{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-filtration-01",
  "mko_id": "mko-finite-filtration",
  "title": {
    "zh-Hant": "有限 filtration：以分割細化核對資訊嵌套，並區分 adapted 與獨立",
    "en": "Finite filtration: checking information nesting by partition refinement, and separating it from adaptedness and independence"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-filtration-01",
    "source": {
      "mko_id": "mko-finite-filtration",
      "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
      "sha256": "1b51e644ad6518d67dc1c6b78e1ff5043eb9a2fe5576f71a9f732b6b1c168a62",
      "english_sha256": "fed8bebcbc4e41746b79d4713668de11f3a5bf2f427093fdad7613878c8a36f3"
    },
    "mode": "definition_walkthrough",
    "audience_prerequisites": [
      "set_operations_def",
      "subset_def",
      "set_extensionality",
      "prob_axioms"
    ],
    "goal": {
      "zh-Hant": "在 Ω={0,1,2,3}（兩次擲幣：0=正正、1=正反、2=反正、3=反反）上，以分割編碼 P_0、P_1、P_2 表示時刻 0、1、2 的 σ 代數。證明「較晚分割細化較早分割 ⇔ σ 代數包含」，逐相鄰時刻完整核對，得出 (F_t) 是 filtration；給出不是 filtration 的分割序列；並證明 filtration 既不使每個過程 adapted，也不推出獨立。",
      "en": "On Ω={0,1,2,3} (two coin tosses: 0=HH, 1=HT, 2=TH, 3=TT), represent the σ-algebras at times 0, 1, 2 by partition rows P_0, P_1, P_2. Prove that 'the later partition refines the earlier one ⇔ the σ-algebras are nested', check every adjacent pair of times completely and conclude that (F_t) is a filtration; give a sequence of partitions that is not a filtration; and show that a filtration neither makes every process adapted nor implies independence."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Ω={0,1,2,3}；結果的意義：0=正正、1=正反、2=反正、3=反反（只作說明，不影響分割的定義）。",
          "en": "Ω={0,1,2,3}; meaning of the outcomes: 0=HH, 1=HT, 2=TH, 3=TT (for orientation only; it does not affect the partitions)."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "分割編碼（依結果 0、1、2、3 給 label）：P_0=(a,a,a,a)、P_1=(a,a,b,b)、P_2=(a,b,c,d)；同一 label 表示同一區塊（MKO 假設 a1）。",
          "en": "Partition rows (labels for outcomes 0, 1, 2, 3): P_0=(a,a,a,a), P_1=(a,a,b,b), P_2=(a,b,c,d); equal labels mean the same block (MKO assumption a1)."
        }
      },
      {
        "id": "G3",
        "statement": {
          "zh-Hant": "定義：F_t=σ(P_t)；(F_t)_{t=0,1,2} 是 filtration，若 s≤t ⇒ F_s⊆F_t。P' 細化 P（記 P'⪯P），若 P' 的每個區塊都含於 P 的某個區塊。",
          "en": "Definitions: F_t=σ(P_t); (F_t)_{t=0,1,2} is a filtration if s≤t ⇒ F_s⊆F_t. P' refines P (written P'⪯P) if every block of P' is contained in some block of P."
        }
      },
      {
        "id": "G4",
        "statement": {
          "zh-Hant": "對照序列：Q_0=(a,a,b,b)、Q_1=(a,b,a,b)。過程：X_1(ω)=1 若 ω∈{0,1}（第一次為正），否則 0；Y_1(ω)=1 若 ω∈{0,2}（第二次為正），否則 0。",
          "en": "Contrast sequence: Q_0=(a,a,b,b), Q_1=(a,b,a,b). Processes: X_1(ω)=1 if ω∈{0,1} (first toss heads), else 0; Y_1(ω)=1 if ω∈{0,2} (second toss heads), else 0."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G2",
          "set_builder"
        ],
        "before_tex": "P_0=(a,a,a,a),\\ P_1=(a,a,b,b),\\ P_2=(a,b,c,d)",
        "after_tex": "P_0=\\{\\Omega\\},\\ P_1=\\{\\{0,1\\},\\{2,3\\}\\},\\ P_2=\\{\\{0\\},\\{1\\},\\{2\\},\\{3\\}\\}",
        "relation": "equivalent",
        "rule": "set_builder",
        "conditions": [],
        "explanation": {
          "zh-Hant": "ω 所在的區塊是 {ω'∈Ω ∣ ω' 與 ω 的 label 相同}。P_0 全部同 label，只有一個區塊 Ω；P_1 中 0、1 同為 a，2、3 同為 b；P_2 四個 label 互異，得四個單點。label 名稱本身沒有意義：把 P_1 寫成 (b,b,a,a) 是同一分割。",
          "en": "The block of ω is {ω'∈Ω ∣ ω' has the same label as ω}. In P_0 all labels agree, giving the single block Ω; in P_1, 0 and 1 share a and 2 and 3 share b; in P_2 the four labels differ, giving four singletons. The label names carry no meaning: writing P_1 as (b,b,a,a) gives the same partition."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "set_builder",
            "G2"
          ],
          "explanation": {
            "zh-Hant": "由區塊可重建一行 label（每區塊給一個新 label），且只差 label 改名；兩種表示決定同一分割。",
            "en": "From the blocks a label row can be rebuilt (one new label per block), unique up to renaming labels; both descriptions determine the same partition."
