{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-function-graph-01",
  "mko_id": "mko-function-graph",
  "title": {
    "zh-Hant": "實值函數圖形：成員判定與有限表格的總函數驗證",
    "en": "Graphs of real-valued functions: membership tests and verifying a finite table as a total function"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-function-graph-01",
    "source": {
      "mko_id": "mko-function-graph",
      "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
      "sha256": "6a0892c9773f3366d44da22a063aa3c99163828135681c4e15719eda4e3638be",
      "english_sha256": "4c1bb95586ab87e347b417ee2f0798ef6a6326dd0c2100e111a3f89dd223f2f2"
    },
    "mode": "definition_walkthrough",
    "audience_prerequisites": [
      "cartesian_product_def",
      "function_def",
      "set_builder",
      "set_extensionality",
      "order_basics",
      "numeric_evaluation"
    ],
    "goal": {
      "zh-Hant": "(一) 設 D=[0,2]、f:D→ℝ、f(x)=x²−1。先證明 (u,v)∈G_f ⇔ u∈D 且 v=f(u)，再判定 (1,0)、(−1,0)、(2,2)、(2,3) 是否在 G_f 上。(二) 設 D_0={0,1,2}、B={0,1,4}。證明一個集合 Γ 是某個總函數 D_0→B 的圖形，當且僅當 Γ⊆D_0×B 且每個輸入恰有一個輸出；據此判定 T={(0,0),(1,1),(2,4)} 是、而 T_1={(0,0),(2,4)}、T_2={(0,0),(1,1),(1,4),(2,4)}、T_3={(0,0),(1,1),(2,4),(3,9)} 都不是。",
      "en": "(1) Let D=[0,2], f:D→ℝ, f(x)=x²−1. First prove (u,v)∈G_f ⇔ u∈D and v=f(u), then decide whether (1,0), (−1,0), (2,2) and (2,3) lie on G_f. (2) Let D_0={0,1,2} and B={0,1,4}. Prove that a set Γ is the graph of some total function D_0→B exactly when Γ⊆D_0×B and every input has exactly one output; use it to decide that T={(0,0),(1,1),(2,4)} is such a graph, while T_1={(0,0),(2,4)}, T_2={(0,0),(1,1),(1,4),(2,4)} and T_3={(0,0),(1,1),(2,4),(3,9)} are not."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "固定平面直角座標系，點以實數有序對表示；先宣告定義域，總函數在定義域的每個輸入恰有一個實數輸出（MKO 假設 a1、a2）。",
          "en": "A Cartesian coordinate system is fixed and points are ordered pairs of reals; the domain is declared first, and a total function has exactly one real output at each input of its domain (MKO assumptions a1, a2)."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "D=[0,2]={x∈ℝ ∣ 0≤x≤2}，f:D→ℝ，f(x)=x²−1；待判定的有序對為 (1,0)、(−1,0)、(2,2)、(2,3)。",
          "en": "D=[0,2]={x∈ℝ ∣ 0≤x≤2}, f:D→ℝ, f(x)=x²−1; the pairs to test are (1,0), (−1,0), (2,2) and (2,3)."
        }
      },
      {
        "id": "G3",
        "statement": {
          "zh-Hant": "D_0={0,1,2}，B={0,1,4}；候選表格 T={(0,0),(1,1),(2,4)}、T_1={(0,0),(2,4)}、T_2={(0,0),(1,1),(1,4),(2,4)}、T_3={(0,0),(1,1),(2,4),(3,9)}。",
          "en": "D_0={0,1,2} and B={0,1,4}; candidate tables T={(0,0),(1,1),(2,4)}, T_1={(0,0),(2,4)}, T_2={(0,0),(1,1),(1,4),(2,4)} and T_3={(0,0),(1,1),(2,4),(3,9)}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2"
        ],
        "before_tex": "f:D\\to\\mathbb{R}\\ \\text{total},\\ D\\subseteq\\mathbb{R}",
        "after_tex": "G_f=\\{(x,f(x))\\mid x\\in D\\}\\subseteq\\mathbb{R}^2",
        "relation": "definition",
        "rule": "mko_graph_def",
        "conditions": [
          "D\\ \\text{declared}"
        ],
        "explanation": {
          "zh-Hant": "依 MKO，圖形是讓 x 走遍宣告的定義域 D 時得到的有序對 (x,f(x)) 的集合。定義域是圖形定義的一部分：同一個公式配上不同的 D，就是不同的函數、不同的圖形。",
          "en": "By the MKO, the graph is the set of ordered pairs (x,f(x)) obtained as x runs through the declared domain D. The domain is part of the definition of the graph: the same formula with a different D is a different function with a different graph."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "mko_graph_def"
          ],
          "explanation": {
            "zh-Hant": "此步只寫出定義；用到 f 是總函數，所以每個 x∈D 都貢獻恰好一點。",
            "en": "This step only states the definition; it uses that f is total, so every x∈D contributes exactly one point."
          }
        },
        "leaf_basis": "mko_graph_def + set_builder"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "(u,v)\\in G_f",
        "after_tex": "u\\in D\\ \\land\\ v=f(u)",
        "relation": "equivalent",
        "rule": "L_graph_member",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依引理 L_graph_member，成員判定要同時檢查兩件事：u 在定義域中，且 v 等於 f(u)。兩個方向在 S02.1、S02.2 證明。",
          "en": "By Lemma L_graph_member, a membership test must check two things together: u lies in the domain, and v equals f(u). The two directions are proved in S02.1 and S02.2."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "L_graph_member"
          ],
          "explanation": {
            "zh-Hant": "引理兩個方向都已在子步證明，所以可作為等價使用。",
            "en": "Both directions of the lemma are proved in the sub-steps, so it may be used as an equivalence."
