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        "id": "-- BOTH:\nexample : max a b = max b a := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  repeat\n    apply max_le\n    apply le_max_right\n    apply le_max_left\n\n-- BOTH:\nexample : min (min a b) c = min a (min b c) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply le_min\n    · apply le_trans\n      apply min_le_left\n      apply min_le_left\n    apply le_min\n    · apply le_trans\n      apply min_le_left\n      apply min_le_right\n    apply min_le_right\n  apply le_min\n  · apply le_min\n    · apply min_le_left\n    apply le_trans\n    apply min_le_right\n    apply min_le_left\n  apply le_trans\n  apply min_le_right\n  apply min_le_right",
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            "text": "example : min (min a b) c = min a (min b c) := by\n  apply le_antisymm\n  · apply le_min\n    · apply le_trans\n      apply min_le_left\n      apply min_le_left\n    apply le_min\n    · apply le_trans\n      apply min_le_left\n      apply min_le_right\n    apply min_le_right\n  apply le_min\n  · apply le_min\n    · apply min_le_left\n    apply le_trans\n    apply min_le_right\n    apply min_le_left\n  apply le_trans\n  apply min_le_right\n  apply min_le_right",
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        "en": "example : min (min a b) c = min a (min b c) := by\n  apply le_antisymm\n  · apply le_min\n    · apply le_trans\n      apply min_le_left\n      apply min_le_left\n    apply le_min\n    · apply le_trans\n      apply min_le_left\n      apply min_le_right\n    apply min_le_right\n  apply le_min\n  · apply le_min\n    · apply min_le_left\n    apply le_trans\n    apply min_le_right\n    apply min_le_left\n  apply le_trans\n  apply min_le_right\n  apply min_le_right"
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            "text": "theorem aux : min a b + c ≤ min (a + c) (b + c) := by\n  apply le_min\n  · apply add_le_add_left\n    apply min_le_left\n  apply add_le_add_left\n  apply min_le_right\n\n-- BOTH:",
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            "sha256": "345246e570cc3f10949879bde483841fad2f2f10f52113409cceb92061eb21dc",
            "text": "theorem aux : min a b + c ≤ min (a + c) (b + c) := by\n  apply le_min\n  · apply add_le_add_left\n    apply min_le_left\n  apply add_le_add_left\n  apply min_le_right\n\n-- BOTH:",
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            "sha256": "e518f5c23ebd8177f7d36db2ede35c4faf23395109e9677cff8eccbb44c94cbf",
            "text": "example : min a b + c = min (a + c) (b + c) := by\n  apply le_antisymm\n  · apply aux\n  have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel]\n  rw [h]\n  apply add_le_add_left\n  rw [sub_eq_add_neg]\n  apply le_trans\n  apply aux\n  rw [add_neg_cancel_right, add_neg_cancel_right]",
            "line_start": 206,
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            "source_region": "example : min a b + c = min (a + c) (b + c) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply aux\n  have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel]\n  rw [h]\n  apply add_le_add_left\n  rw [sub_eq_add_neg]\n  apply le_trans\n  apply aux\n  rw [add_neg_cancel_right, add_neg_cancel_right]"
          },
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            "sha256": "e518f5c23ebd8177f7d36db2ede35c4faf23395109e9677cff8eccbb44c94cbf",
            "text": "example : min a b + c = min (a + c) (b + c) := by\n  apply le_antisymm\n  · apply aux\n  have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel]\n  rw [h]\n  apply add_le_add_left\n  rw [sub_eq_add_neg]\n  apply le_trans\n  apply aux\n  rw [add_neg_cancel_right, add_neg_cancel_right]",
            "line_start": 207,
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            "source_region": "example : min a b + c = min (a + c) (b + c) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply aux\n  have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel]\n  rw [h]\n  apply add_le_add_left\n  rw [sub_eq_add_neg]\n  apply le_trans\n  apply aux\n  rw [add_neg_cancel_right, add_neg_cancel_right]"
          }
        },
        "id": "example : min a b + c = min (a + c) (b + c) := by\n  apply le_antisymm\n  · apply aux\n  have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel]\n  rw [h]\n  apply add_le_add_left\n  rw [sub_eq_add_neg]\n  apply le_trans\n  apply aux\n  rw [add_neg_cancel_right, add_neg_cancel_right]",
        "en": "example : min a b + c = min (a + c) (b + c) := by\n  apply le_antisymm\n  · apply aux\n  have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel]\n  rw [h]\n  apply add_le_add_left\n  rw [sub_eq_add_neg]\n  apply le_trans\n  apply aux\n  rw [add_neg_cancel_right, add_neg_cancel_right]"
      },
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      "concept_ids": [
        "urn:mil:concept:contraposition",
        "urn:mil:concept:push-neg-tactic",
        "urn:mil:concept:tactic"
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      "code": {
        "id": "example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by\n  contrapose! h\n  exact h\n\nexample (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by\n  contrapose! h\n  use x / 2\n  constructor <;> linarith",
        "en": "example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by\n  contrapose! h\n  exact h\n\nexample (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by\n  contrapose! h\n  use x / 2\n  constructor <;> linarith",
        "sha256": {
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        "data": {
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        "schema_version": "1.0.0",
        "source_local_id": "S03_Negation:QUOTE:14",
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          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C03_Logic.html#negation",
          "anchor": "negation",
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      "rights_ids": [
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        "id": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
        "en": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
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      "concept_ids": [],
      "code": {
        "id": "example (h : a < b) : ¬b < a := by\n  intro h'\n  have : a < a := lt_trans h h'\n  apply lt_irrefl a this",
        "en": "example (h : a < b) : ¬b < a := by\n  intro h'\n  have : a < a := lt_trans h h'\n  apply lt_irrefl a this",
        "sha256": {
          "id": "7641c96bdbe748d39be033f21e9efd37c371588abd35487b4d50d6e5e934e180",
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        "translation_state": "structurally_verified",
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        "en": "example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by\n  intro fnub\n  rcases fnub with ⟨a, fnuba⟩\n  rcases h a with ⟨x, hx⟩\n  have : f x ≤ a := fnuba x\n  linarith",
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        "en": "example (h : ¬∀ x, P x) : ∃ x, ¬P x := by\n  by_contra h'\n  apply h\n  intro x\n  show P x\n  by_contra h''\n  exact h' ⟨x, h''⟩",
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        "schema_version": "1.0.0",
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      "rights_ids": [
        "mil.repo.code"
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        "id": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
        "en": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
        "sha256": {
          "id": "1afddad4df461c717b83553a2aa3f117857e4220d90f4637d90862afd226dcce",
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        "id": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
        "en": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
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        "id": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
        "en": "example (h : 0 < 0) : a > 37 := by\n  exfalso\n  apply lt_irrefl 0 h\n\nexample (h : 0 < 0) : a > 37 :=\n  absurd h (lt_irrefl 0)\n\nexample (h : 0 < 0) : a > 37 := by\n  have h' : ¬0 < 0 := lt_irrefl 0\n  contradiction",
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            "text": "theorem abs_ltαα : |x| < y ↔ -y < x ∧ x < y := by\n  rcases le_or_gt 0 x with h | h\n  · rw [abs_of_nonneg h]\n    constructor\n    · intro h'\n      constructor\n      · linarith\n      exact h'\n    · intro h'\n      rcases h' with ⟨h1, h2⟩\n      exact h2\n  · rw [abs_of_neg h]\n    constructor\n    · intro h'\n      constructor\n      · linarith\n      · linarith\n    · intro h'\n      linarith\n\n-- BOTH:\nend MyAbs\n\nend",
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            "text": "example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by\n  have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self]\n  have h'' : (x + y) * (x - y) = 0 := by\n    rw [← h']\n    ring\n  rcases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 | h1\n  · right\n    exact eq_neg_iff_add_eq_zero.mpr h1\n  · left\n    exact eq_of_sub_eq_zero h1\n\n-- BOTH:\nend",
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        "en": "theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n + t n) (a + b) := by\n  intro ε εpos\n  dsimp -- this line is not needed but cleans up the goal a bit.\n  have ε2pos : 0 < ε / 2 := by linarith\n  rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩\n  rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩\n  use max Ns Nt\n  sorry",
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        "id": "-- BOTH:\ntheorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n * t n) (a * b) := by\n  have h₁ : ConvergesTo (fun n ↦ s n * (t n + -b)) 0 := by\n    apply aux cs\n    convert convergesTo_add ct (convergesTo_const (-b))\n    ring\n  have := convergesTo_add h₁ (convergesTo_mul_const b cs)\n  convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1\n  · ext; ring\n  ring",
        "en": "-- BOTH:\ntheorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n * t n) (a * b) := by\n  have h₁ : ConvergesTo (fun n ↦ s n * (t n + -b)) 0 := by\n    apply aux cs\n    convert convergesTo_add ct (convergesTo_const (-b))\n    ring\n  have := convergesTo_add h₁ (convergesTo_mul_const b cs)\n  convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1\n  · ext; ring\n  ring",