          }
        },
        "leaf_basis": "set_builder + def_unfold"
      },
      {
        "id": "S02",
        "uses": [
          "S01",
          "sigma_partition"
        ],
        "before_tex": "F_t=\\sigma(P_t)",
        "after_tex": "F_0=\\{\\varnothing,\\Omega\\},\\ F_1=\\{\\varnothing,\\{0,1\\},\\{2,3\\},\\Omega\\},\\ F_2=2^{\\Omega}",
        "relation": "equivalent",
        "rule": "sigma_partition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "σ 代數由區塊的聯集組成：F_0 有 2^1=2 個事件，F_1 有 2^2=4 個，F_2 有 2^4=16 個，即 Ω 的全部子集（N1 到 N3）。事件 A 在時刻 t 可測，當且僅當它是 P_t 區塊的聯集（MKO 段落 2）。",
          "en": "A σ-algebra consists of the unions of blocks: F_0 has 2^1=2 events, F_1 has 2^2=4, and F_2 has 2^4=16, all subsets of Ω (N1 to N3). An event A is measurable at time t exactly when it is a union of blocks of P_t (MKO paragraph 2)."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "sigma_partition"
          ],
          "explanation": {
            "zh-Hant": "列出的集合正是區塊的全部聯集；雙向成立。",
            "en": "The listed sets are exactly all unions of blocks; the step holds both ways."
          }
        },
        "leaf_basis": "sigma_partition + set_operations_def + numeric_evaluation"
      },
      {
        "id": "S03",
        "uses": [
          "G3",
          "sigma_partition"
        ],
        "before_tex": "P'\\preceq P",
        "after_tex": "\\sigma(P)\\subseteq\\sigma(P')",
        "relation": "equivalent",
        "rule": "L_refine_iff",
        "conditions": [
          "P,P'\\ \\text{partitions of the same}\\ \\Omega"
        ],
        "explanation": {
          "zh-Hant": "依引理 L_refine_iff（在 S03.1、S03.2 證明），細化與 σ 代數包含等價。這把有限事件族之間的包含核對，改寫成較精簡的區塊包含核對：本題 |Ω|=4，事件至多 2^4=16 個；有 b 個區塊的分割恰生成 2^b 個聯集事件，並非無限多種事件。",
          "en": "By Lemma L_refine_iff (proved in S03.1 and S03.2), refinement is equivalent to inclusion of σ-algebras. This recasts inclusion between finite event families as a more compact complete check of block inclusions. Here |Ω|=4, so there are at most 2^4=16 events; a partition with b blocks generates exactly 2^b union events, not infinitely many events."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "L_refine_iff"
          ],
          "explanation": {
            "zh-Hant": "兩個方向分別在 S03.1、S03.2 證明；兩個分割必須在同一 Ω 上。",
            "en": "The two directions are proved in S03.1 and S03.2; both partitions must be on the same Ω."
          }
        },
        "leaf_basis": "sigma_partition + set_extensionality + subset_def"
      },
      {
        "id": "S03.1",
        "uses": [
          "G3",
          "sigma_partition",
          "set_operations_def"
        ],
        "before_tex": "P'\\preceq P,\\ B\\in P",
        "after_tex": "B=\\bigcup\\{C\\in P'\\mid C\\cap B\\neq\\varnothing\\},\\ \\text{hence}\\ \\sigma(P)\\subseteq\\sigma(P')",
        "relation": "implies",
        "rule": "set_extensionality",
        "conditions": [],
        "explanation": {
          "zh-Hant": "（⇒）取 B∈P。⊆：每個 ω∈B 屬於 P' 的某個區塊 C，且 C∩B 含 ω。⊇：若 C∩B≠∅，由細化 C⊆B' 對某個 B'∈P；則 B'∩B⊇C∩B≠∅，而 P 的區塊相等或互斥，所以 B'=B，C⊆B。因此每個 P 區塊是 P' 區塊的聯集；P 區塊的聯集也就是 P' 區塊的聯集，σ(P)⊆σ(P')。",
          "en": "(⇒) Take B∈P. ⊆: each ω∈B lies in some block C of P', and C∩B contains ω. ⊇: if C∩B≠∅, refinement gives C⊆B' for some B'∈P; then B'∩B⊇C∩B≠∅, and blocks of P are equal or disjoint, so B'=B and C⊆B. Hence each block of P is a union of blocks of P'; unions of P-blocks are then unions of P'-blocks, so σ(P)⊆σ(P')."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G3",
            "set_extensionality",
            "sigma_partition"
          ],
          "explanation": {
            "zh-Hant": "用到 P' 覆蓋 Ω 與 P 的區塊互斥；只證一個方向。",
            "en": "It uses that P' covers Ω and that the blocks of P are disjoint; only one direction is proved here."