          }
        },
        "leaf_basis": "set_builder + cartesian_product_def + exists_intro + exists_elim"
      },
      {
        "id": "S02.1",
        "uses": [
          "S01",
          "cartesian_product_def"
        ],
        "before_tex": "(u,v)\\in G_f",
        "after_tex": "u\\in D\\ \\land\\ v=f(u)",
        "relation": "implies",
        "rule": "cartesian_product_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "(u,v)∈G_f 表示有某個 x∈D 使 (u,v)=(x,f(x))。依有序對相等條件，u=x 且 v=f(x)。把 x 換成 u：u∈D，且 v=f(u)。這是 ⇒ 方向。",
          "en": "(u,v)∈G_f means there is some x∈D with (u,v)=(x,f(x)). By the equality condition for ordered pairs, u=x and v=f(x). Replacing x by u: u∈D and v=f(u). This is the ⇒ direction."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "cartesian_product_def",
            "exists_elim",
            "eq_replace"
          ],
          "explanation": {
            "zh-Hant": "取出見證 x 後，結論不含 x，符合 ∃消去的規則。",
            "en": "After naming the witness x, the conclusion no longer mentions x, as ∃-elimination requires."
          }
        },
        "leaf_basis": "exists_elim + cartesian_product_def + eq_replace"
      },
      {
        "id": "S02.2",
        "uses": [
          "S01",
          "exists_intro"
        ],
        "before_tex": "u\\in D\\ \\land\\ v=f(u)",
        "after_tex": "(u,v)=(u,f(u))\\in G_f",
        "relation": "implies",
        "rule": "exists_intro",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若 u∈D 且 v=f(u)，則 (u,v)=(u,f(u))，取 x=u∈D 作見證，(u,v) 屬於 G_f。這是 ⇐ 方向。S02.1 與 S02.2 合起來證得引理 L_graph_member。",
          "en": "If u∈D and v=f(u), then (u,v)=(u,f(u)), and with the witness x=u∈D, (u,v) belongs to G_f. This is the ⇐ direction. S02.1 and S02.2 together prove Lemma L_graph_member."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "exists_intro",
            "cartesian_product_def"
          ],
          "explanation": {
            "zh-Hant": "見證 x=u 必須在 D 中；若 u∉D，即使 v 等於公式的值，也找不到見證。",
            "en": "The witness x=u must lie in D; if u∉D there is no witness even when v equals the value of the formula."
          }
        },
        "leaf_basis": "exists_intro + cartesian_product_def"
      },
      {
        "id": "S03",
        "uses": [
          "S02",
          "G2"
        ],
        "before_tex": "(1,0)",
        "after_tex": "1\\in D,\\ f(1)=1^2-1=0\\ \\Rightarrow\\ (1,0)\\in G_f",
        "relation": "implies",
        "rule": "L_graph_member",
        "conditions": [
          "0\\le 1\\le 2"
        ],
        "explanation": {
          "zh-Hant": "第一個條件：0≤1≤2，所以 1∈D。第二個條件：f(1)=1²−1=0，等於第二坐標 0。兩者都成立，依 S02，(1,0)∈G_f。",
          "en": "First condition: 0≤1≤2, so 1∈D. Second condition: f(1)=1²−1=0, which equals the second coordinate 0. Both hold, so (1,0)∈G_f by S02."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "numeric_evaluation",
            "order_basics"
          ],
          "explanation": {
            "zh-Hant": "兩個條件都核對了（N1、N2、N3），缺一不可。",
            "en": "Both conditions are checked (N1, N2, N3); neither may be skipped."
          }
        },
        "leaf_basis": "L_graph_member + order_basics + numeric_evaluation"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "G2"
        ],
        "before_tex": "(-1,0)",
        "after_tex": "-1\\notin D\\ \\Rightarrow\\ (-1,0)\\notin G_f",
        "relation": "implies",
        "rule": "L_graph_member",
        "conditions": [
          "-1<0"
        ],
        "explanation": {
          "zh-Hant": "公式在 x=−1 也能求值：(−1)²−1=0，與第二坐標相等。但 −1<0，所以 −1∉D=[0,2]，S02 的第一個條件不成立，(−1,0)∉G_f。這正是 MKO 提醒的情形：公式在域外可求值，不代表該點在宣告的圖形內。",
          "en": "The formula can be evaluated at x=−1: (−1)²−1=0, equal to the second coordinate. But −1<0, so −1∉D=[0,2]; the first condition of S02 fails and (−1,0)∉G_f. This is exactly the case the MKO warns about: a formula that can be evaluated outside the domain does not put the point on the declared graph."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "order_basics",
            "numeric_evaluation"
          ],
          "explanation": {
            "zh-Hant": "只要一個條件不成立就推出不屬於（N4、N5）；不需要也不應只看等式。",
            "en": "Failure of one condition is enough for non-membership (N4, N5); the equation alone does not and should not decide it."