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        "id": "theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ}\n      (sa : ConvergesTo s a) (sb : ConvergesTo s b) :\n    a = b := by\n  by_contra abne\n  have : |a - b| > 0 := by sorry\n  let ε := |a - b| / 2\n  have εpos : ε > 0 := by\n    change |a - b| / 2 > 0\n    linarith\n  rcases sa ε εpos with ⟨Na, hNa⟩\n  rcases sb ε εpos with ⟨Nb, hNb⟩\n  let N := max Na Nb\n  have absa : |s N - a| < ε := by sorry\n  have absb : |s N - b| < ε := by sorry\n  have : |a - b| < |a - b| := by sorry\n  exact lt_irrefl _ this",
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            "text": "theorem convergesTo_addαα {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n + t n) (a + b) := by\n  intro ε εpos\n  dsimp\n  have ε2pos : 0 < ε / 2 := by linarith\n  rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩\n  rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩\n  use max Ns Nt\n  intro n hn\n  have ngeNs : n ≥ Ns := le_of_max_le_left hn\n  have ngeNt : n ≥ Nt := le_of_max_le_right hn\n  calc\n    |s n + t n - (a + b)| = |s n - a + (t n - b)| := by\n      congr\n      ring\n    _ ≤ |s n - a| + |t n - b| := (abs_add_le _ _)\n    _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt))\n    _ = ε := by norm_num",
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            "sha256": "1bd8fb3646d6f9313fb5894fc51c2f4d58de872ae3b68fa387de200a5903a961",
            "text": "theorem convergesTo_addαα {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n + t n) (a + b) := by\n  intro ε εpos\n  dsimp\n  have ε2pos : 0 < ε / 2 := by linarith\n  rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩\n  rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩\n  use max Ns Nt\n  intro n hn\n  have ngeNs : n ≥ Ns := le_of_max_le_left hn\n  have ngeNt : n ≥ Nt := le_of_max_le_right hn\n  calc\n    |s n + t n - (a + b)| = |s n - a + (t n - b)| := by\n      congr\n      ring\n    _ ≤ |s n - a| + |t n - b| := (abs_add_le _ _)\n    _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt))\n    _ = ε := by norm_num",
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        "id": "theorem convergesTo_addαα {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n + t n) (a + b) := by\n  intro ε εpos\n  dsimp\n  have ε2pos : 0 < ε / 2 := by linarith\n  rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩\n  rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩\n  use max Ns Nt\n  intro n hn\n  have ngeNs : n ≥ Ns := le_of_max_le_left hn\n  have ngeNt : n ≥ Nt := le_of_max_le_right hn\n  calc\n    |s n + t n - (a + b)| = |s n - a + (t n - b)| := by\n      congr\n      ring\n    _ ≤ |s n - a| + |t n - b| := (abs_add_le _ _)\n    _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt))\n    _ = ε := by norm_num",
        "en": "theorem convergesTo_addαα {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n + t n) (a + b) := by\n  intro ε εpos\n  dsimp\n  have ε2pos : 0 < ε / 2 := by linarith\n  rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩\n  rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩\n  use max Ns Nt\n  intro n hn\n  have ngeNs : n ≥ Ns := le_of_max_le_left hn\n  have ngeNt : n ≥ Nt := le_of_max_le_right hn\n  calc\n    |s n + t n - (a + b)| = |s n - a + (t n - b)| := by\n      congr\n      ring\n    _ ≤ |s n - a| + |t n - b| := (abs_add_le _ _)\n    _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt))\n    _ = ε := by norm_num"
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            "text": "theorem convergesTo_mul_constαα {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) :\n    ConvergesTo (fun n ↦ c * s n) (c * a) := by\n  by_cases h : c = 0\n  · convert convergesTo_const 0\n    · rw [h]\n      ring\n    rw [h]\n    ring\n  have acpos : 0 < |c| := abs_pos.mpr h\n  intro ε εpos\n  dsimp\n  have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos\n  rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩\n  use Ns\n  intro n ngt\n  calc\n    |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub]\n    _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos)\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt acpos).symm",
            "line_start": 211,
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            "sha256": "a3b3f3a44c40c04b654b33e31d67ae2e00b3d52ee006988dffe174ddbc3b8dfa",
            "text": "theorem convergesTo_mul_constαα {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) :\n    ConvergesTo (fun n ↦ c * s n) (c * a) := by\n  by_cases h : c = 0\n  · convert convergesTo_const 0\n    · rw [h]\n      ring\n    rw [h]\n    ring\n  have acpos : 0 < |c| := abs_pos.mpr h\n  intro ε εpos\n  dsimp\n  have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos\n  rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩\n  use Ns\n  intro n ngt\n  calc\n    |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub]\n    _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos)\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt acpos).symm",
            "line_start": 212,
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        "id": "theorem convergesTo_mul_constαα {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) :\n    ConvergesTo (fun n ↦ c * s n) (c * a) := by\n  by_cases h : c = 0\n  · convert convergesTo_const 0\n    · rw [h]\n      ring\n    rw [h]\n    ring\n  have acpos : 0 < |c| := abs_pos.mpr h\n  intro ε εpos\n  dsimp\n  have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos\n  rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩\n  use Ns\n  intro n ngt\n  calc\n    |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub]\n    _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos)\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt acpos).symm",
        "en": "theorem convergesTo_mul_constαα {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) :\n    ConvergesTo (fun n ↦ c * s n) (c * a) := by\n  by_cases h : c = 0\n  · convert convergesTo_const 0\n    · rw [h]\n      ring\n    rw [h]\n    ring\n  have acpos : 0 < |c| := abs_pos.mpr h\n  intro ε εpos\n  dsimp\n  have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos\n  rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩\n  use Ns\n  intro n ngt\n  calc\n    |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub]\n    _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos)\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt acpos).symm"
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            "sha256": "92f74342ee2380dadc1cb070488dcf705a247aa6635a86da20bb2bd5c39bdc4e",
            "text": "theorem exists_abs_le_of_convergesToαα {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) :\n    ∃ N b, ∀ n, N ≤ n → |s n| < b := by\n  rcases cs 1 zero_lt_one with ⟨N, h⟩\n  use N, |a| + 1\n  intro n ngt\n  calc\n    |s n| = |s n - a + a| := by\n      congr\n      abel\n    _ ≤ |s n - a| + |a| := (abs_add_le _ _)\n    _ < |a| + 1 := by linarith [h n ngt]",
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            "sha256": "92f74342ee2380dadc1cb070488dcf705a247aa6635a86da20bb2bd5c39bdc4e",
            "text": "theorem exists_abs_le_of_convergesToαα {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) :\n    ∃ N b, ∀ n, N ≤ n → |s n| < b := by\n  rcases cs 1 zero_lt_one with ⟨N, h⟩\n  use N, |a| + 1\n  intro n ngt\n  calc\n    |s n| = |s n - a + a| := by\n      congr\n      abel\n    _ ≤ |s n - a| + |a| := (abs_add_le _ _)\n    _ < |a| + 1 := by linarith [h n ngt]",
            "line_start": 247,
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        },
        "id": "theorem exists_abs_le_of_convergesToαα {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) :\n    ∃ N b, ∀ n, N ≤ n → |s n| < b := by\n  rcases cs 1 zero_lt_one with ⟨N, h⟩\n  use N, |a| + 1\n  intro n ngt\n  calc\n    |s n| = |s n - a + a| := by\n      congr\n      abel\n    _ ≤ |s n - a| + |a| := (abs_add_le _ _)\n    _ < |a| + 1 := by linarith [h n ngt]",
        "en": "theorem exists_abs_le_of_convergesToαα {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) :\n    ∃ N b, ∀ n, N ≤ n → |s n| < b := by\n  rcases cs 1 zero_lt_one with ⟨N, h⟩\n  use N, |a| + 1\n  intro n ngt\n  calc\n    |s n| = |s n - a + a| := by\n      congr\n      abel\n    _ ≤ |s n - a| + |a| := (abs_add_le _ _)\n    _ < |a| + 1 := by linarith [h n ngt]"
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            "sha256": "bac24cff79786d087597ba7dafbfef9dfa4bc8f0dd6a273726b16dd0b699acaa",
            "text": "theorem auxαα {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) :\n    ConvergesTo (fun n ↦ s n * t n) 0 := by\n  intro ε εpos\n  dsimp\n  rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩\n  have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _))\n  have pos₀ : ε / B > 0 := div_pos εpos Bpos\n  rcases ct _ pos₀ with ⟨N₁, h₁⟩\n  use max N₀ N₁\n  intro n ngt\n  have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt\n  have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt\n  calc\n    |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero]\n    _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _))\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt Bpos).symm",
            "line_start": 286,
            "file_sha256": "bdc8336abef78e121d9adb554e77fd2b0004c8bab9f62fca0c04b195c53e8fa5",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "bac24cff79786d087597ba7dafbfef9dfa4bc8f0dd6a273726b16dd0b699acaa",
            "text": "theorem auxαα {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) :\n    ConvergesTo (fun n ↦ s n * t n) 0 := by\n  intro ε εpos\n  dsimp\n  rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩\n  have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _))\n  have pos₀ : ε / B > 0 := div_pos εpos Bpos\n  rcases ct _ pos₀ with ⟨N₁, h₁⟩\n  use max N₀ N₁\n  intro n ngt\n  have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt\n  have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt\n  calc\n    |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero]\n    _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _))\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt Bpos).symm",
            "line_start": 287,
            "file_sha256": "b79f0dfcd83504fbf19c1626e64d1a30a0ed47049f1ed2e59be8d10dd38065d6",
            "method": "exact_contiguous_span"
          }
        },
        "id": "theorem auxαα {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) :\n    ConvergesTo (fun n ↦ s n * t n) 0 := by\n  intro ε εpos\n  dsimp\n  rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩\n  have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _))\n  have pos₀ : ε / B > 0 := div_pos εpos Bpos\n  rcases ct _ pos₀ with ⟨N₁, h₁⟩\n  use max N₀ N₁\n  intro n ngt\n  have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt\n  have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt\n  calc\n    |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero]\n    _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _))\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt Bpos).symm",