          }
        },
        "leaf_basis": "set_extensionality + set_operations_def + sigma_partition"
      },
      {
        "id": "S03.2",
        "uses": [
          "G3",
          "sigma_partition"
        ],
        "before_tex": "\\sigma(P)\\subseteq\\sigma(P'),\\ C\\in P',\\ \\omega\\in C",
        "after_tex": "C\\subseteq B\\ \\text{for the block}\\ B\\in P\\ \\text{containing}\\ \\omega",
        "relation": "implies",
        "rule": "subset_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "（⇐）設 B 是含 ω 的 P 區塊。B∈σ(P)⊆σ(P')，所以 B 是若干 P' 區塊的聯集；ω∈B 落在其中某個 P' 區塊，而 ω 只屬於 P' 的區塊 C，所以 C 是其中之一，C⊆B。每個 C 都如此，所以 P'⪯P。S03.1 與 S03.2 合起來證得引理 L_refine_iff。",
          "en": "(⇐) Let B be the block of P containing ω. B∈σ(P)⊆σ(P'), so B is a union of blocks of P'; ω∈B lies in one of them, and ω belongs only to the P'-block C, so C is one of them and C⊆B. This holds for every C, so P'⪯P. S03.1 and S03.2 together prove Lemma L_refine_iff."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G3",
            "sigma_partition",
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "關鍵是 P' 的區塊互斥，使 ω 所在的 P' 區塊唯一。",
            "en": "The key is that the blocks of P' are disjoint, so the P'-block containing ω is unique."
          }
        },
        "leaf_basis": "subset_def + sigma_partition + set_operations_def"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "G3"
        ],
        "before_tex": "P_1\\ \\text{vs}\\ P_0",
        "after_tex": "\\{0,1\\}\\subseteq\\Omega,\\ \\{2,3\\}\\subseteq\\Omega:\\ P_1\\preceq P_0",
        "relation": "implies",
        "rule": "subset_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐一核對 P_1 的全部 2 個區塊：{0,1} 與 {2,3} 都含於 P_0 的唯一區塊 Ω。核對是完整的，每個區塊都檢查了。",
          "en": "Check both blocks of P_1: {0,1} and {2,3} each lie inside Ω, the only block of P_0. The check is complete; every block was examined."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "細化要求 P_1 的每個區塊都通過；這裡 2 個都通過。",
            "en": "Refinement requires every block of P_1 to pass; here both do."
          }
        },
        "leaf_basis": "subset_def + forall_intro"
      },
      {
        "id": "S05",
        "uses": [
          "S01",
          "G3"
        ],
        "before_tex": "P_2\\ \\text{vs}\\ P_1",
        "after_tex": "\\{0\\},\\{1\\}\\subseteq\\{0,1\\},\\ \\{2\\},\\{3\\}\\subseteq\\{2,3\\}:\\ P_2\\preceq P_1",
        "relation": "implies",
        "rule": "subset_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐一核對 P_2 的全部 4 個區塊：{0}、{1} 含於 {0,1}，{2}、{3} 含於 {2,3}。核對完整。",
          "en": "Check all 4 blocks of P_2: {0} and {1} lie in {0,1}, and {2} and {3} lie in {2,3}. The check is complete."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "4 個區塊全部通過；任一失敗就不是細化。",
            "en": "All 4 blocks pass; a single failure would rule out refinement."
          }
        },
        "leaf_basis": "subset_def + forall_intro"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "P_1\\preceq P_0,\\ P_2\\preceq P_1",
        "after_tex": "F_0\\subseteq F_1\\subseteq F_2,\\ \\text{so}\\ (F_t)_{t=0,1,2}\\ \\text{is a filtration}",
        "relation": "implies",
        "rule": "L_refine_iff",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依 L_refine_iff，S04 給 F_0⊆F_1，S05 給 F_1⊆F_2；包含關係可傳遞，所以 F_0⊆F_2。於是對所有 s≤t 都有 F_s⊆F_t（s=t 時自明），(F_t) 是 filtration。只需核相鄰時刻，正是因為包含可傳遞（MKO 假設 a2）。",
          "en": "By L_refine_iff, S04 gives F_0⊆F_1 and S05 gives F_1⊆F_2; inclusion is transitive, so F_0⊆F_2. Hence F_s⊆F_t for all s≤t (trivially when s=t), and (F_t) is a filtration. Checking adjacent times suffices precisely because inclusion is transitive (MKO assumption a2)."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04",
            "S05",
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "需要每一對相鄰時刻都通過；不宣稱這個 filtration 是某個過程的 natural filtration。",
            "en": "Every adjacent pair of times must pass; it is not claimed that this filtration is the natural filtration of any process."