          }
        },
        "leaf_basis": "L_graph_member + order_basics + numeric_evaluation"
      },
      {
        "id": "S05",
        "uses": [
          "S02",
          "G2"
        ],
        "before_tex": "(2,2),\\ (2,3)",
        "after_tex": "f(2)=3:\\ (2,2)\\notin G_f,\\ (2,3)\\in G_f",
        "relation": "implies",
        "rule": "L_graph_member",
        "conditions": [
          "0\\le 2\\le 2"
        ],
        "explanation": {
          "zh-Hant": "0≤2≤2，所以 2∈D（端點包含在閉區間內）。f(2)=2²−1=3。(2,2) 的第二坐標 2≠3，所以 (2,2)∉G_f；(2,3) 兩個條件都成立，(2,3)∈G_f。G_f 中第一坐標為 2 的點恰有 (2,3) 一個，因為 f 在 2 只有一個輸出。",
          "en": "0≤2≤2, so 2∈D (the endpoint belongs to the closed interval). f(2)=2²−1=3. The second coordinate of (2,2) is 2≠3, so (2,2)∉G_f; (2,3) satisfies both conditions, so (2,3)∈G_f. The only point of G_f with first coordinate 2 is (2,3), because f has a single output at 2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "numeric_evaluation",
            "function_def"
          ],
          "explanation": {
            "zh-Hant": "u∈D 不夠，v=f(u) 也必須成立（N6、N7）。",
            "en": "u∈D is not enough; v=f(u) must hold as well (N6, N7)."
          }
        },
        "leaf_basis": "L_graph_member + function_def + numeric_evaluation"
      },
      {
        "id": "S06",
        "uses": [
          "G3",
          "S02"
        ],
        "before_tex": "\\Gamma\\ \\text{is the graph of some total}\\ F:D\\to B",
        "after_tex": "\\Gamma\\subseteq D\\times B\\ \\land\\ \\forall u\\in D\\ \\exists!\\,v:\\ (u,v)\\in\\Gamma",
        "relation": "equivalent",
        "rule": "L_table_graph",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依引理 L_table_graph，有限表格是不是一個總函數 D→B 的圖形，可用兩個條件判定：不多出 D×B 之外的點，且每個輸入恰有一個輸出。這是 MKO para1 的表格驗證；兩個方向在 S06.1、S06.2 證明。",
          "en": "By Lemma L_table_graph, whether a table is the graph of a total function D→B is decided by two conditions: no point outside D×B, and exactly one output for every input. This is the table verification of MKO para1; the two directions are proved in S06.1 and S06.2."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "L_table_graph",
            "mko_table_check"
          ],
          "explanation": {
            "zh-Hant": "引理兩個方向都在子步證明，所以可作為等價使用；後面用 ⇐ 接受 T，用 ⇒ 的逆否拒絕 T_1、T_2、T_3。",
            "en": "Both directions of the lemma are proved in the sub-steps, so it may be used as an equivalence; ⇐ accepts T below, and the contrapositive of ⇒ rejects T_1, T_2 and T_3."
          }
        },
        "leaf_basis": "function_def + set_extensionality + L_graph_member"
      },
      {
        "id": "S06.1",
        "uses": [
          "S02",
          "function_def",
          "set_extensionality"
        ],
        "before_tex": "\\Gamma\\subseteq D\\times B,\\ \\forall u\\in D\\ \\exists!\\,v:(u,v)\\in\\Gamma",
        "after_tex": "F(u):=\\text{the unique}\\ v,\\quad F:D\\to B\\ \\text{total},\\quad G_F=\\Gamma",
        "relation": "implies",
        "rule": "function_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "⇐ 方向：對每個 u∈D，令 F(u) 為那個唯一的 v；因 Γ⊆D×B，F(u)∈B，所以 F:D→B 是總函數。再證 G_F=Γ：由 L_graph_member，(u,v)∈G_F ⇔ u∈D 且 v=F(u)。若 u∈D 且 v=F(u)，依 F 的取法 (u,v)∈Γ；反之若 (u,v)∈Γ，則 u∈D（Γ⊆D×B），且依唯一性 v=F(u)。依 set_extensionality，G_F=Γ。",
          "en": "The ⇐ direction: for each u∈D let F(u) be that unique v; since Γ⊆D×B, F(u)∈B, so F:D→B is a total function. Now G_F=Γ: by L_graph_member, (u,v)∈G_F ⇔ u∈D and v=F(u). If u∈D and v=F(u), then (u,v)∈Γ by the choice of F; conversely, if (u,v)∈Γ then u∈D (as Γ⊆D×B), and v=F(u) by uniqueness. By set_extensionality, G_F=Γ."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "function_def",
            "set_extensionality",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "存在性給出 F 在每點有定義（總），唯一性給出 F 是函數且 Γ⊆G_F；Γ⊆D×B 保證輸出落在 B 且沒有域外點。",
            "en": "Existence makes F defined at every point (total), uniqueness makes F a function and gives Γ⊆G_F, and Γ⊆D×B keeps outputs in B with no point outside the domain."