        "en": "theorem auxαα {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) :\n    ConvergesTo (fun n ↦ s n * t n) 0 := by\n  intro ε εpos\n  dsimp\n  rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩\n  have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _))\n  have pos₀ : ε / B > 0 := div_pos εpos Bpos\n  rcases ct _ pos₀ with ⟨N₁, h₁⟩\n  use max N₀ N₁\n  intro n ngt\n  have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt\n  have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt\n  calc\n    |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero]\n    _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _))\n    _ = ε := mul_div_cancel₀ _ (ne_of_lt Bpos).symm"
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            "state": "exact_hash_span",
            "sha256": "02624fbc8100f51e4cf7deb3b6865e20998684e42823d193223ebf7e7b0ff521",
            "text": "theorem convergesTo_uniqueαα {s : ℕ → ℝ} {a b : ℝ}\n      (sa : ConvergesTo s a) (sb : ConvergesTo s b) :\n    a = b := by\n  by_contra abne\n  have : |a - b| > 0 := by\n    apply lt_of_le_of_ne\n    · apply abs_nonneg\n    intro h''\n    apply abne\n    apply eq_of_abs_sub_eq_zero h''.symm\n  let ε := |a - b| / 2\n  have εpos : ε > 0 := by\n    change |a - b| / 2 > 0\n    linarith\n  rcases sa ε εpos with ⟨Na, hNa⟩\n  rcases sb ε εpos with ⟨Nb, hNb⟩\n  let N := max Na Nb\n  have absa : |s N - a| < ε := by\n    apply hNa\n    apply le_max_left\n  have absb : |s N - b| < ε := by\n    apply hNb\n    apply le_max_right\n  have : |a - b| < |a - b|\n  calc\n    |a - b| = |(-(s N - a)) + (s N - b)| := by\n      congr\n      ring\n    _ ≤ |(-(s N - a))| + |s N - b| := (abs_add_le _ _)\n    _ = |s N - a| + |s N - b| := by rw [abs_neg]\n    _ < ε + ε := (add_lt_add absa absb)\n    _ = |a - b| := by norm_num [ε]\n\n  exact lt_irrefl _ this",
            "line_start": 350,
            "file_sha256": "bdc8336abef78e121d9adb554e77fd2b0004c8bab9f62fca0c04b195c53e8fa5",
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          "en": {
            "state": "exact_hash_span",
            "sha256": "02624fbc8100f51e4cf7deb3b6865e20998684e42823d193223ebf7e7b0ff521",
            "text": "theorem convergesTo_uniqueαα {s : ℕ → ℝ} {a b : ℝ}\n      (sa : ConvergesTo s a) (sb : ConvergesTo s b) :\n    a = b := by\n  by_contra abne\n  have : |a - b| > 0 := by\n    apply lt_of_le_of_ne\n    · apply abs_nonneg\n    intro h''\n    apply abne\n    apply eq_of_abs_sub_eq_zero h''.symm\n  let ε := |a - b| / 2\n  have εpos : ε > 0 := by\n    change |a - b| / 2 > 0\n    linarith\n  rcases sa ε εpos with ⟨Na, hNa⟩\n  rcases sb ε εpos with ⟨Nb, hNb⟩\n  let N := max Na Nb\n  have absa : |s N - a| < ε := by\n    apply hNa\n    apply le_max_left\n  have absb : |s N - b| < ε := by\n    apply hNb\n    apply le_max_right\n  have : |a - b| < |a - b|\n  calc\n    |a - b| = |(-(s N - a)) + (s N - b)| := by\n      congr\n      ring\n    _ ≤ |(-(s N - a))| + |s N - b| := (abs_add_le _ _)\n    _ = |s N - a| + |s N - b| := by rw [abs_neg]\n    _ < ε + ε := (add_lt_add absa absb)\n    _ = |a - b| := by norm_num [ε]\n\n  exact lt_irrefl _ this",
            "line_start": 351,
            "file_sha256": "b79f0dfcd83504fbf19c1626e64d1a30a0ed47049f1ed2e59be8d10dd38065d6",
            "method": "exact_contiguous_span"
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        "id": "-- BOTH:\ndef primes : Set ℕ :=\n  { x | Nat.Prime x }\n\n-- EXAMPLES:\nexample : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by\n  ext\n  rw [mem_iUnion₂]\n  simp\n\nexample : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by\n  ext\n  simp\n\nexample : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by\n  intro x\n  contrapose!\n  simp\n  apply Nat.exists_prime_and_dvd",
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        "id": "example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by\n  rintro x ⟨xs, xt | xu⟩\n  · left; exact ⟨xs, xt⟩\n  · right; exact ⟨xs, xu⟩",
        "en": "example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by\n  rintro x ⟨xs, xt | xu⟩\n  · left; exact ⟨xs, xt⟩\n  · right; exact ⟨xs, xu⟩",
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        "id": "example : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  intro x xstu\n  have xs : x ∈ s := xstu.1.1\n  have xnt : x ∉ t := xstu.1.2\n  have xnu : x ∉ u := xstu.2\n  constructor\n  · exact xs\n  intro xtu\n  -- x ∈ t ∨ x ∈ u\n  rcases xtu with xt | xu\n  · show False; exact xnt xt\n  · show False; exact xnu xu\n\nexample : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  rintro x ⟨⟨xs, xnt⟩, xnu⟩\n  use xs\n  rintro (xt | xu) <;> contradiction",
        "en": "example : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  intro x xstu\n  have xs : x ∈ s := xstu.1.1\n  have xnt : x ∉ t := xstu.1.2\n  have xnu : x ∉ u := xstu.2\n  constructor\n  · exact xs\n  intro xtu\n  -- x ∈ t ∨ x ∈ u\n  rcases xtu with xt | xu\n  · show False; exact xnt xt\n  · show False; exact xnu xu\n\nexample : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  rintro x ⟨⟨xs, xnt⟩, xnu⟩\n  use xs\n  rintro (xt | xu) <;> contradiction",
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        "id": "example : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  intro x xstu\n  have xs : x ∈ s := xstu.1.1\n  have xnt : x ∉ t := xstu.1.2\n  have xnu : x ∉ u := xstu.2\n  constructor\n  · exact xs\n  intro xtu\n  -- x ∈ t ∨ x ∈ u\n  rcases xtu with xt | xu\n  · show False; exact xnt xt\n  · show False; exact xnu xu\n\nexample : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  rintro x ⟨⟨xs, xnt⟩, xnu⟩\n  use xs\n  rintro (xt | xu) <;> contradiction",
        "en": "example : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  intro x xstu\n  have xs : x ∈ s := xstu.1.1\n  have xnt : x ∉ t := xstu.1.2\n  have xnu : x ∉ u := xstu.2\n  constructor\n  · exact xs\n  intro xtu\n  -- x ∈ t ∨ x ∈ u\n  rcases xtu with xt | xu\n  · show False; exact xnt xt\n  · show False; exact xnu xu\n\nexample : (s \\ t) \\ u ⊆ s \\ (t ∪ u) := by\n  rintro x ⟨⟨xs, xnt⟩, xnu⟩\n  use xs\n  rintro (xt | xu) <;> contradiction",
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        "id": "theorem sb_injective (hf : Injective f) : Injective (sbFun f g) := by\n  set A := sbSet f g with A_def\n  set h := sbFun f g with h_def\n  intro x₁ x₂ (hxeq : h x₁ = h x₂)\n  show x₁ = x₂\n  simp only [h_def, sbFun, ← A_def] at hxeq\n  by_cases xA : x₁ ∈ A ∨ x₂ ∈ A\n  · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA\n    · symm\n      apply this hxeq.symm xA.symm (xA.resolve_left x₁A)\n    have x₂A : x₂ ∈ A := by\n      apply _root_.not_imp_self.mp\n      intro (x₂nA : x₂ ∉ A)\n      rw [if_pos x₁A, if_neg x₂nA] at hxeq\n      rw [A_def, sbSet, mem_iUnion] at x₁A\n      have x₂eq : x₂ = g (f x₁) := by\n/- EXAMPLES:\n        sorry\nSOLUTIONS: -/\n        rw [hxeq, sb_right_inv f g x₂nA]\n-- BOTH:\n      rcases x₁A with ⟨n, hn⟩\n      rw [A_def, sbSet, mem_iUnion]\n      use n + 1\n      simp [sbAux]\n      exact ⟨x₁, hn, x₂eq.symm⟩\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [if_pos x₁A, if_pos x₂A] at hxeq\n    exact hf hxeq\n-- BOTH:\n  push_neg at xA\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [if_neg xA.1, if_neg xA.2] at hxeq\n  rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2]\n-- BOTH:",
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      "code": {
        "id": "example {m n : ℕ} (coprime_mn : m.Coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by\n  intro sqr_eq\n  have : 2 ∣ m := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply even_of_even_sqr\n    rw [sqr_eq]\n    apply dvd_mul_right\n-- BOTH:\n  obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this\n  have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by\n    rw [← sqr_eq, meq]\n    ring\n  have : 2 * k ^ 2 = n ^ 2 :=\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    (mul_right_inj' (by norm_num)).mp this\n-- BOTH:\n  have : 2 ∣ n := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply even_of_even_sqr\n    rw [← this]\n    apply dvd_mul_right\n-- BOTH:\n  have : 2 ∣ m.gcd n := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply Nat.dvd_gcd <;>\n    assumption\n-- BOTH:\n  have : 2 ∣ 1 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    convert this\n    symm\n    exact coprime_mn\n-- BOTH:\n  norm_num at this",
        "en": "example {m n : ℕ} (coprime_mn : m.Coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by\n  intro sqr_eq\n  have : 2 ∣ m := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply even_of_even_sqr\n    rw [sqr_eq]\n    apply dvd_mul_right\n-- BOTH:\n  obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this\n  have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by\n    rw [← sqr_eq, meq]\n    ring\n  have : 2 * k ^ 2 = n ^ 2 :=\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    (mul_right_inj' (by norm_num)).mp this\n-- BOTH:\n  have : 2 ∣ n := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply even_of_even_sqr\n    rw [← this]\n    apply dvd_mul_right\n-- BOTH:\n  have : 2 ∣ m.gcd n := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply Nat.dvd_gcd <;>\n    assumption\n-- BOTH:\n  have : 2 ∣ 1 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    convert this\n    symm\n    exact coprime_mn\n-- BOTH:\n  norm_num at this",
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        "id": "example {m n p : ℕ} (coprime_mn : m.Coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  intro sqr_eq\n  have : p ∣ m := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [sqr_eq]\n    apply dvd_mul_right\n  obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this\n  have : p * (p * k ^ 2) = p * n ^ 2 := by\n    rw [← sqr_eq, meq]\n    ring\n  have : p * k ^ 2 = n ^ 2 := by\n    apply (mul_right_inj' _).mp this\n    exact prime_p.ne_zero\n  have : p ∣ n := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [← this]\n    apply dvd_mul_right\n  have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption\n  have : p ∣ 1 := by\n    convert this\n    symm\n    exact coprime_mn\n  have : 2 ≤ 1 := by\n    apply prime_p.two_le.trans\n    exact Nat.le_of_dvd zero_lt_one this\n  norm_num at this",