          }
        },
        "leaf_basis": "subset_def + logic_chain + mko_filtration_def"
      },
      {
        "id": "S07",
        "uses": [
          "S02"
        ],
        "before_tex": "A=\\{0,1,2\\}",
        "after_tex": "A\\in F_2,\\ A\\notin F_1",
        "relation": "implies",
        "rule": "sigma_partition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "A 是 F_2 中單點的聯集，所以 A∈F_2。若 A 是 P_1 區塊的聯集，因 2∈A 而 2 所在區塊為 {2,3}，A 就必含 3，但 3∉A；所以 A∉F_1。資訊隨時間增加：時刻 2 可判定的事件，時刻 1 不一定可判定。",
          "en": "A is a union of singletons, so A∈F_2. If A were a union of P_1-blocks, then since 2∈A and the block of 2 is {2,3}, A would contain 3; but 3∉A, so A∉F_1. Information grows with time: an event decidable at time 2 need not be decidable at time 1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "sigma_partition"
          ],
          "explanation": {
            "zh-Hant": "只需找到一個被切開的區塊 {2,3}；不宣稱反向。",
            "en": "One split block, {2,3}, suffices; no converse is claimed."
          }
        },
        "leaf_basis": "sigma_partition + contradiction"
      },
      {
        "id": "S08",
        "uses": [
          "G4",
          "S03",
          "set_builder"
        ],
        "before_tex": "Q_0=\\{\\{0,1\\},\\{2,3\\}\\},\\ Q_1=\\{\\{0,2\\},\\{1,3\\}\\}",
        "after_tex": "Q_1\\not\\preceq Q_0,\\ \\text{so}\\ \\sigma(Q_0)\\not\\subseteq\\sigma(Q_1)",
        "relation": "implies",
        "rule": "L_refine_iff",
        "conditions": [],
        "explanation": {
          "zh-Hant": "Q_1 的區塊 {0,2} 既不含於 {0,1}（2∉{0,1}），也不含於 {2,3}（0∉{2,3}），所以 Q_1 不細化 Q_0；依 L_refine_iff，σ(Q_0)⊄σ(Q_1)。具體見證：{0,1}∈σ(Q_0)，但它含 0 而不含 2，而 {0,2} 是 Q_1 的區塊，所以 {0,1} 不是 Q_1 區塊的聯集。這兩個時刻的資訊不嵌套：任意的分割序列不是 filtration。",
          "en": "The Q_1-block {0,2} lies neither in {0,1} (2∉{0,1}) nor in {2,3} (0∉{2,3}), so Q_1 does not refine Q_0; by L_refine_iff, σ(Q_0)⊄σ(Q_1). A concrete witness: {0,1}∈σ(Q_0), but it contains 0 and not 2, while {0,2} is a Q_1-block, so {0,1} is not a union of Q_1-blocks. The information at the two times is not nested: an arbitrary sequence of partitions is not a filtration."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G4",
            "L_refine_iff"
          ],
          "explanation": {
            "zh-Hant": "一個不通過的區塊就足以否定細化；見證事件獨立地確認包含失敗。",
            "en": "One failing block suffices to refute refinement; the witness event independently confirms that inclusion fails."
          }
        },
        "leaf_basis": "subset_def + quantifier_negation + sigma_partition"
      },
      {
        "id": "S09",
        "uses": [
          "G4",
          "S01",
          "meas_const"
        ],
        "before_tex": "X_1=(1,1,0,0)\\ \\text{on}\\ (0,1,2,3)",
        "after_tex": "X_1\\ F_1\\text{-measurable}",
        "relation": "implies",
        "rule": "meas_const",
        "conditions": [],
        "explanation": {
          "zh-Hant": "X_1 在 {0,1} 上恆為 1、在 {2,3} 上恆為 0，即在 P_1 的每個區塊上為常數，所以 X_1 是 F_1 可測：第一次擲幣的結果在時刻 1 已知。",
          "en": "X_1 equals 1 throughout {0,1} and 0 throughout {2,3}, i.e. it is constant on every block of P_1, so X_1 is F_1-measurable: the first toss is known at time 1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "meas_const",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "要逐區塊核對常數，這裡 2 個區塊都通過。",
            "en": "Constancy must be checked block by block; here both blocks pass."