          }
        },
        "leaf_basis": "function_def + set_extensionality + L_graph_member"
      },
      {
        "id": "S06.2",
        "uses": [
          "S02",
          "function_def"
        ],
        "before_tex": "\\Gamma=G_F,\\ F:D\\to B\\ \\text{total}",
        "after_tex": "\\Gamma\\subseteq D\\times B\\ \\land\\ \\forall u\\in D\\ \\exists!\\,v:(u,v)\\in\\Gamma",
        "relation": "implies",
        "rule": "L_graph_member",
        "conditions": [],
        "explanation": {
          "zh-Hant": "⇒ 方向：若 Γ=G_F，Γ 的每個元素都是 (x,F(x))，x∈D、F(x)∈B，所以 Γ⊆D×B。對 u∈D，依 L_graph_member，(u,v)∈Γ ⇔ v=F(u)，所以恰有一個 v，即 F(u)。S06.1 與 S06.2 合起來證得引理 L_table_graph。",
          "en": "The ⇒ direction: if Γ=G_F, every element of Γ is (x,F(x)) with x∈D and F(x)∈B, so Γ⊆D×B. For u∈D, by L_graph_member, (u,v)∈Γ ⇔ v=F(u), so there is exactly one v, namely F(u). S06.1 and S06.2 together prove Lemma L_table_graph."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "function_def"
          ],
          "explanation": {
            "zh-Hant": "這個方向的逆否是拒絕表格的依據：只要缺一個輸出、多一個輸出或多出域外點，就不是總函數 D→B 的圖形。",
            "en": "The contrapositive of this direction is the basis for rejecting tables: a missing output, an extra output, or a point outside the domain each rules out being the graph of a total function D→B."
          }
        },
        "leaf_basis": "L_graph_member + function_def + subset_def"
      },
      {
        "id": "S07",
        "uses": [
          "G3",
          "cartesian_product_def"
        ],
        "before_tex": "T=\\{(0,0),(1,1),(2,4)\\}",
        "after_tex": "T\\subseteq D_0\\times B",
        "relation": "implies",
        "rule": "cartesian_product_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐一檢查 T 的三個有序對：第一坐標 0、1、2 都在 D_0={0,1,2} 中，第二坐標 0、1、4 都在 B={0,1,4} 中。所以每個元素都屬於 D_0×B。",
          "en": "Check the three pairs of T one by one: the first coordinates 0, 1, 2 all lie in D_0={0,1,2}, and the second coordinates 0, 1, 4 all lie in B={0,1,4}. So every element belongs to D_0×B."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "cartesian_product_def",
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "有限集合可逐一列舉檢查，這是完整的檢查而非抽樣。",
            "en": "A finite set can be checked element by element; this is a complete check, not a sample."
          }
        },
        "leaf_basis": "cartesian_product_def + subset_def"
      },
      {
        "id": "S08",
        "uses": [
          "G3",
          "S07"
        ],
        "before_tex": "u\\in D_0=\\{0,1,2\\}",
        "after_tex": "u=0\\mapsto 0,\\ u=1\\mapsto 1,\\ u=2\\mapsto 4,\\ \\text{each exactly once}",
        "relation": "case_split",
        "rule": "case_analysis",
        "conditions": [],
        "explanation": {
          "zh-Hant": "D_0 只有三個元素，逐一檢查：第一坐標為 0 的有序對只有 (0,0)，為 1 的只有 (1,1)，為 2 的只有 (2,4)。每個輸入恰有一個輸出：既不缺（存在），也不多（唯一）。",
          "en": "D_0 has only three elements, so check each: the only pair with first coordinate 0 is (0,0), with 1 only (1,1), and with 2 only (2,4). Every input has exactly one output: none is missing (existence) and none is doubled (uniqueness)."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "case_analysis",
            "G3"
          ],
          "explanation": {
            "zh-Hant": "三個情形涵蓋 D_0 的全部元素；存在與唯一兩項都要檢查。",
            "en": "The three cases cover every element of D_0; both existence and uniqueness are checked."
          }
        },
        "leaf_basis": "case_analysis + and_intro_elim"
      },
      {
        "id": "S09",
        "uses": [
          "S06",
          "S07",
          "S08"
        ],
        "before_tex": "T\\subseteq D_0\\times B,\\ \\forall u\\in D_0\\ \\exists!\\,v",
        "after_tex": "T=G_F,\\ F:D_0\\to B,\\ F(0)=0,\\ F(1)=1,\\ F(2)=4",
        "relation": "theorem_application",
        "rule": "L_table_graph",
        "conditions": [],
        "explanation": {
          "zh-Hant": "S07 與 S08 正是 L_table_graph 的兩個條件，依 ⇐ 方向（S06.1），T 是總函數 F:D_0→B 的圖形，F(0)=0、F(1)=1、F(2)=4。這三個值恰與 x² 在 0、1、2 的值相同，但本稿不宣稱 F 在 D_0 之外有任何公式。",
          "en": "S07 and S08 are exactly the two conditions of L_table_graph, so by the ⇐ direction (S06.1) T is the graph of the total function F:D_0→B with F(0)=0, F(1)=1 and F(2)=4. These three values happen to match x² at 0, 1 and 2, but no formula for F outside D_0 is claimed."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "S07",
            "S08",
            "L_table_graph"
          ],
          "explanation": {
            "zh-Hant": "兩個條件都已正向證成；三點與 x² 吻合（N8、N11、N12）只是觀察，不是全域公式的證據。",
            "en": "Both conditions were proved forwards; the agreement with x² at three points (N8, N11, N12) is an observation, not evidence for a global formula."