        "en": "example {m n p : ℕ} (coprime_mn : m.Coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  intro sqr_eq\n  have : p ∣ m := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [sqr_eq]\n    apply dvd_mul_right\n  obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this\n  have : p * (p * k ^ 2) = p * n ^ 2 := by\n    rw [← sqr_eq, meq]\n    ring\n  have : p * k ^ 2 = n ^ 2 := by\n    apply (mul_right_inj' _).mp this\n    exact prime_p.ne_zero\n  have : p ∣ n := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [← this]\n    apply dvd_mul_right\n  have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption\n  have : p ∣ 1 := by\n    convert this\n    symm\n    exact coprime_mn\n  have : 2 ≤ 1 := by\n    apply prime_p.two_le.trans\n    exact Nat.le_of_dvd zero_lt_one this\n  norm_num at this",
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        "id": "example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by\n  intro sqr_eq\n  have nsqr_nez : n ^ 2 ≠ 0 := by simpa\n  have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow',\n      add_comm]\n-- BOTH:\n  have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by\n    rw [← eq1, sqr_eq, eq2]\n  rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this\n  norm_num at this",
        "en": "example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by\n  intro sqr_eq\n  have nsqr_nez : n ^ 2 ≠ 0 := by simpa\n  have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow',\n      add_comm]\n-- BOTH:\n  have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by\n    rw [← eq1, sqr_eq, eq2]\n  rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this\n  norm_num at this",
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        "id": "example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} :\n    k ∣ r.factorization p := by\n  rcases r with _ | r\n  · simp\n  have npow_nz : n ^ k ≠ 0 := fun npowz ↦ nnz (eq_zero_of_pow_eq_zero npowz)\n  have eq1 : (m ^ k).factorization p = k * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : ((r + 1) * n ^ k).factorization p =\n      k * n.factorization p + (r + 1).factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm]\n-- BOTH:\n  have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by\n    rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel]\n  rw [this]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply Nat.dvd_sub <;>\n  apply Nat.dvd_mul_right\n-- BOTH:",
        "en": "example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} :\n    k ∣ r.factorization p := by\n  rcases r with _ | r\n  · simp\n  have npow_nz : n ^ k ≠ 0 := fun npowz ↦ nnz (eq_zero_of_pow_eq_zero npowz)\n  have eq1 : (m ^ k).factorization p = k * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : ((r + 1) * n ^ k).factorization p =\n      k * n.factorization p + (r + 1).factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm]\n-- BOTH:\n  have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by\n    rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel]\n  rw [this]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply Nat.dvd_sub <;>\n  apply Nat.dvd_mul_right\n-- BOTH:",
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        "id": "#check Nat.Prime.dvd_mul\n#check Nat.Prime.dvd_mul Nat.prime_two\n#check Nat.prime_two.dvd_mul\n\n-- BOTH:\ntheorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by\n  rw [pow_two, Nat.prime_two.dvd_mul] at h\n  cases h <;> assumption\n\n-- EXAMPLES:\nexample {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m :=\n  Nat.Prime.dvd_of_dvd_pow Nat.prime_two h",
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        "en": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
        "sha256": {
          "id": "c457b60a8daceb4d4f9dcc0202db6ee38b53d3faa3985125fa6f24e404f2e14f",
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      "companion": false,
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        "concept_ids": [
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          "urn:mil:concept:gcd",
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        "data": {
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          "declaration_kind": "example",
          "formal_equal": true,
          "is_exercise": false,
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          "sorry_holes": 0,
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        "qa_event_ids": [
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          "title": "Akar Irasional",
          "state": "enclosing_section",
          "page_sha256": "27dcf6062fe73be2e1189ebd5cfbcf11bb4dd0c0f0f3a0d00238226fb7df1d52",
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        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C05_Elementary_Number_Theory.html#irrational-roots",
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          "title": "Irrational Roots",
          "state": "enclosing_section",
          "page_sha256": "cd4a5a540a409ac83cef0e8b264afabe3ebd4847825b086552b2fbe5396a644d",
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        "urn:mil:concept:coprimality",
        "urn:mil:concept:gcd",
        "urn:mil:concept:natural-number"
      ],
      "code": {
        "id": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
        "en": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
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      "companion": false,
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        "concept_ids": [
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        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
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    },
    {
      "id": "urn:mil:unit:c05:s01:example:003",
      "kind": "example",
      "practice": false,
      "chapter_id": "urn:mil:unit:c05",
      "section_id": "urn:mil:unit:c05:s01",
      "title": {
        "id": "anonymous-example-003",
        "en": "anonymous-example-003"
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      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C05_Elementary_Number_Theory.html#akar-irasional",
          "anchor": "akar-irasional",
          "title": "Akar Irasional",
          "state": "enclosing_section",
          "page_sha256": "27dcf6062fe73be2e1189ebd5cfbcf11bb4dd0c0f0f3a0d00238226fb7df1d52",
          "bound_unit_id": "urn:mil:unit:c05:s01",
          "code_equivalence_claimed": false
        },
        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C05_Elementary_Number_Theory.html#irrational-roots",
          "anchor": "irrational-roots",
          "title": "Irrational Roots",
          "state": "enclosing_section",
          "page_sha256": "cd4a5a540a409ac83cef0e8b264afabe3ebd4847825b086552b2fbe5396a644d",
          "bound_unit_id": "urn:mil:unit:c05:s01",
          "code_equivalence_claimed": false
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      },
      "source_locator": "MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean#line-50",
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        "urn:mil:concept:coprimality",
        "urn:mil:concept:gcd",
        "urn:mil:concept:natural-number"
      ],
      "code": {
        "id": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
        "en": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
        "sha256": {
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      "companion": false,
      "source_record": {
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        "concept_ids": [
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          "urn:mil:concept:gcd",
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        ],
        "data": {
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          "declaration_kind": "example",
          "formal_equal": true,
          "is_exercise": false,
          "lean_name": "",
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          "sorry_holes": 0,
          "source_line": 50,
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        "provenance": [],
        "qa_event_ids": [
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      "practice": false,
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          "title": "Akar Irasional",
          "state": "enclosing_section",
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          "anchor": "irrational-roots",
          "title": "Irrational Roots",
          "state": "enclosing_section",
          "page_sha256": "cd4a5a540a409ac83cef0e8b264afabe3ebd4847825b086552b2fbe5396a644d",
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          "code_equivalence_claimed": false
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      "source_locator": "MIL/C05_Elementary_Number_Theory/S01_Irrational_Roots.lean#line-52",
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        "urn:mil:concept:coprimality",
        "urn:mil:concept:gcd",
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      "code": {
        "id": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
        "en": "#print Nat.Coprime\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 :=\n  h\n\nexample (m n : Nat) (h : m.Coprime n) : m.gcd n = 1 := by\n  rw [Nat.Coprime] at h\n  exact h\n\nexample : Nat.Coprime 12 7 := by norm_num\n\nexample : Nat.gcd 12 8 = 4 := by norm_num",
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      "practice": false,
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          "title": "Akar Irasional",
          "state": "enclosing_section",
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          "anchor": "irrational-roots",
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          "state": "enclosing_section",
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      "rights_ids": [
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      "concept_ids": [
        "urn:mil:concept:prime-number"
      ],
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        "en": "example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by\n  intro sqr_eq\n  have nsqr_nez : n ^ 2 ≠ 0 := by simpa\n  have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow',\n      add_comm]\n-- BOTH:\n  have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by\n    rw [← eq1, sqr_eq, eq2]\n  rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this\n  norm_num at this",
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        "id": "example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} :\n    k ∣ r.factorization p := by\n  rcases r with _ | r\n  · simp\n  have npow_nz : n ^ k ≠ 0 := fun npowz ↦ nnz (eq_zero_of_pow_eq_zero npowz)\n  have eq1 : (m ^ k).factorization p = k * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : ((r + 1) * n ^ k).factorization p =\n      k * n.factorization p + (r + 1).factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm]\n-- BOTH:\n  have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by\n    rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel]\n  rw [this]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply Nat.dvd_sub <;>\n  apply Nat.dvd_mul_right\n-- BOTH:",
