f as onto."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "有單射A→B且|A|=2<3=|B|；反向滿射非單射。",
      "en": "There is an injection A to B with |A|=2<3=|B|; a reverse surjection is not injective."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "f是雙射。",
          "en": "f is bijective."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "3沒有前像。",
          "en": "3 has no preimage."
        }
      },
      {
        "claim": {
          "zh-Hant": "g是單射。",
          "en": "g is injective."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "g(1)=g(3)=a且1≠3。",
          "en": "g(1)=g(3)=a with 1 distinct from 3."
        }
      },
      {
        "claim": {
          "zh-Hant": "由這兩個有限例子已證無限滿射都有section。",
          "en": "These finite examples prove every infinite surjection has a section."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "證書只涵蓋A、B、C，未處理無限集合或選擇原理。",
          "en": "The certificates concern A,B,C only and handle neither infinite sets nor choice principles."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設有限集合計數、單射與滿射的定義。",
          "en": "Assume finite counting and the definitions of injection and surjection."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "本輪限完整有限集合。A到B單射證|A|<=|B|，滿射證|A|>=|B|，雙射證相等；若|A|>|B|則不存在單射。無限集合的surjection逆向比較需要額外choice／section責任，不在本輪。",
          "en": "This slice is limited to complete finite sets. An injection A→B certifies |A|≤|B|, a surjection certifies |A|≥|B|, and a bijection certifies equality. If |A|>|B| there is no injection. Reverse comparison from a surjection for infinite sets requires additional choice or section obligations and is outside this slice."
        },
        "source": {
          "mko_id": "mko-cardinality-comparison",
          "sha256": "50efb74b57a576c8cec0b8230fc37bf7cd9c7ecd6303efe09810085d9a570227",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "sizes",
        "lhs": "2",
        "rhs": "3",
        "step": "S04",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只處理明示有限集合與完整映射，不作無限選擇主張。",
      "en": "Explicit finite sets and full maps only; no infinite-choice claim."
    },
    "md_symbols": [
      [
        "f,g",
        "方向不同的映射",
        "maps with opposite directions"
      ],
      [
        "A,B,C",
        "完整有限集合",
        "complete finite sets"
      ]
    ],
    "md_error_controls": [
      {
        "title": "總表 / Total table",
        "zh": "f須有兩個entries，g須有三個；來源長度不可混用。",
        "en": "f needs two entries and g three; source lengths cannot be interchanged."
      },
      {
        "title": "比較方向 / Comparison direction",
        "zh": "單射A→B給|A|≤|B|，滿射同方向給≥。",
        "en": "An injection A to B gives ≤, whereas a surjection in that direction gives ≥."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "f7395fd1e6ecb7e29d7182b804fa89d6a0e5440629fc2b259fbe84acd59ce55b",
    "original_source_sha256": "f7395fd1e6ecb7e29d7182b804fa89d6a0e5440629fc2b259fbe84acd59ce55b",
    "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": [],
    "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."
  }
}