          }
        },
        "leaf_basis": "meas_const"
      },
      {
        "id": "S10",
        "uses": [
          "G4",
          "S01",
          "S06",
          "meas_const"
        ],
        "before_tex": "Y_1=(1,0,1,0)\\ \\text{on}\\ (0,1,2,3)",
        "after_tex": "Y_1\\ \\text{not}\\ F_1\\text{-measurable}",
        "relation": "implies",
        "rule": "meas_const",
        "conditions": [],
        "explanation": {
          "zh-Hant": "Y_1(0)=1、Y_1(1)=0，而 0、1 在 P_1 的同一區塊，所以 Y_1 在該區塊上不是常數，不是 F_1 可測。因此以 Y_1 作為時刻 1 值的過程不 adapted 於 (F_t)，即使 S06 已證明 (F_t) 是 filtration。Y_1 是 F_2 可測（任何函數在單點上都是常數）。",
          "en": "Y_1(0)=1 and Y_1(1)=0, while 0 and 1 lie in the same block of P_1, so Y_1 is not constant there and is not F_1-measurable. Hence a process whose time-1 value is Y_1 is not adapted to (F_t), even though S06 proved that (F_t) is a filtration. Y_1 is F_2-measurable (every function is constant on singletons)."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "meas_const",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "adapted 是過程與 filtration 之間的另一個條件，必須另核（MKO 段落 2）；filtration 的存在不提供它。",
            "en": "Adaptedness is a separate condition linking a process to a filtration and must be checked on its own (MKO paragraph 2); the existence of a filtration does not supply it."
          }
        },
        "leaf_basis": "meas_const + quantifier_negation"
      },
      {
        "id": "S11",
        "uses": [
          "S06",
          "G4",
          "prob_axioms"
        ],
        "before_tex": "P(\\{0\\})=P(\\{3\\})=\\tfrac12,\\ P(\\{1\\})=P(\\{2\\})=0",
        "after_tex": "P(X_1=1,\\,Y_1=1)=\\tfrac12\\neq\\tfrac14=P(X_1=1)P(Y_1=1)",
        "relation": "implies",
        "rule": "prob_axioms",
        "conditions": [],
        "explanation": {
          "zh-Hant": "(F_t) 的定義完全沒有用到機率。取一個合法的機率（總和 1/2+0+0+1/2=1，N5）：P(X_1=1)=P({0,1})=1/2，P(Y_1=1)=P({0,2})=1/2，但 P(X_1=1,Y_1=1)=P({0})=1/2≠1/4（N4）。兩次擲幣不獨立，而 filtration 不變。所以 filtration 不推出獨立；同理也不推出 Markov 性。",
          "en": "The definition of (F_t) uses no probability at all. Take a valid probability (total 1/2+0+0+1/2=1, N5): P(X_1=1)=P({0,1})=1/2 and P(Y_1=1)=P({0,2})=1/2, but P(X_1=1,Y_1=1)=P({0})=1/2≠1/4 (N4). The two tosses are not independent, while the filtration is unchanged. So a filtration does not imply independence; likewise it implies no Markov property."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "prob_axioms"
          ],
          "explanation": {
            "zh-Hant": "一個反例就足以否定「filtration 推出獨立」；不宣稱在其他機率下也不獨立。",
            "en": "One counterexample suffices to refute 'a filtration implies independence'; it is not claimed that independence fails under every probability."
          }
        },
        "leaf_basis": "prob_axioms + numeric_evaluation + quantifier_negation"
      },
      {
        "id": "S12",
        "uses": [
          "S06",
          "S08",
          "S10",
          "S11"
        ],
        "before_tex": "\\text{S06, S08, S10, S11}",
        "after_tex": "(F_t)\\ \\text{filtration};\\ (\\sigma(Q_t))\\ \\text{not};\\ \\text{adaptedness and independence not implied}",
        "relation": "implies",
        "rule": "mko_filtration_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整的相鄰細化核對證明 (F_t) 是 filtration；Q 序列因一個區塊不細化而不是；filtration 只表資訊嵌套，以 Y_1 為時刻 1 值的過程不 adapted，擲幣也可以不獨立。",
          "en": "The complete adjacent refinement check proves that (F_t) is a filtration; the sequence Q is not, because one block fails to refine; a filtration expresses information nesting only, a process with Y_1 as its time-1 value is not adapted, and the tosses need not be independent."
        },
        "reverse_audit": {
          "kind": "sufficient_goal_reduction",
          "required_ids": [
            "S06",
            "S08",
            "S10",
            "S11"
          ],
          "explanation": {
            "zh-Hant": "目標 G：本稿目標的四個結論。足夠條件 H：S06、S08、S10、S11；H⇒G 由合取。H 由 G1 到 G4 正向證成，沒有把 G 當作前提。",
            "en": "Goal G: the four conclusions stated in the goal. Sufficient condition H: S06, S08, S10 and S11; H⇒G by conjunction. H is proved forwards from G1 to G4; G is never used as a premise."