        "en": "example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} :\n    k ∣ r.factorization p := by\n  rcases r with _ | r\n  · simp\n  have npow_nz : n ^ k ≠ 0 := fun npowz ↦ nnz (eq_zero_of_pow_eq_zero npowz)\n  have eq1 : (m ^ k).factorization p = k * m.factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_pow']\n-- BOTH:\n  have eq2 : ((r + 1) * n ^ k).factorization p =\n      k * n.factorization p + (r + 1).factorization p := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm]\n-- BOTH:\n  have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by\n    rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel]\n  rw [this]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply Nat.dvd_sub <;>\n  apply Nat.dvd_mul_right\n-- BOTH:",
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            "text": "    apply Nat.dvd_gcd <;>\n    assumption",
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            "sha256": "d9cf6cf31d2c35dcbd03ea0c495e009856a0a49f3c5d801fa0e12ebb4dd16fcd",
            "text": "  intro sqr_eq\n  have : p ∣ m := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [sqr_eq]\n    apply dvd_mul_right\n  obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this\n  have : p * (p * k ^ 2) = p * n ^ 2 := by\n    rw [← sqr_eq, meq]\n    ring\n  have : p * k ^ 2 = n ^ 2 := by\n    apply (mul_right_inj' _).mp this\n    exact prime_p.ne_zero\n  have : p ∣ n := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [← this]\n    apply dvd_mul_right\n  have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption\n  have : p ∣ 1 := by\n    convert this\n    symm\n    exact coprime_mn\n  have : 2 ≤ 1 := by\n    apply prime_p.two_le.trans\n    exact Nat.le_of_dvd zero_lt_one this\n  norm_num at this",
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            "sha256": "d9cf6cf31d2c35dcbd03ea0c495e009856a0a49f3c5d801fa0e12ebb4dd16fcd",
            "text": "  intro sqr_eq\n  have : p ∣ m := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [sqr_eq]\n    apply dvd_mul_right\n  obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this\n  have : p * (p * k ^ 2) = p * n ^ 2 := by\n    rw [← sqr_eq, meq]\n    ring\n  have : p * k ^ 2 = n ^ 2 := by\n    apply (mul_right_inj' _).mp this\n    exact prime_p.ne_zero\n  have : p ∣ n := by\n    apply prime_p.dvd_of_dvd_pow\n    rw [← this]\n    apply dvd_mul_right\n  have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption\n  have : p ∣ 1 := by\n    convert this\n    symm\n    exact coprime_mn\n  have : 2 ≤ 1 := by\n    apply prime_p.two_le.trans\n    exact Nat.le_of_dvd zero_lt_one this\n  norm_num at this",
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        "id": "inductive MyNat where\n  | zero : MyNat\n  | succ : MyNat → MyNat\n\nnamespace MyNat\n\ndef add : MyNat → MyNat → MyNat\n  | x, zero => x\n  | x, succ y => succ (add x y)\n\ndef mul : MyNat → MyNat → MyNat\n  | _, zero => zero\n  | x, succ y => add (mul x y) x\n\ntheorem zero_add (n : MyNat) : add zero n = n := by\n  induction' n with n ih\n  · rfl\n  rw [add, ih]\n\ntheorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by\n  induction' n with n ih\n  · rfl\n  rw [add, ih]\n  rfl\n\ntheorem add_comm (m n : MyNat) : add m n = add n m := by\n  induction' n with n ih\n  · rw [zero_add]\n    rfl\n  rw [add, succ_add, ih]\n\ntheorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  induction' k with k ih\n  · rfl\n  rw [add, ih]\n  rfl\n\n-- BOTH:\ntheorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  induction' k with k ih\n  · rfl\n  rw [add, mul, mul, ih, add_assoc]\n\n-- BOTH:\ntheorem zero_mul (n : MyNat) : mul zero n = zero := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  induction' n with n ih\n  · rfl\n  rw [mul, ih]\n  rfl\n\n-- BOTH:\ntheorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  induction' n with n ih\n  · rfl\n  rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add]\n  rfl\n\n-- BOTH:\ntheorem mul_comm (m n : MyNat) : mul m n = mul n m := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  induction' n with n ih\n  · rw [zero_mul]\n    rfl\n  rw [mul, ih, succ_mul]\n\n-- BOTH:\nend MyNat",
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        "id": "theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by\n  by_cases np : n.Prime\n  · use n, np\n  induction' n using Nat.strong_induction_on with n ih\n  rw [Nat.prime_def_lt] at np\n  push_neg at np\n  rcases np h with ⟨m, mltn, mdvdn, mne1⟩\n  have : m ≠ 0 := by\n    intro mz\n    rw [mz, zero_dvd_iff] at mdvdn\n    linarith\n  have mgt2 : 2 ≤ m := two_le this mne1\n  by_cases mp : m.Prime\n  · use m, mp\n  · rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩\n    use p, pp\n    apply pdvd.trans mdvdn",
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        "id": "open Finset\n\n-- EXAMPLES:\nsection\nvariable {α : Type*} [DecidableEq α] (r s t : Finset α)\n\nexample : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by\n  rw [subset_iff]\n  intro x\n  rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter]\n  tauto\n\nexample : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by\n  simp [subset_iff]\n  intro x\n  tauto\n\nexample : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by\n  simp [subset_iff]\n  intro x\n  tauto\n\nexample : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by\n  ext x\n  simp\n  tauto\n\nend",
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        "id": "example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext x\n  rw [mem_inter, mem_union, mem_union, mem_union, mem_inter]\n  tauto\n\nexample : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by\n  ext x\n  simp\n  tauto\n\n-- BOTH:\nexample : (r \\ s) \\ t = r \\ (s ∪ t) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext x\n  rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union]\n  tauto\n\nexample : (r \\ s) \\ t = r \\ (s ∪ t) := by\n  ext x\n  simp\n  tauto",
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        "id": "theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by\n  by_contra h\n  push_neg at h\n  rcases h with ⟨n, hn⟩\n  have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by\n    apply ex_finset_of_bounded\n    use n\n    contrapose! hn\n    rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩\n    exact ⟨p, pltn, pp, p4⟩\n  rcases this with ⟨s, hs⟩\n  have h₁ : ((4 * ∏ i ∈ erase s 3, i) + 3) % 4 = 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [add_comm, Nat.add_mul_mod_self_left]\n-- BOTH:\n  rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩\n  have ps : p ∈ s := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [← hs p]\n    exact ⟨pp, p4eq⟩\n-- BOTH:\n  have pne3 : p ≠ 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this\n-- BOTH:\n  have : p ∣ 4 * ∏ i ∈ erase s 3, i := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption\n-- BOTH:\n  have : p ∣ 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    convert Nat.dvd_sub pdvd this\n    simp\n-- BOTH:\n  have : p = 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply pp.eq_of_dvd_of_prime Nat.prime_three this\n-- BOTH:\n  contradiction",
        "en": "theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by\n  by_contra h\n  push_neg at h\n  rcases h with ⟨n, hn⟩\n  have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by\n    apply ex_finset_of_bounded\n    use n\n    contrapose! hn\n    rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩\n    exact ⟨p, pltn, pp, p4⟩\n  rcases this with ⟨s, hs⟩\n  have h₁ : ((4 * ∏ i ∈ erase s 3, i) + 3) % 4 = 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [add_comm, Nat.add_mul_mod_self_left]\n-- BOTH:\n  rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩\n  have ps : p ∈ s := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [← hs p]\n    exact ⟨pp, p4eq⟩\n-- BOTH:\n  have pne3 : p ≠ 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this\n-- BOTH:\n  have : p ∣ 4 * ∏ i ∈ erase s 3, i := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption\n-- BOTH:\n  have : p ∣ 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    convert Nat.dvd_sub pdvd this\n    simp\n-- BOTH:\n  have : p = 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply pp.eq_of_dvd_of_prime Nat.prime_three this\n-- BOTH:\n  contradiction",
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        "id": "theorem bounded_of_ex_finset (Q : ℕ → Prop) :\n    (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by\n  rintro ⟨s, hs⟩\n  use s.sup id + 1\n  intro k Qk\n  apply Nat.lt_succ_of_le\n  show id k ≤ s.sup id\n  apply le_sup (hs k Qk)\n\ntheorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] :\n    (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by\n  rintro ⟨n, hn⟩\n  use (range (n + 1)).filter Q\n  intro k\n  simpa using hn k",
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        "id": "theorem bounded_of_ex_finset (Q : ℕ → Prop) :\n    (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by\n  rintro ⟨s, hs⟩\n  use s.sup id + 1\n  intro k Qk\n  apply Nat.lt_succ_of_le\n  show id k ≤ s.sup id\n  apply le_sup (hs k Qk)\n\ntheorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] :\n    (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by\n  rintro ⟨n, hn⟩\n  use (range (n + 1)).filter Q\n  intro k\n  simpa using hn k",
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        "id": "theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by\n  by_cases np : n.Prime\n  · use n, np\n  induction' n using Nat.strong_induction_on with n ih\n  rw [Nat.prime_def_lt] at np\n  push_neg at np\n  rcases np h with ⟨m, mltn, mdvdn, mne1⟩\n  have : m ≠ 0 := by\n    intro mz\n    rw [mz, zero_dvd_iff] at mdvdn\n    linarith\n  have mgt2 : 2 ≤ m := two_le this mne1\n  by_cases mp : m.Prime\n  · use m, mp\n  · rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩\n    use p, pp\n    apply pdvd.trans mdvdn",