          }
        },
        "leaf_basis": "L_table_graph + function_def"
      },
      {
        "id": "S10",
        "uses": [
          "S06",
          "G3"
        ],
        "before_tex": "T_1=\\{(0,0),(2,4)\\}",
        "after_tex": "\\neg\\exists v:(1,v)\\in T_1\\ \\Rightarrow\\ T_1\\ \\text{is not the graph of a total}\\ D_0\\to B",
        "relation": "implies",
        "rule": "contrapositive",
        "conditions": [
          "1\\in D_0"
        ],
        "explanation": {
          "zh-Hant": "T_1 中每個第一坐標至多出現一次（0 對應 0、2 對應 4），所以它通過垂直線測試。但 1∈D_0，而 T_1 中沒有第一坐標為 1 的有序對，存在性失敗。依 S06.2 的逆否，T_1 不是任何總函數 D_0→B 的圖形。垂直線測試只檢查「至多一個」，不檢查宣告域上的「至少一個」。",
          "en": "In T_1 each first coordinate occurs at most once (0 goes to 0, 2 goes to 4), so it passes the vertical line test. But 1∈D_0 and T_1 has no pair with first coordinate 1, so existence fails. By the contrapositive of S06.2, T_1 is not the graph of any total function D_0→B. The vertical line test checks 'at most one', not 'at least one' on the declared domain."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "contrapositive",
            "G3"
          ],
          "explanation": {
            "zh-Hant": "結論只針對宣告的 D_0；T_1 是定義在 {0,2} 上的函數圖形，但那是另一個定義域。",
            "en": "The conclusion concerns the declared D_0 only; T_1 is the graph of a function on {0,2}, but that is a different domain."
          }
        },
        "leaf_basis": "contrapositive + L_table_graph + quantifier_negation"
      },
      {
        "id": "S11",
        "uses": [
          "S06",
          "G3"
        ],
        "before_tex": "T_2=\\{(0,0),(1,1),(1,4),(2,4)\\}",
        "after_tex": "(1,1),(1,4)\\in T_2,\\ 1\\neq 4\\ \\Rightarrow\\ T_2\\ \\text{is not a function graph}",
        "relation": "implies",
        "rule": "contrapositive",
        "conditions": [],
        "explanation": {
          "zh-Hant": "輸入 1 在 T_2 中有兩個不同的輸出 1 與 4，唯一性失敗（垂直線 x=1 與 T_2 交於兩點）。依 S06.2 的逆否，T_2 不是任何總函數 D_0→B 的圖形；雖然它的四個點都在 D_0×B 中。",
          "en": "Input 1 has two different outputs, 1 and 4, in T_2, so uniqueness fails (the vertical line x=1 meets T_2 twice). By the contrapositive of S06.2, T_2 is not the graph of any total function D_0→B, even though all four of its points lie in D_0×B."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "contrapositive"
          ],
          "explanation": {
            "zh-Hant": "1≠4（N9）即足以推翻唯一性；不需要檢查其他輸入。",
            "en": "1≠4 (N9) is enough to break uniqueness; no other input needs checking."
          }
        },
        "leaf_basis": "contrapositive + L_table_graph + numeric_evaluation"
      },
      {
        "id": "S12",
        "uses": [
          "S06",
          "G3"
        ],
        "before_tex": "T_3=\\{(0,0),(1,1),(2,4),(3,9)\\}",
        "after_tex": "(3,9)\\notin D_0\\times B\\ \\Rightarrow\\ T_3\\ \\text{is not the graph of a total}\\ D_0\\to B",
        "relation": "implies",
        "rule": "contrapositive",
        "conditions": [
          "3\\notin D_0"
        ],
        "explanation": {
          "zh-Hant": "T_3 在 D_0 的每個輸入恰有一個輸出，但多了一點 (3,9)：3∉D_0（而且 9∉B），所以 T_3⊄D_0×B。依 S06.2 的逆否，T_3 不是總函數 D_0→B 的圖形。MKO 要求表格不能多出 D 外的圖形點。",
          "en": "T_3 has exactly one output at each input of D_0, but it has the extra point (3,9): 3∉D_0 (and 9∉B), so T_3⊄D_0×B. By the contrapositive of S06.2, T_3 is not the graph of a total function D_0→B. The MKO requires that a table have no graph point outside D."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06",
            "contrapositive",
            "cartesian_product_def"
          ],
          "explanation": {
            "zh-Hant": "「每個輸入恰有一個輸出」只是條件之一；Γ⊆D×B 也必須成立。",
            "en": "'Exactly one output per input' is only one of the conditions; Γ⊆D×B must hold too."
          }
        },
        "leaf_basis": "contrapositive + L_table_graph + cartesian_product_def"
      },
      {
        "id": "S13",
        "uses": [
          "S03",
          "S04",
          "S05",
          "S09",
          "S10",
          "S11",
          "S12"
        ],
        "before_tex": "\\text{four pairs and four tables}",
        "after_tex": "(1,0),(2,3)\\in G_f;\\ (-1,0),(2,2)\\notin G_f;\\ T\\ \\text{accepted};\\ T_1,T_2,T_3\\ \\text{rejected}",
        "relation": "implies",
        "rule": "and_intro_elim",
        "conditions": [],
        "explanation": {
          "zh-Hant": "彙整：(1,0) 與 (2,3) 在 G_f 上；(−1,0) 因 −1∉D、(2,2) 因 2≠f(2) 而不在。T 是總函數 D_0→B 的圖形；T_1 缺輸出、T_2 輸出不唯一、T_3 有域外點，都不是。",
          "en": "Summary: (1,0) and (2,3) lie on G_f; (−1,0) does not because −1∉D, and (2,2) does not because 2≠f(2). T is the graph of a total function D_0→B; T_1 misses an output, T_2 has a non-unique output and T_3 has a point outside the domain, so none of them is."