          }
        },
        "leaf_basis": "and_intro_elim"
      }
    ],
    "result": {
      "zh-Hant": "對同一有限 Ω 的分割，較晚分割細化較早分割當且僅當 σ 代數包含。P_1 的 2 個區塊含於 Ω，P_2 的 4 個單點含於 P_1 的區塊，所以 F_0⊆F_1⊆F_2，(F_t) 是 filtration；Q_1 的區塊 {0,2} 不含於任何 Q_0 區塊，{0,1}∈σ(Q_0)∖σ(Q_1)，不是 filtration。第一次擲幣 X_1 是 F_1 可測，第二次擲幣 Y_1 不是；在 P({0})=P({3})=1/2 下兩次擲幣不獨立。filtration 只表示資訊嵌套。",
      "en": "For partitions of the same finite Ω, the later partition refines the earlier one exactly when the σ-algebras are nested. The 2 blocks of P_1 lie in Ω and the 4 singletons of P_2 lie in blocks of P_1, so F_0⊆F_1⊆F_2 and (F_t) is a filtration; the Q_1-block {0,2} lies in no Q_0-block, {0,1}∈σ(Q_0)∖σ(Q_1), and Q is not a filtration. The first toss X_1 is F_1-measurable, the second toss Y_1 is not; under P({0})=P({3})=1/2 the two tosses are not independent. A filtration expresses information nesting only."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "σ(Q_0)、σ(Q_1) 都是同一 Ω 上的 σ 代數，所以 (σ(Q_0),σ(Q_1)) 是 filtration",
          "en": "σ(Q_0) and σ(Q_1) are σ-algebras on the same Ω, so (σ(Q_0),σ(Q_1)) is a filtration"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "{0,1}∈σ(Q_0) 但不在 σ(Q_1)（S08）。",
          "en": "{0,1}∈σ(Q_0) but it is not in σ(Q_1) (S08)."
        }
      },
      {
        "claim": {
          "zh-Hant": "(F_t) 是 filtration，所以第二次擲幣 Y_1 是 F_1 可測",
          "en": "(F_t) is a filtration, so the second toss Y_1 is F_1-measurable"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "Y_1 在區塊 {0,1} 上取 1 與 0（S10）。",
          "en": "Y_1 takes the values 1 and 0 on the block {0,1} (S10)."
        }
      },
      {
        "claim": {
          "zh-Hant": "(F_t) 是 filtration，所以兩次擲幣獨立",
          "en": "(F_t) is a filtration, so the two tosses are independent"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "P({0})=P({3})=1/2 時 P(兩次皆正)=1/2≠1/4（S11）。",
          "en": "With P({0})=P({3})=1/2, P(both heads)=1/2≠1/4 (S11)."
        }
      },
      {
        "claim": {
          "zh-Hant": "∅、Ω 都屬於 σ(Q_1)，所以 σ(Q_0)⊆σ(Q_1)",
          "en": "∅ and Ω both belong to σ(Q_1), so σ(Q_0)⊆σ(Q_1)"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "包含要對 σ(Q_0) 的每個事件成立；{0,1} 失敗。",
          "en": "Inclusion must hold for every event of σ(Q_0); {0,1} fails."
        }
      },
      {
        "claim": {
          "zh-Hant": "把 P_1 改寫成 (b,b,a,a) 就得到不同的分割與不同的 F_1",
          "en": "Rewriting P_1 as (b,b,a,a) gives a different partition and a different F_1"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "label 只表示區塊等價；區塊仍是 {0,1}、{2,3}（S01）。",
          "en": "Labels only indicate block equivalence; the blocks are still {0,1} and {2,3} (S01)."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "set_operations_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合運算：x∈A∪B ⇔ x∈A 或 x∈B；x∈A∩B ⇔ x∈A 且 x∈B；x∈A∖B ⇔ x∈A 且 x∉B；在全集 U 中 Aᶜ=U∖A。",
          "en": "Set operations: x∈A∪B ⇔ x∈A or x∈B; x∈A∩B ⇔ x∈A and x∈B; x∈A∖B ⇔ x∈A and x∉B; in a universe U, Aᶜ=U∖A."
        }
      },
      {
        "id": "subset_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "子集：A⊆B 表示 A 的每個元素都是 B 的元素，即 ∀x (x∈A⇒x∈B)。",
          "en": "Subset: A⊆B means every element of A is an element of B, i.e. ∀x (x∈A⇒x∈B)."
        }
      },
      {
        "id": "set_extensionality",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合相等：A=B 當且僅當 A⊆B 且 B⊆A，即兩者有相同元素。",
          "en": "Set equality: A=B if and only if A⊆B and B⊆A, i.e. they have the same elements."
