{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-lagrange-theorem-finite-groups-01",
  "mko_id": "mko-lagrange-theorem-finite-groups",
  "title": {
    "zh-Hant": "全部左陪集的Lagrange計數",
    "en": "Lagrange counting from all left cosets"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-lagrange-theorem-finite-groups-01",
    "source": {
      "mko_id": "mko-lagrange-theorem-finite-groups",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c283a1941e05986f16993e7482210024c2ac94218c77a43426996a572e70d274",
      "english_sha256": "3d916cf7153c04f3479aad8ce4165679e517db05ea5d26d87d9169d55c29d921"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "H={e,s}有三個互異左陪集，每個大小2，所以6=[G:H]|H|=3·2。",
      "en": "H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "G={e,r,r²,s,rs,r²s}，依序編碼(0,0),(1,0),(2,0),(0,1),(1,1),(2,1)。乘法(i,j)(k,l)=(i+(−1)^j k mod3,j+l mod2)。",
          "en": "G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "G=\\{e,r,r^2,s,rs,r^2s\\}",
        "after_tex": "\\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\\\\hline e&e&r&r^2&s&rs&r^2s\\\\r&r&r^2&e&rs&r^2s&s\\\\r^2&r^2&e&r&r^2s&s&rs\\\\s&s&r^2s&rs&e&r^2&r\\\\rs&rs&s&r^2s&r&e&r^2\\\\r^2s&r^2s&rs&s&r^2&r&e\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "將給定配對公式逐36對展開，全部輸出在同一六元素載體，沒有缺格。",
          "en": "Expand the pair formula for all 36 products; every output belongs to the same six-element carrier and no cell is missing."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "labels依序綁定六個pair，故同名r²不代表額外第七元素。",
            "en": "The six labels bind the six pairs; r^2 is not an additional seventh element."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(i,j)(k,l)=(i+(-1)^jk,j+l)\\quad(\\bmod(3,2))",
        "after_tex": "((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\\quad e=(0,0);\\quad(i,j)^{-1}=(-(-1)^ji,j)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩種括號的座標公式相同，證全部216 triples結合律；(0,0)兩側中性，列出的逆元兩側乘回(0,0)。",
          "en": "Both bracketings have identical coordinates, proving associativity for all 216 triples; (0,0) is a two-sided identity and the inverse formula multiplies back on both sides."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "指數j+l只依mod2，因(−1)²=1，公式對所有代表一致；逆元清單e,r²,r,s,rs,r²s完整。",
            "en": "The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "H=\\{e,s\\}",
        "after_tex": "e\\in H;\\quad ee=e,es=s,se=s,ss=e;\\quad e^{-1}=e,s^{-1}=s",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部四乘積與兩逆元留在H，先直接認證子群。",
          "en": "All four products and both inverses stay in H, directly qualifying the subgroup first."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "大小2整除6本身不是subgroup證書；此處已核closure。",
            "en": "Size 2 dividing 6 is not itself a subgroup certificate; closure is checked here."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S03"
        ],
        "before_tex": "g=e,r,r^2,s,rs,r^2s",
        "after_tex": "eH=sH=\\{e,s\\};\\quad rH=(rs)H=\\{r,rs\\};\\quad r^2H=(r^2s)H=\\{r^2,r^2s\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部六個g的左陪集都列出，再去重為三組。",
          "en": "List the left coset for every one of the six g values, then deduplicate to three sets."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "不能只列eH和rH而漏掉r²H的兩個group元素。",
            "en": "Listing only eH and rH would miss the two elements in r^2H."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S02",
          "S03",
          "S04"
        ],
        "before_tex": "C_0=\\{e,s\\},C_1=\\{r,rs\\},C_2=\\{r^2,r^2s\\}",
        "after_tex": "C_i\\cap C_j=\\varnothing\\ (i\\ne j),\\quad C_0\\cup C_1\\cup C_2=G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個明列sets互斥覆蓋；一般重疊gh₁=kh₂亦推出k⁻¹g=h₂h₁⁻¹∈H，故兩cosets相等。",
          "en": "The explicit sets are disjoint and cover G; generally, overlap gh1=kh2 gives k^-1*g=h2*h1^-1 in H and hence equality of the cosets."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "相等或不交用H的乘法及逆元封閉，不依隨意分組。",
            "en": "Equal-or-disjoint uses product and inverse closure of H rather than arbitrary grouping."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S02",
          "S03",
          "S04"
        ],
        "before_tex": "L_g:H\\to gH,\\quad h\\mapsto gh",
        "after_tex": "L_g^{-1}(k)=g^{-1}k\\Rightarrow |gH|=|H|=2",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "左平移的逆映射明確，因此三個cosets均與H等大。",
          "en": "The inverse of left translation is explicit, so all three cosets have H's size."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "g⁻¹k回到H源自k∈gH，且兩方向復合皆identity。",
            "en": "g^-1*k lies in H because k lies in gH, and both compositions are identities."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S05",
          "S06"
        ],
        "before_tex": "G=C_0\\sqcup C_1\\sqcup C_2,\\quad |C_i|=2",