        },
        "reverse_audit": {
          "kind": "sufficient_goal_reduction",
          "required_ids": [
            "S03",
            "S04",
            "S05",
            "S09",
            "S10",
            "S11",
            "S12",
            "and_intro_elim"
          ],
          "explanation": {
            "zh-Hant": "目標 G：八個判定。足夠條件 H：S03 至 S12 的各別結論；H⇒G 只是合取。H 由定義與兩個引理正向證成，沒有把 G 當作前提。",
            "en": "Goal G: the eight decisions. Sufficient condition H: the separate conclusions of S03 to S12; H⇒G is just conjunction. H is proved forwards from the definitions and the two lemmas; G is never used as a premise."
          }
        },
        "leaf_basis": "and_intro_elim"
      }
    ],
    "result": {
      "zh-Hant": "(u,v)∈G_f 當且僅當 u∈D 且 v=f(u)。對 D=[0,2]、f(x)=x²−1：(1,0)、(2,3) 在圖形上；(−1,0) 不在（−1∉D，雖然公式值為 0）；(2,2) 不在（f(2)=3）。一個集合是總函數 D_0→B 的圖形，當且僅當它是 D_0×B 的子集合且每個輸入恰有一個輸出：T={(0,0),(1,1),(2,4)} 是；T_1 缺輸入 1 的輸出（雖通過垂直線測試）、T_2 在 1 有兩個輸出、T_3 多出 (3,9)，都不是。",
      "en": "(u,v)∈G_f exactly when u∈D and v=f(u). For D=[0,2] and f(x)=x²−1: (1,0) and (2,3) lie on the graph; (−1,0) does not (−1∉D, although the formula gives 0); (2,2) does not (f(2)=3). A set is the graph of a total function D_0→B exactly when it is a subset of D_0×B and every input has exactly one output: T={(0,0),(1,1),(2,4)} is one; T_1 lacks an output at 1 (though it passes the vertical line test), T_2 has two outputs at 1, and T_3 has the extra point (3,9), so none of these is."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "(−1,0)∈G_f，因為 (−1)²−1=0",
          "en": "(−1,0)∈G_f because (−1)²−1=0"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "−1∉D=[0,2]；成員判定還要求第一坐標在定義域中。",
          "en": "−1∉D=[0,2]; membership also requires the first coordinate to lie in the domain."
        }
      },
      {
        "claim": {
          "zh-Hant": "(2,2)∈G_f，因為 2∈D",
          "en": "(2,2)∈G_f because 2∈D"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "f(2)=3≠2；第二坐標必須等於 f(2)。",
          "en": "f(2)=3≠2; the second coordinate must equal f(2)."
        }
      },
      {
        "claim": {
          "zh-Hant": "T_1={(0,0),(2,4)} 通過垂直線測試，所以是總函數 D_0→B 的圖形",
          "en": "T_1={(0,0),(2,4)} passes the vertical line test, so it is the graph of a total function D_0→B"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "1∈D_0 沒有輸出；垂直線測試不檢查總定義性。",
          "en": "1∈D_0 has no output; the vertical line test does not check totality."
        }
      },
      {
        "claim": {
          "zh-Hant": "T_3 在 D_0 的每個輸入都恰有一個輸出，所以是總函數 D_0→B 的圖形",
          "en": "T_3 has exactly one output at every input of D_0, so it is the graph of a total function D_0→B"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "T_3 含 (3,9)，3∉D_0、9∉B，所以 T_3⊄D_0×B。",
          "en": "T_3 contains (3,9) with 3∉D_0 and 9∉B, so T_3⊄D_0×B."
        }
      },
      {
        "claim": {
          "zh-Hant": "T 與 x² 在 0、1、2 吻合，所以 F 就是 ℝ 上的函數 x²",
          "en": "T agrees with x² at 0, 1 and 2, so F is the function x² on ℝ"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "F 只定義在 D_0 上；有限資料不決定定義域外的值或全域公式。",
          "en": "F is defined only on D_0; finite data determine neither values outside the domain nor a global formula."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "cartesian_product_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "笛卡兒積：A×B 是所有有序對 (a,b)（a∈A、b∈B）的集合；(a,b)=(c,d) ⇔ a=c 且 b=d。",
          "en": "Cartesian product: A×B is the set of ordered pairs (a,b) with a∈A and b∈B; (a,b)=(c,d) ⇔ a=c and b=d."
        }
      },
      {
        "id": "function_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "函數：f:A→B 對每個 a∈A 指定恰好一個 f(a)∈B。兩函數相等 ⇔ 定義域、對應域相同且對每個 a 都有相同值。",
          "en": "Function: f:A→B assigns to each a∈A exactly one f(a)∈B. Two functions are equal ⇔ they have the same domain and codomain and agree at every 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": "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": "order_basics",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數的序：任兩數恰有 u<v、u=v、u>v 之一；u<v 且 v<w 推出 u<w；u<v 推出 u+w<v+w；u<v 且 w>0 推出 uw<vw；u<v 且 w<0 推出 uw>vw。",
          "en": "Order of the reals: exactly one of u<v, u=v, u>v holds; u<v and v<w give u<w; u<v gives u+w<v+w; u<v and w>0 give uw<vw; u<v and w<0 give uw>vw."