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        "en": "theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by\n  by_contra h\n  push_neg at h\n  rcases h with ⟨n, hn⟩\n  have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by\n    apply ex_finset_of_bounded\n    use n\n    contrapose! hn\n    rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩\n    exact ⟨p, pltn, pp, p4⟩\n  rcases this with ⟨s, hs⟩\n  have h₁ : ((4 * ∏ i ∈ erase s 3, i) + 3) % 4 = 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [add_comm, Nat.add_mul_mod_self_left]\n-- BOTH:\n  rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩\n  have ps : p ∈ s := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rw [← hs p]\n    exact ⟨pp, p4eq⟩\n-- BOTH:\n  have pne3 : p ≠ 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this\n-- BOTH:\n  have : p ∣ 4 * ∏ i ∈ erase s 3, i := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption\n-- BOTH:\n  have : p ∣ 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    convert Nat.dvd_sub pdvd this\n    simp\n-- BOTH:\n  have : p = 3 := by\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    apply pp.eq_of_dvd_of_prime Nat.prime_three this\n-- BOTH:\n  contradiction",
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            "text": "    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this",
            "line_start": 614,
            "file_sha256": "8d07aeed6fc62576db1faa5d45dd7b15f092fc7ed75f7b0c23a7249a7f85ad75",
            "method": "exact_contiguous_span"
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            "sha256": "18d7e1c37299a7472d2bb664bdb853e2b05bfaa7bdd5021156dfee87f1ec52e8",
            "text": "    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this",
            "line_start": 615,
            "file_sha256": "b1d8d557b3817d59ee29d57d868d4e0064c1b8c81cfcb481713a624c30afb655",
            "method": "exact_contiguous_span"
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        "id": "    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this",
        "en": "    intro peq\n    rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd\n    rw [Nat.prime_three.dvd_mul] at pdvd\n    norm_num at pdvd\n    have : 3 ∈ s.erase 3 := by\n      apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd\n      intro n\n      simp [← hs n]\n      tauto\n    simp at this"
      },
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            "sha256": "7121c1d899a2e9bc2dbd4a90356bd5448656bfa9b151f817cfb8183e54ae2075",
            "text": "    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption",
            "line_start": 629,
            "file_sha256": "8d07aeed6fc62576db1faa5d45dd7b15f092fc7ed75f7b0c23a7249a7f85ad75",
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            "state": "exact_hash_span",
            "sha256": "7121c1d899a2e9bc2dbd4a90356bd5448656bfa9b151f817cfb8183e54ae2075",
            "text": "    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption",
            "line_start": 630,
            "file_sha256": "b1d8d557b3817d59ee29d57d868d4e0064c1b8c81cfcb481713a624c30afb655",
            "method": "exact_contiguous_span"
          }
        },
        "id": "    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption",
        "en": "    apply dvd_trans _ (dvd_mul_left _ _)\n    apply dvd_prod_of_mem\n    simp\n    constructor <;> assumption"
      },
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          "hole_ranks": [
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          "is_alternative": false,
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          "source_unit_id": "urn:mil:unit:c05:s03:exercise:primes-mod-4-eq-3-infinite",
          "statement_signature": "apply dvd_trans _ (dvd_mul_left _ _) apply dvd_prod_of_mem simp constructor <;> assumption",
          "target_is_exercise": true,
          "target_source_line": 586,
          "target_text_sha256": "7121c1d899a2e9bc2dbd4a90356bd5448656bfa9b151f817cfb8183e54ae2075",
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            "text": "    convert Nat.dvd_sub pdvd this\n    simp",
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        "en": "    convert Nat.dvd_sub pdvd this\n    simp"
      },
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        "en": "    apply pp.eq_of_dvd_of_prime Nat.prime_three this"
      },
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        "id": "example (n : ℕ) : #(triangle n) = (n + 1) * n / 2 := by\n  apply Nat.eq_div_of_mul_eq_right (by norm_num)\n  let turn (p : ℕ × ℕ) : ℕ × ℕ := (n - 1 - p.1, n - p.2)\n  calc 2 * #(triangle n)\n      = #(triangle n) + #(triangle n) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          ring\nBOTH: -/\n    _ = #(triangle n) + #(triangle n |>.image turn) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          rw [Finset.card_image_of_injOn]\n          rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [turn]\n          simp_all [triangle]; omega\nBOTH: -/\n    _ = #(range n ×ˢ range (n + 1)) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          rw [←Finset.card_union_of_disjoint]; swap\n          . rw [Finset.disjoint_iff_ne]\n            rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [turn] at *\n            simp_all [triangle]; omega\n          congr; ext p; rcases p with ⟨p1, p2⟩\n          simp [triangle, turn]\n          constructor\n          . rintro (h | h) <;> omega\n          rcases Nat.lt_or_ge p1 p2 with h | h\n          . omega\n          . intro h'\n            right\n            use n - 1 - p1, n - p2\n            omega\nBOTH: -/\n    _ = (n + 1) * n := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          simp [mul_comm]\nBOTH: -/",
        "en": "example (n : ℕ) : #(triangle n) = (n + 1) * n / 2 := by\n  apply Nat.eq_div_of_mul_eq_right (by norm_num)\n  let turn (p : ℕ × ℕ) : ℕ × ℕ := (n - 1 - p.1, n - p.2)\n  calc 2 * #(triangle n)\n      = #(triangle n) + #(triangle n) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          ring\nBOTH: -/\n    _ = #(triangle n) + #(triangle n |>.image turn) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          rw [Finset.card_image_of_injOn]\n          rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [turn]\n          simp_all [triangle]; omega\nBOTH: -/\n    _ = #(range n ×ˢ range (n + 1)) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          rw [←Finset.card_union_of_disjoint]; swap\n          . rw [Finset.disjoint_iff_ne]\n            rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [turn] at *\n            simp_all [triangle]; omega\n          congr; ext p; rcases p with ⟨p1, p2⟩\n          simp [triangle, turn]\n          constructor\n          . rintro (h | h) <;> omega\n          rcases Nat.lt_or_ge p1 p2 with h | h\n          . omega\n          . intro h'\n            right\n            use n - 1 - p1, n - p2\n            omega\nBOTH: -/\n    _ = (n + 1) * n := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          simp [mul_comm]\nBOTH: -/",
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        "id": "def triangle' (n : ℕ) : Finset (ℕ × ℕ) := {p ∈ range n ×ˢ range n | p.1 ≤ p.2}\n\n-- EXAMPLES:\nexample (n : ℕ) : #(triangle' n) = #(triangle n) := by sorry\n/- SOLUTIONS:\nexample (n : ℕ) : #(triangle' n) = #(triangle n) := by\n  let f (p : ℕ × ℕ) : ℕ × ℕ := (p.1, p.2 + 1)\n  have : triangle n = (triangle' n |>.image f) := by\n    ext p; rcases p with ⟨p1, p2⟩\n    simp [triangle, triangle', f]\n    constructor\n    . intro h\n      use p2 - 1\n      omega\n    . simp; omega\n  rw [this, card_image_of_injOn]\n  rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [f]\nBOTH: -/",
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        "en": "open Classical\nvariable (s t : Finset ℕ) (a b : ℕ)\n\ntheorem doubleCounting {α β : Type*} (s : Finset α) (t : Finset β)\n    (r : α → β → Prop)\n    (h_left : ∀ a ∈ s, 3 ≤ #{b ∈ t | r a b})\n    (h_right : ∀ b ∈ t, #{a ∈ s | r a b} ≤ 1) :\n    3 * #(s) ≤ #(t) := by\n  calc 3 * #(s)\n      = ∑ a ∈ s, 3                               := by simp [mul_comm]\n    _ ≤ ∑ a ∈ s, #({b ∈ t | r a b})              := sum_le_sum h_left\n    _ = ∑ a ∈ s, ∑ b ∈ t, if r a b then 1 else 0 := by simp\n    _ = ∑ b ∈ t, ∑ a ∈ s, if r a b then 1 else 0 := sum_comm\n    _ = ∑ b ∈ t, #({a ∈ s | r a b})              := by simp\n    _ ≤ ∑ b ∈ t, 1                               := sum_le_sum h_right\n    _ ≤ #(t)                                     := by simp",
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        "id": "def triangle' (n : ℕ) : Finset (ℕ × ℕ) := {p ∈ range n ×ˢ range n | p.1 ≤ p.2}\n\n-- EXAMPLES:\nexample (n : ℕ) : #(triangle' n) = #(triangle n) := by sorry\n/- SOLUTIONS:\nexample (n : ℕ) : #(triangle' n) = #(triangle n) := by\n  let f (p : ℕ × ℕ) : ℕ × ℕ := (p.1, p.2 + 1)\n  have : triangle n = (triangle' n |>.image f) := by\n    ext p; rcases p with ⟨p1, p2⟩\n    simp [triangle, triangle', f]\n    constructor\n    . intro h\n      use p2 - 1\n      omega\n    . simp; omega\n  rw [this, card_image_of_injOn]\n  rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [f]\nBOTH: -/",
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        "id": "example (n : ℕ) : #(triangle n) = (n + 1) * n / 2 := by\n  apply Nat.eq_div_of_mul_eq_right (by norm_num)\n  let turn (p : ℕ × ℕ) : ℕ × ℕ := (n - 1 - p.1, n - p.2)\n  calc 2 * #(triangle n)\n      = #(triangle n) + #(triangle n) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          ring\nBOTH: -/\n    _ = #(triangle n) + #(triangle n |>.image turn) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          rw [Finset.card_image_of_injOn]\n          rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [turn]\n          simp_all [triangle]; omega\nBOTH: -/\n    _ = #(range n ×ˢ range (n + 1)) := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          rw [←Finset.card_union_of_disjoint]; swap\n          . rw [Finset.disjoint_iff_ne]\n            rintro ⟨p1, p2⟩ hp ⟨q1, q2⟩ hq; simp [turn] at *\n            simp_all [triangle]; omega\n          congr; ext p; rcases p with ⟨p1, p2⟩\n          simp [triangle, turn]\n          constructor\n          . rintro (h | h) <;> omega\n          rcases Nat.lt_or_ge p1 p2 with h | h\n          . omega\n          . intro h'\n            right\n            use n - 1 - p1, n - p2\n            omega\nBOTH: -/\n    _ = (n + 1) * n := by\n-- EXAMPLES:\n          sorry\n/- SOLUTIONS:\n          simp [mul_comm]\nBOTH: -/",
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        "en": "example {n : ℕ} (A : Finset ℕ)\n    (hA : #(A) = n + 1)\n    (hA' : A ⊆ range (2 * n)) :\n    ∃ m ∈ A, ∃ k ∈ A, Nat.Coprime m k := by\n  have : ∃ t ∈ range n, 1 < #({u ∈ A | u / 2 = t}) := by\n    apply exists_lt_card_fiber_of_mul_lt_card_of_maps_to\n-- EXAMPLES:\n    · sorry\n/- SOLUTIONS:\n    · intro u hu\n      specialize hA' hu\n      simp only [mem_range] at *\n      exact Nat.div_lt_of_lt_mul hA'\nEXAMPLES: -/\n    · sorry\n/- SOLUTIONS:\n    · simp [hA]\nBOTH: -/\n  rcases this with ⟨t, ht, ht'⟩\n  simp only [one_lt_card, mem_filter] at ht'\n-- EXAMPLES:\n  sorry\n/- SOLUTIONS:\n  rcases ht' with ⟨m, ⟨hm, hm'⟩, k, ⟨hk, hk'⟩, hmk⟩\n  use m, hm, k, hk\n  have : m = k + 1 ∨ k = m + 1 := by omega\n  rcases this with h | h <;> simp [h]\nBOTH: -/",