        "after_tex": "[G:H]=3,\\quad|G|=3\\cdot2=6,\\quad|H|\\mid|G|",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整不交partition與等大證書給Lagrange計數。",
          "en": "The complete disjoint partition and equal-size certificates yield Lagrange counting."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "整除來自實際subgroup cosets，不是僅看數字6和2。",
            "en": "Divisibility comes from actual subgroup cosets, not merely numbers 6 and 2."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S01",
          "S03",
          "S07"
        ],
        "before_tex": "K=\\{e,r,s\\},\\quad|K|=3\\mid6",
        "after_tex": "rr=r^2\\notin K\\Rightarrow K\\text{ not subgroup};\\quad\\{e,r,r^2\\}\\le G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "指定三元素subset雖有divisor大小卻不封閉；同群仍有另一個order3 subgroup，避免錯誤存在性推論。",
          "en": "The specified divisor-sized subset fails closure, while the same group has another order-3 subgroup, preventing a false nonexistence conclusion."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S03",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "K失敗只否證K；不能宣稱所有三元素候選都失敗。",
            "en": "Failure refutes K only and not all three-element candidates."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "H={e,s}有三個互異左陪集，每個大小2，所以6=[G:H]|H|=3·2。",
      "en": "H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "K大小3整除6所以K是subgroup。",
          "en": "K has size 3 dividing 6, so K is a subgroup."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "r²不在K但r·r=r²。",
          "en": "r*r=r^2 is outside K."
        }
      },
      {
        "claim": {
          "zh-Hant": "K失敗證G無order3 subgroup。",
          "en": "Failure of K proves G has no order-3 subgroup."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{e,r,r²}由完整表確是order3 subgroup。",
          "en": "The full table qualifies {e,r,r^2} as an order-3 subgroup."
        }
      },
      {
        "claim": {
          "zh-Hant": "只列兩個cosets也完成partition。",
          "en": "Two listed cosets suffice for the partition."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "r²與r²s仍未覆蓋。",
          "en": "r^2 and r^2s would remain uncovered."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設精確模3、模2整數算術、有限集合外延相等與群／作用的定義；不預設此運算已為群。",
          "en": "Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "H須由完整Cayley table認證為subgroup。全部left cosets gH必須相等或不交並完整覆蓋G；h↦gh給出H到gH的bijection，所以每個coset大小為|H|。結論|G|=[G:H]|H|。本輪converse family只證d整除|G|不表示任意指定d-element subset是subgroup；不以該witness證明或否定某個order-d subgroup是否存在。",
          "en": "The complete Cayley table must certify H as a subgroup. Every left coset gH is equal to or disjoint from every other and all cosets cover G. Left translation h→g*h bijects H with gH, so every coset has size |H|, giving |G|=[G:H]*|H|. The converse family here only shows that a specified divisor-sized subset need not be a subgroup; its witness neither proves nor refutes existence of some subgroup of that order."
        },
        "source": {
          "mko_id": "mko-lagrange-theorem-finite-groups",
          "sha256": "c283a1941e05986f16993e7482210024c2ac94218c77a43426996a572e70d274",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "lagrange",
        "lhs": "3*2",
        "rhs": "6",
        "step": "S07"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整六元素表與完整coset partition；subset反例不聲稱同階subgroup不存在。",
      "en": "A full six-element table and full coset partition; a failed subset does not prove absence of subgroups of that order."
    },
    "md_symbols": [
      [
        "G",
        "六元素群載體",
        "six-element group carrier"
      ],
      [
        "r,s",
        "r³=e,s²=e,sr=r²s的labels",
        "labels with r^3=e,s^2=e,sr=r^2s"
      ],
      [
        "H",
        "指定order2 subgroup",
        "specified order-2 subgroup"
      ],
      [
        "[G:H]",
        "互異左陪集個數",
        "number of distinct left cosets"
      ]
    ],
    "md_error_controls": [
      {
        "title": "同一群表 / Same group table",
        "zh": "全部積、逆元與cosets綁定這個完整Cayley表。",
        "en": "Bind every product, inverse and coset to this complete Cayley table."
      },
      {
        "title": "完整枚舉 / Complete enumeration",
        "zh": "任何全稱有限性質都需全載體或全稱公式，不用proper sample。",
        "en": "Every universal finite property needs the full carrier or a universal formula, not a proper sample."
      }
    ],
    "md_open_items": []
  },
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    "original_source_sha256": "69a1f66a1aae765143e62e40fb90126333ae697e9503310f59d7ba0e97a5a2b2",
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    "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": [],
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  }
}