        }
      },
      {
        "id": "prob_axioms",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "機率公理：P(A)≥0；P(Ω)=1；兩兩互斥事件的（可數）聯集機率等於各機率之和。由此 P(∅)=0、P(Aᶜ)=1−P(A)。",
          "en": "Probability axioms: P(A)≥0; P(Ω)=1; the probability of a (countable) union of pairwise disjoint events is the sum of their probabilities. Hence P(∅)=0 and P(Aᶜ)=1−P(A)."
        }
      },
      {
        "id": "set_builder",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合建構式：x∈{y∈A ∣ P(y)} 當且僅當 x∈A 且 P(x)。",
          "en": "Set-builder notation: x∈{y∈A ∣ P(y)} if and only if x∈A and P(x)."
        }
      },
      {
        "id": "def_unfold",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "展開定義：把一個已定義的詞換成它的定義內容（或反向收回），命題的真假不變。",
          "en": "Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true."
        }
      },
      {
        "id": "numeric_evaluation",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "明確整數或有理數算式的精確求值；可以獨立重算。",
          "en": "Exact evaluation of an explicit integer or rational expression; independently recomputable."
        }
      },
      {
        "id": "forall_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∀引入（全稱推廣）：若某命題對一個「任意的」元素 x 證成，而證明沒有對 x 加任何額外假設，則它對論域中所有 x 成立。",
          "en": "∀-introduction (generalization): if a statement is proved for an arbitrary element x, with no extra assumption on x, it holds for every x in the domain."
        }
      },
      {
        "id": "logic_chain",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "推論的傳遞：由 P⇒Q 與 Q⇒R 得 P⇒R；⇔ 同理。",
          "en": "Chaining: from P⇒Q and Q⇒R infer P⇒R; likewise for ⇔."
        }
      },
      {
        "id": "contradiction",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "反證法：若假設 ¬P 推出矛盾（某命題與其否定同時成立），則 P 成立。",
          "en": "Proof by contradiction: if assuming ¬P leads to a contradiction (a statement and its negation both hold), then P holds."
        }
      },
      {
        "id": "quantifier_negation",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "量詞否定：¬(∀x, P(x)) 等價於 ∃x, ¬P(x)；¬(∃x, P(x)) 等價於 ∀x, ¬P(x)。",
          "en": "Negating quantifiers: ¬(∀x, P(x)) is equivalent to ∃x, ¬P(x), and ¬(∃x, P(x)) is equivalent to ∀x, ¬P(x)."
        }
      },
      {
        "id": "and_intro_elim",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "合取：由 P 與 Q 得 P∧Q；由 P∧Q 可分別得 P 與 Q。",
          "en": "Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q."
        }
      },
      {
        "id": "mko_filtration_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-finite-filtration",
          "sha256": "1b51e644ad6518d67dc1c6b78e1ff5043eb9a2fe5576f71a9f732b6b1c168a62",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "以同一有限樣本空間上的 partition 表示各時刻 sigma 代數。F_s 包含於 F_t 等價於較晚 partition 細化較早 partition。filtration 只表資訊嵌套，不自動使過程 adapted，也不宣稱 natural filtration、獨立或 Markov 性。",
          "en": "Represent each time's sigma-algebra by a partition of the same finite sample space. F_s being contained in F_t means the later partition refines the earlier one. A filtration specifies information nesting only: a process is not automatically adapted, and no natural-filtration, independence or Markov claim follows."
        }
      },
      {
        "id": "sigma_partition",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限集合 Ω 的分割 𝒫 生成的 σ 代數 σ(𝒫) 恰由區塊的聯集組成（含 ∅）；共有 2^{|𝒫|} 個事件。",
          "en": "The σ-algebra σ(𝒫) generated by a partition 𝒫 of a finite set Ω consists exactly of the unions of blocks (including ∅); it has 2^{|𝒫|} events."
        }
      },
      {
        "id": "meas_const",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "Ω 上的實函數為 σ(𝒫) 可測，當且僅當它在 𝒫 的每個區塊上為常數。",
          "en": "A real function on Ω is σ(𝒫)-measurable exactly when it is constant on each block of 𝒫."
        }
      },
      {
        "id": "L_refine_iff",
        "kind": "lemma",
        "proved_in": [
          "S03.1",
          "S03.2"
        ],
        "statement": {
          "zh-Hant": "對同一有限 Ω 的分割 P、P'：P'⪯P ⇔ σ(P)⊆σ(P')。",
          "en": "For partitions P and P' of the same finite Ω: P'⪯P ⇔ σ(P)⊆σ(P')."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N1",
        "step": "S02",
        "lhs": "2**1",
        "rhs": "2"
      },
      {
        "id": "N2",
        "step": "S02",
        "lhs": "2**2",
        "rhs": "4"
      },
      {
        "id": "N3",
        "step": "S02",
        "lhs": "2**4",
        "rhs": "16"
      },
      {
        "id": "N4",
        "step": "S11",
        "lhs": "1/2",
        "relation": "!=",
        "rhs": "(1/2)*(1/2)"
      },
      {
        "id": "N5",
        "step": "S11",
        "lhs": "1/2+0+0+1/2",
        "rhs": "1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只在 MKO 範圍內工作：所有分割使用同一樣本載體 0 到 3，每行的 label 只表示區塊等價（假設 a1）；細化逐相鄰時刻完整核對（假設 a2）。不宣稱 natural filtration、獨立或 Markov 性質；不處理 martingale；不改動 canonical MKO 或英文 edition。",
      "en": "Works inside the MKO's scope: every partition uses the same outcomes 0 to 3, and labels within a row indicate only block equivalence (assumption a1); refinement is checked completely at every pair of adjacent times (assumption a2). No natural-filtration, independence or Markov claim is made; martingales are not treated; the canonical MKO and its English edition are not changed."