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        "id": "theorem size_le : ∀ t : BinTree, size t ≤ 2^depth t - 1\n  | empty    => Nat.zero_le _\n  | node l r => by\n    simp only [depth, size]\n    calc l.size + r.size + 1\n      ≤ (2^l.depth - 1) + (2^r.depth - 1) + 1 := by\n          gcongr <;> apply size_le\n    _ ≤ (2 ^ max l.depth r.depth - 1) + (2 ^ max l.depth r.depth - 1) + 1 := by\n          gcongr <;> simp\n    _ ≤ 2 ^ (max l.depth r.depth + 1) - 1 := by\n          have : 0 < 2 ^ max l.depth r.depth := by simp\n          omega",
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            "text": "theorem subst_eq_of_not_mem_vars :\n    ∀ (A : PropForm) (n : ℕ) (C : PropForm), n ∉ A.vars → A.subst n C = A\n  | var m, n, C, h => by simp_all [subst, vars]; tauto\n  | fls, n, C, _ => by rw [subst]\n  | conj A B, n, C, h => by\n    simp_all [subst, vars, subst_eq_of_not_mem_vars A, subst_eq_of_not_mem_vars B]\n  | disj A B, n, C, h => by\n    simp_all [subst, vars, subst_eq_of_not_mem_vars A, subst_eq_of_not_mem_vars B]\n  | impl A B, n, C, h => by\n    simp_all [subst, vars, subst_eq_of_not_mem_vars A, subst_eq_of_not_mem_vars B]\n\n-- pembuktian alternatif:",
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        "en": "theorem subst_eq_of_not_mem_vars :\n    ∀ (A : PropForm) (n : ℕ) (C : PropForm), n ∉ A.vars → A.subst n C = A\n  | var m, n, C, h => by simp_all [subst, vars]; tauto\n  | fls, n, C, _ => by rw [subst]\n  | conj A B, n, C, h => by\n    simp_all [subst, vars, subst_eq_of_not_mem_vars A, subst_eq_of_not_mem_vars B]\n  | disj A B, n, C, h => by\n    simp_all [subst, vars, subst_eq_of_not_mem_vars A, subst_eq_of_not_mem_vars B]\n  | impl A B, n, C, h => by\n    simp_all [subst, vars, subst_eq_of_not_mem_vars A, subst_eq_of_not_mem_vars B]\n\n-- alternative proof:"
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        "id": "def Point'' :=\n  ℝ × ℝ × ℝ\n\ndef IsLinear' (f : ℝ → ℝ) :=\n  (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x",
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        "id": "def StandardTwoSimplex' :=\n  { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 }\n\ndef StandardSimplex' (n : ℕ) :=\n  { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 }",
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            "text": "def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1)\n  (a b : StandardTwoSimplex) : StandardTwoSimplex\nwhere\n  x := lambda * a.x + (1 - lambda) * b.x\n  y := lambda * a.y + (1 - lambda) * b.y\n  z := lambda * a.z + (1 - lambda) * b.z\n  x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg)\n  y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg)\n  z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg)\n  sum_eq := by\n    trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda)\n    · ring\n    simp [a.sum_eq, b.sum_eq]",
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            "text": "def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1)\n  (a b : StandardTwoSimplex) : StandardTwoSimplex\nwhere\n  x := lambda * a.x + (1 - lambda) * b.x\n  y := lambda * a.y + (1 - lambda) * b.y\n  z := lambda * a.z + (1 - lambda) * b.z\n  x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg)\n  y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg)\n  z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg)\n  sum_eq := by\n    trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda)\n    · ring\n    simp [a.sum_eq, b.sum_eq]",
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        "id": "theorem norm_mod_lt (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) :\n    (x % y).norm < y.norm := by\n  have norm_y_pos : 0 < norm y := by rwa [norm_pos]\n  have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩\n  · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring\n  have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y\n  · calc\n      norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj]\n      _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2\n          + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs]\n      _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos\n      _ = norm y / 2 * (norm y / 2 * 2) := by ring\n      _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num\n  calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos\n    _ < norm y := by\n        apply Int.ediv_lt_of_lt_mul\n        · norm_num\n        · linarith",
        "en": "theorem norm_mod_lt (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) :\n    (x % y).norm < y.norm := by\n  have norm_y_pos : 0 < norm y := by rwa [norm_pos]\n  have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩\n  · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring\n  have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y\n  · calc\n      norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj]\n      _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2\n          + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs]\n      _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos\n      _ = norm y / 2 * (norm y / 2 * 2) := by ring\n      _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num\n  calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos\n    _ < norm y := by\n        apply Int.ediv_lt_of_lt_mul\n        · norm_num\n        · linarith",
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        "id": "theorem norm_mod_lt (x : GaussInt) {y : GaussInt} (hy : y ≠ 0) :\n    (x % y).norm < y.norm := by\n  have norm_y_pos : 0 < norm y := by rwa [norm_pos]\n  have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩\n  · ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring\n  have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y\n  · calc\n      norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj]\n      _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2\n          + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs]\n      _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos\n      _ = norm y / 2 * (norm y / 2 * 2) := by ring\n      _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num\n  calc norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right H2 norm_y_pos\n    _ < norm y := by\n        apply Int.ediv_lt_of_lt_mul\n        · norm_num\n        · linarith",
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        "id": "\nclass LE₁ (α : Type) where\n  /-- Relasi kurang dari atau sama dengan. -/\n  le : α → α → Prop\n\n@[inherit_doc] infix:50 \" ≤₁ \" => LE₁.le\n\nclass Preorder₁ (α : Type)\n-- SOLUTIONS:\n  extends LE₁ α where\n  le_refl : ∀ a : α, a ≤₁ a\n  le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c\n-- BOTH:\n\nclass PartialOrder₁ (α : Type)\n-- SOLUTIONS:\n  extends Preorder₁ α where\n  le_antisymm : ∀ a b : α, a ≤₁ b → b ≤₁ a → a = b\n-- BOTH:\n\nclass OrderedCommMonoid₁ (α : Type)\n-- SOLUTIONS:\n  extends PartialOrder₁ α, CommMonoid₃ α where\n  mul_of_le : ∀ a b : α, a ≤₁ b → ∀ c : α, c * a ≤₁ c * b\n-- BOTH:\n\ninstance : OrderedCommMonoid₁ ℕ where\n-- SOLUTIONS:\n  le := (· ≤ ·)\n  le_refl := fun _ ↦ le_rfl\n  le_trans := fun _ _ _ ↦ le_trans\n  le_antisymm := fun _ _ ↦ le_antisymm\n  mul := (· * ·)\n  mul_assoc₃ := mul_assoc\n  one := 1\n  one_mul := one_mul\n  mul_one := mul_one\n  mul_comm := mul_comm\n  mul_of_le := fun _ _ h c ↦ Nat.mul_le_mul_left c h",
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        "id": "class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where\n  /-- Perkalian distributif kiri terhadap penjumlahan -/\n  left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c\n  /-- Perkalian distributif kanan terhadap penjumlahan -/\n  right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c\n\ninstance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n/- EXAMPLES:\n    sorry }\nSOLUTIONS: -/\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }",
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            "text": "instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }",
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            "source_region": "instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n/- EXAMPLES:\n    sorry }\nSOLUTIONS: -/\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }"
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            "sha256": "25f066dcb4566e8ff1d9fc415258d2a761a6dc3038f64e8b80c472c4f6027217",
            "text": "instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }",
            "line_start": 476,
            "file_sha256": "41efe77a09adb30c51732ef8154ae6fec9cde235ed33970ffa9d7d2f19c42eb4",
            "method": "native_conditional_first_marker_projection",
            "source_region": "instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n/- EXAMPLES:\n    sorry }\nSOLUTIONS: -/\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }"
          }
        },
        "id": "instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }",
        "en": "instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=\n{ add_comm := by\n    intro a b\n    have : a + (a + b + b) = a + (b + a + b) := calc\n      a + (a + b + b) = (a + a) + (b + b) := by simp [add_assoc₃, add_assoc₃]\n      _ = (1 * a + 1 * a) + (1 * b + 1 * b) := by simp\n      _ = (1 + 1) * a + (1 + 1) * b := by simp [Ring₃.right_distrib]\n      _ = (1 + 1) * (a + b) := by simp [Ring₃.left_distrib]\n      _ = 1 * (a + b) + 1 * (a + b) := by simp [Ring₃.right_distrib]\n      _ = (a + b) + (a + b) := by simp\n      _ = a + (b + a + b) := by simp [add_assoc₃]\n    exact add_right_cancel₃ (add_left_cancel₃ this) }"
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        "id": "@[ext]\nstructure OrderPresHom (α β : Type) [LE α] [LE β] where\n  toFun : α → β\n  le_of_le : ∀ a a', a ≤ a' → toFun a ≤ toFun a'\n\n@[ext]\nstructure OrderPresMonoidHom (M N : Type) [Monoid M] [LE M] [Monoid N] [LE N] extends\nMonoidHom₁ M N, OrderPresHom M N\n\nclass OrderPresHomClass (F : Type) (α β : outParam Type) [LE α] [LE β]\n-- SOLUTIONS:\nextends DFunLike F α (fun _ ↦ β) where\n  le_of_le : ∀ (f : F) a a', a ≤ a' → f a ≤ f a'\n-- BOTH:\n\ninstance (α β : Type) [LE α] [LE β] : OrderPresHomClass (OrderPresHom α β) α β where\n-- SOLUTIONS:\n  coe := OrderPresHom.toFun\n  coe_injective' _ _ := OrderPresHom.ext\n  le_of_le := OrderPresHom.le_of_le\n-- BOTH:\n\ninstance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] :\n    OrderPresHomClass (OrderPresMonoidHom α β) α β where\n-- SOLUTIONS:\n  coe := fun f ↦ f.toOrderPresHom.toFun\n  coe_injective' _ _ := OrderPresMonoidHom.ext\n  le_of_le := fun f ↦ f.toOrderPresHom.le_of_le\n-- BOTH:\n\ninstance (α β : Type) [LE α] [Monoid α] [LE β] [Monoid β] :\n    MonoidHomClass₃ (OrderPresMonoidHom α β) α β\n/- EXAMPLES:\n  := sorry\nSOLUTIONS: -/\nwhere\n  coe := fun f ↦ f.toOrderPresHom.toFun\n  coe_injective' _ _ := OrderPresMonoidHom.ext\n  map_one := fun f ↦ f.toMonoidHom₁.map_one\n  map_mul := fun f ↦ f.toMonoidHom₁.map_mul",