        }
      },
      {
        "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": "exists_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∃引入：若找到 a∈D 使 P(a) 成立，則 ∃x∈D, P(x)。",
          "en": "∃-introduction: if some a∈D satisfies P(a), then ∃x∈D, P(x)."
        }
      },
      {
        "id": "exists_elim",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∃消去：由 ∃x∈D, P(x)，可取一個新名字 a 表示某個滿足 P(a) 的元素；只要最後結論不含 a，就可用 a 推理。",
          "en": "∃-elimination: from ∃x∈D, P(x) we may name some element a with P(a) by a fresh name; reasoning with a is allowed as long as the final conclusion does not mention a."
        }
      },
      {
        "id": "eq_replace",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "等值代換：命題中的一個式子可換成與它相等的式子，命題的真假不變。",
          "en": "Replacement of equals: an expression inside a statement may be replaced by an equal expression without changing whether the statement is true."
        }
      },
      {
        "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": "case_analysis",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "分情況論證：若 P∨Q 成立，且 P⇒R、Q⇒R 都已證，則 R 成立。所有情況必須涵蓋全部可能。",
          "en": "Proof by cases: if P∨Q holds and both P⇒R and Q⇒R are proved, then R holds. The cases must cover every possibility."
        }
      },
      {
        "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": "contrapositive",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "逆否命題：P⇒Q 與 ¬Q⇒¬P 等價。逆命題 Q⇒P 一般不與 P⇒Q 等價。",
          "en": "Contrapositive: P⇒Q is equivalent to ¬Q⇒¬P. The converse Q⇒P is in general not equivalent to P⇒Q."
        }
      },
      {
        "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": "mko_graph_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-function-graph",
          "sha256": "6a0892c9773f3366d44da22a063aa3c99163828135681c4e15719eda4e3638be",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "對 D⊆R 及總函數 f:D→R，在固定平面直角座標系中定義 G_f={(x,f(x)):x∈D}。有序對 (u,v) 屬於 G_f，當且僅當 u∈D 且 v=f(u)。",
          "en": "For D⊆R and a total function f:D→R, define G_f={(x,f(x)):x∈D} in a fixed planar Cartesian coordinate system. An ordered pair (u,v) belongs to G_f exactly when u∈D and v=f(u)."
        }
      },
      {
        "id": "mko_table_check",
        "kind": "cited",
        "source": {
          "mko_id": "mko-function-graph",
          "sha256": "6a0892c9773f3366d44da22a063aa3c99163828135681c4e15719eda4e3638be",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "有限圖形可依第一座標排序，重複有序對只列一次。給定 D 和 B 時，要逐輸入檢查恰有一個 B 中的輸出，也不能多出 D 外的圖形點；這是指定總函數的表格驗證。",
          "en": "A finite graph may be sorted by first coordinate, listing a repeated ordered pair once. Given D and B, check input by input that there is exactly one output in B, and that there is no graph point outside D; this is the table verification of a specified total function."
        }
      },
      {
        "id": "L_graph_member",
        "kind": "lemma",
        "proved_in": [
          "S02.1",
          "S02.2"
        ],
        "statement": {
          "zh-Hant": "對任何總函數 f:D→ℝ 與實數 u、v：(u,v)∈G_f ⇔ u∈D 且 v=f(u)。",
          "en": "For any total function f:D→ℝ and reals u, v: (u,v)∈G_f ⇔ u∈D and v=f(u)."
        }
      },
      {
        "id": "L_table_graph",
        "kind": "lemma",
        "proved_in": [
          "S06.1",
          "S06.2"
        ],
        "statement": {
          "zh-Hant": "對集合 D、B⊆ℝ 與 Γ⊆ℝ²：Γ 是某個總函數 F:D→B 的圖形，當且僅當 Γ⊆D×B 且每個 u∈D 恰有一個 v 使 (u,v)∈Γ。",
          "en": "For sets D, B⊆ℝ and Γ⊆ℝ²: Γ is the graph of some total function F:D→B exactly when Γ⊆D×B and every u∈D has exactly one v with (u,v)∈Γ."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N1",
        "step": "S03",
        "lhs": "0",
        "relation": "<=",
        "rhs": "1"
      },
      {
        "id": "N2",
        "step": "S03",
        "lhs": "1",
        "relation": "<=",
        "rhs": "2"
      },
      {
        "id": "N3",
        "step": "S03",
        "lhs": "1**2-1",
        "rhs": "0"
      },
      {
        "id": "N4",
        "step": "S04",
        "lhs": "(-1)**2-1",
        "rhs": "0"
      },
      {
        "id": "N5",
        "step": "S04",
        "lhs": "-1",
        "relation": "<",
        "rhs": "0"
      },
      {
        "id": "N6",
        "step": "S05",
        "lhs": "2**2-1",
        "rhs": "3"
      },
      {
        "id": "N7",
        "step": "S05",
        "lhs": "2",
        "relation": "!=",
        "rhs": "3"
      },
      {
        "id": "N8",
        "step": "S09",
        "lhs": "0**2",
        "rhs": "0"
      },
      {
        "id": "N9",
        "step": "S11",
        "lhs": "1",
        "relation": "!=",
        "rhs": "4"
      },
      {
        "id": "N10",
        "step": "S12",
        "lhs": "3",
        "relation": ">",
        "rhs": "2"
      },
      {
        "id": "N11",
        "step": "S09",
        "lhs": "2**2",
        "rhs": "4"
      },
      {
        "id": "N12",
        "step": "S09",
        "lhs": "1**2",
        "rhs": "1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "在 MKO 的範圍內：固定直角座標系、實數有序對，先宣告定義域；總函數在每個輸入恰有一個輸出，有限表格另需輸出落在宣告的陪域 B。不從示意圖讀坐標，不從有限資料推出反函數、滿射或全域公式；不改動 canonical MKO 或英文 edition。",
      "en": "Within the MKO's scope: a fixed Cartesian system, ordered real pairs, and a domain declared first; a total function has exactly one output at each input, and a finite table must also keep its outputs in the declared codomain B. No coordinates are read from sketches, and no inverse, surjectivity or global formula is inferred from finite data; the canonical MKO and its English edition are not changed."