    },
    "md_symbols": [
      [
        "P_t",
        "時刻 t 的分割",
        "the partition at time t"
      ],
      [
        "F_t=\\sigma(P_t)",
        "P_t 生成的 σ 代數（區塊的聯集）",
        "the σ-algebra generated by P_t (unions of blocks)"
      ],
      [
        "P'\\preceq P",
        "P' 細化 P：P' 的每個區塊含於 P 的某個區塊",
        "P' refines P: each block of P' lies inside some block of P"
      ],
      [
        "X_1,\\ Y_1",
        "第一次、第二次擲幣是否為正的指示函數",
        "indicators that the first, respectively second, toss is heads"
      ],
      [
        "Q_0,Q_1",
        "不構成 filtration 的分割序列",
        "a sequence of partitions that is not a filtration"
      ]
    ],
    "md_error_controls": [
      {
        "title": "任意分割序列不一定是 filtration / An arbitrary partition sequence need not be a filtration",
        "zh": "MKO 指出的誤解：把任意 partition 序列叫 filtration。Q_0、Q_1 都是同一 Ω 上的分割，但 Q_1 不細化 Q_0，{0,1} 在時刻 0 可判定、在時刻 1 卻不可判定（S08）。必須逐相鄰時刻完整核對每個區塊。",
        "en": "The MKO's misconception: calling any sequence of partitions a filtration. Q_0 and Q_1 are both partitions of the same Ω, but Q_1 does not refine Q_0, and {0,1} is decidable at time 0 but not at time 1 (S08). Every block must be checked at every adjacent pair of times."
      },
      {
        "title": "filtration 不推出 adapted、獨立或 Markov / A filtration implies no adaptedness, independence or Markov property",
        "zh": "MKO 的第二個誤解。S06 證明 (F_t) 是 filtration，但 Y_1 不是 F_1 可測（S10），且在 P({0})=P({3})=1/2 下擲幣不獨立（S11）。adapted 要逐過程另核；獨立與 Markov 依賴機率，filtration 的定義不含機率。",
        "en": "The MKO's second misconception. S06 proves that (F_t) is a filtration, yet Y_1 is not F_1-measurable (S10), and under P({0})=P({3})=1/2 the tosses are not independent (S11). Adaptedness must be checked separately for each process; independence and Markov properties depend on the probability, which the definition of a filtration does not involve."
      },
      {
        "title": "抽查事件不能證明包含 / Sampled events do not prove inclusion",
        "zh": "MKO 假設 a2：有限抽樣事件不足以證 σ 代數包含。對 Q 序列，事件 ∅ 與 Ω 都同時在 σ(Q_0) 與 σ(Q_1) 中，只抽這兩個會誤判包含成立；完整的區塊細化核對（S03 的引理）才是證書。",
        "en": "MKO assumption a2: finitely sampled events do not establish inclusion of σ-algebras. For the sequence Q, the events ∅ and Ω lie in both σ(Q_0) and σ(Q_1); sampling only these would wrongly suggest inclusion. The complete block-refinement check (the lemma of S03) is the certificate."
      }
    ],
    "md_open_items": [
      {
        "zh-Hant": "本稿不宣稱 (F_t) 是某個過程的 natural filtration，也不處理 martingale（MKO 的下一步，需與條件期望一起加入）。",
        "en": "This guide does not claim that (F_t) is the natural filtration of any process, and it does not treat martingales (the MKO's next step, to be added together with conditional expectation)."
      }
    ]
  },
  "revision": {
    "id": "production-20261019-adopted-96",
    "source_commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
    "adopted_source_sha256": "e6820be99eb83c1217b02247f1bc0e38f7ba1bd24ea85ba09fd8e6937818f6c0",
    "original_source_sha256": "ab8a8b045d8752cb59b7afd124424df8d7210731678c8341756298606f15afcf",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "independent_AI_concurrence": "NOT_MEASURED",
    "issue_ids": [
      "BR-M095"
    ],
    "scope": "Owner-adopted natural-language teaching guide; not a formal proof, canonical MKO rewrite or evidence-status promotion."
  }
}