    },
    "md_symbols": [
      [
        "D,\\ D_0",
        "宣告的定義域：D=[0,2]，D_0={0,1,2}",
        "declared domains: D=[0,2] and D_0={0,1,2}"
      ],
      [
        "B",
        "有限表格宣告的陪域 {0,1,4}",
        "the codomain {0,1,4} declared for the finite tables"
      ],
      [
        "G_f=\\{(x,f(x))\\mid x\\in D\\}",
        "f 的圖形",
        "the graph of f"
      ],
      [
        "T,\\ T_1,\\ T_2,\\ T_3",
        "四個候選表格（有序對的有限集合）",
        "four candidate tables (finite sets of ordered pairs)"
      ],
      [
        "\\Gamma",
        "一般的候選有序對集合",
        "a general candidate set of ordered pairs"
      ]
    ],
    "md_error_controls": [
      {
        "title": "只檢查 y=f(x) 會漏掉定義域 / Checking only y=f(x) forgets the domain",
        "zh": "MKO 指出的第一個誤解是只檢查等式而漏掉定義域。(−1,0) 滿足 0=(−1)²−1，但 −1∉[0,2]，所以不在 G_f 上（S04）。成員判定必須同時檢查 u∈D 與 v=f(u)（S02）。",
        "en": "The first error named in the MKO is to check the equation and forget the domain. (−1,0) satisfies 0=(−1)²−1, but −1∉[0,2], so it is not on G_f (S04). A membership test must check both u∈D and v=f(u) (S02)."
      },
      {
        "title": "垂直線測試不等於總定義 / The vertical line test is not totality",
        "zh": "MKO 指出的第二個誤解是把「垂直線至多交一點」當成「宣告域上每點恰有輸出」。T_1 每條垂直線至多交一點，但輸入 1 沒有輸出，所以不是總函數 D_0→B 的圖形（S10）。完整的表格驗證還要檢查沒有域外點（S12）。",
        "en": "The second error named in the MKO is to take 'every vertical line meets at most one point' for 'every point of the declared domain has exactly one output'. Each vertical line meets T_1 at most once, but input 1 has no output, so T_1 is not the graph of a total function D_0→B (S10). A full table check must also exclude points outside the domain (S12)."
      },
      {
        "title": "有限表格不決定域外延伸，示意圖不替代精確資料 / Finite tables do not determine off-domain extensions, and sketches do not replace exact data",
        "zh": "T 在 0、1、2 的值與 x² 相同，但這不推出 F 在其他地方的公式，F 甚至只定義在 D_0 上（S09）。示意圖不提供精確坐標；有限資料也不自動建立反函數或滿射（MKO para2）。",
        "en": "T agrees with x² at 0, 1 and 2, but this gives no formula for F elsewhere; F is defined only on D_0 (S09). A sketch provides no exact coordinates, and finite data do not by themselves establish an inverse or surjectivity (MKO para2)."
      }
    ],
    "md_open_items": [
      {
        "zh-Hant": "本稿不討論圖形的幾何性質（連續、形狀），也不從圖形推出反函數或滿射。",
        "en": "This guide does not discuss geometric properties of graphs (continuity, shape), and infers no inverse or surjectivity from a graph."
      },
      {
        "zh-Hant": "N8、N11、N12 只記錄 x² 在 0、2、1 的值為 0、4、1，與 T 在這三點的輸出相同；它們不支持任何全域公式。",
        "en": "N8, N11 and N12 only record that x² takes the values 0, 4 and 1 at 0, 2 and 1, matching the outputs of T there; they support no global formula."
      }
    ]
  },
  "revision": {
    "id": "production-20261019-adopted-96",
    "source_commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
    "adopted_source_sha256": "fd7c2071f9e6afa63b288de5141c6fb96961f0e1fe8adfbf5194e5cf438c7930",
    "original_source_sha256": "0b42b0a6e4a678932e19deb35bc9d4abbca16e6381f1a4c619ddbdb34f21a2b6",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "independent_AI_concurrence": "NOT_MEASURED",
    "issue_ids": [
      "BR-M009"
    ],
    "scope": "Owner-adopted natural-language teaching guide; not a formal proof, canonical MKO rewrite or evidence-status promotion.",
    "english_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c"
    }
  }
}