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        "id": "def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M  where\n  r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z\n  iseqv := {\n    refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩\n    symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩\n    trans := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩\n      refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n      rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc]\n-- BOTH:\n  }\n\ninstance [CommMonoid M] : HasQuotient M (Submonoid M) where\n  Quotient := fun N ↦ Quotient N.Setoid\n\ndef QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid\n\ninstance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where\n  mul := Quotient.map₂ (· * ·) (by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩\n    refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n    rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha,\n        mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc]\n-- BOTH:\n        )\n  mul_assoc := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩\n    apply Quotient.sound\n    dsimp only\n    rw [mul_assoc]\n-- BOTH:\n  one := QuotientMonoid.mk N 1\n  one_mul := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul]\n-- BOTH:\n  mul_one := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one]",
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        "id": "def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M  where\n  r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z\n  iseqv := {\n    refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩\n    symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩\n    trans := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩\n      refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n      rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc]\n-- BOTH:\n  }\n\ninstance [CommMonoid M] : HasQuotient M (Submonoid M) where\n  Quotient := fun N ↦ Quotient N.Setoid\n\ndef QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid\n\ninstance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where\n  mul := Quotient.map₂ (· * ·) (by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩\n    refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n    rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha,\n        mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc]\n-- BOTH:\n        )\n  mul_assoc := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩\n    apply Quotient.sound\n    dsimp only\n    rw [mul_assoc]\n-- BOTH:\n  one := QuotientMonoid.mk N 1\n  one_mul := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul]\n-- BOTH:\n  mul_one := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one]",
        "en": "def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M  where\n  r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z\n  iseqv := {\n    refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩\n    symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩\n    trans := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩\n      refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n      rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc]\n-- BOTH:\n  }\n\ninstance [CommMonoid M] : HasQuotient M (Submonoid M) where\n  Quotient := fun N ↦ Quotient N.Setoid\n\ndef QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid\n\ninstance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where\n  mul := Quotient.map₂ (· * ·) (by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩\n    refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n    rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha,\n        mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc]\n-- BOTH:\n        )\n  mul_assoc := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩\n    apply Quotient.sound\n    dsimp only\n    rw [mul_assoc]\n-- BOTH:\n  one := QuotientMonoid.mk N 1\n  one_mul := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul]\n-- BOTH:\n  mul_one := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one]",
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        "id": "def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M  where\n  r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z\n  iseqv := {\n    refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩\n    symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩\n    trans := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩\n      refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n      rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc]\n-- BOTH:\n  }\n\ninstance [CommMonoid M] : HasQuotient M (Submonoid M) where\n  Quotient := fun N ↦ Quotient N.Setoid\n\ndef QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid\n\ninstance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where\n  mul := Quotient.map₂ (· * ·) (by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩\n    refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n    rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha,\n        mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc]\n-- BOTH:\n        )\n  mul_assoc := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩\n    apply Quotient.sound\n    dsimp only\n    rw [mul_assoc]\n-- BOTH:\n  one := QuotientMonoid.mk N 1\n  one_mul := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul]\n-- BOTH:\n  mul_one := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one]",
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        "id": "def Submonoid.Setoid [CommMonoid M] (N : Submonoid M) : Setoid M  where\n  r := fun x y ↦ ∃ w ∈ N, ∃ z ∈ N, x*w = y*z\n  iseqv := {\n    refl := fun x ↦ ⟨1, N.one_mem, 1, N.one_mem, rfl⟩\n    symm := fun ⟨w, hw, z, hz, h⟩ ↦ ⟨z, hz, w, hw, h.symm⟩\n    trans := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      rintro a b c ⟨w, hw, z, hz, h⟩ ⟨w', hw', z', hz', h'⟩\n      refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n      rw [← mul_assoc, h, mul_comm b, mul_assoc, h', ← mul_assoc, mul_comm z, mul_assoc]\n-- BOTH:\n  }\n\ninstance [CommMonoid M] : HasQuotient M (Submonoid M) where\n  Quotient := fun N ↦ Quotient N.Setoid\n\ndef QuotientMonoid.mk [CommMonoid M] (N : Submonoid M) : M → M ⧸ N := Quotient.mk N.Setoid\n\ninstance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where\n  mul := Quotient.map₂ (· * ·) (by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro a₁ b₁ ⟨w, hw, z, hz, ha⟩ a₂ b₂ ⟨w', hw', z', hz', hb⟩\n    refine ⟨w*w', N.mul_mem hw hw', z*z', N.mul_mem hz hz', ?_⟩\n    rw [mul_comm w, ← mul_assoc, mul_assoc a₁, hb, mul_comm, ← mul_assoc, mul_comm w, ha,\n        mul_assoc, mul_comm z, mul_assoc b₂, mul_comm z', mul_assoc]\n-- BOTH:\n        )\n  mul_assoc := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩\n    apply Quotient.sound\n    dsimp only\n    rw [mul_assoc]\n-- BOTH:\n  one := QuotientMonoid.mk N 1\n  one_mul := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul]\n-- BOTH:\n  mul_one := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n    rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one]",
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        "id": "section exercises\nvariable {G H : Type*} [Group G] [Group H]\n\nopen Subgroup\n\nexample (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  intro x hx\n  rw [mem_comap] at * -- Lean tidak memerlukan baris ini\n  exact hST hx\n-- BOTH:\n\nexample (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  intro x hx\n  rw [mem_map] at * -- Lean tidak memerlukan baris ini\n  rcases hx with ⟨y, hy, rfl⟩\n  use y, hST hy\n-- BOTH:\n\nvariable {K : Type*} [Group K]\n\n-- Ingat bahwa Anda dapat memakai taktik `ext` untuk membuktikan kesamaan subgrup.\nexample (φ : G →* H) (ψ : H →* K) (U : Subgroup K) :\n    comap (ψ.comp φ) U = comap φ (comap ψ U) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  -- Seluruh bukti dapat berupa ``rfl``, tetapi mari kita uraikan sedikit.\n  ext x\n  simp only [mem_comap]\n  rfl\n-- BOTH:\n\n-- Mengambil citra langsung subgrup melalui satu homomorfisme lalu homomorfisme lain\n-- sama dengan mengambil citra langsungnya melalui komposisi kedua homomorfisme.\nexample (φ : G →* H) (ψ : H →* K) (S : Subgroup G) :\n    map (ψ.comp φ) S = map ψ (S.map φ) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext x\n  simp only [mem_map]\n  constructor\n  · rintro ⟨y, y_in, hy⟩\n    exact ⟨φ y, ⟨y, y_in, rfl⟩, hy⟩\n  · rintro ⟨y, ⟨z, z_in, hz⟩, hy⟩\n    use z, z_in\n    calc ψ.comp φ z = ψ (φ z) := rfl\n    _               = ψ y := by congr\n    _               = x := hy\n-- BOTH:\n\nend exercises",
        "en": "section exercises\nvariable {G H : Type*} [Group G] [Group H]\n\nopen Subgroup\n\nexample (φ : G →* H) (S T : Subgroup H) (hST : S ≤ T) : comap φ S ≤ comap φ T := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  intro x hx\n  rw [mem_comap] at * -- Lean does not need this line\n  exact hST hx\n-- BOTH:\n\nexample (φ : G →* H) (S T : Subgroup G) (hST : S ≤ T) : map φ S ≤ map φ T := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  intro x hx\n  rw [mem_map] at * -- Lean does not need this line\n  rcases hx with ⟨y, hy, rfl⟩\n  use y, hST hy\n-- BOTH:\n\nvariable {K : Type*} [Group K]\n\n-- Remember you can use the `ext` tactic to prove an equality of subgroups.\nexample (φ : G →* H) (ψ : H →* K) (U : Subgroup K) :\n    comap (ψ.comp φ) U = comap φ (comap ψ U) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  -- The whole proof could be ``rfl``, but let's decompose it a bit.\n  ext x\n  simp only [mem_comap]\n  rfl\n-- BOTH:\n\n-- Pushing a subgroup along one homomorphism and then another is equal to\n-- pushing it forward along the composite of the homomorphisms.\nexample (φ : G →* H) (ψ : H →* K) (S : Subgroup G) :\n    map (ψ.comp φ) S = map ψ (S.map φ) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext x\n  simp only [mem_map]\n  constructor\n  · rintro ⟨y, y_in, hy⟩\n    exact ⟨φ y, ⟨y, y_in, rfl⟩, hy⟩\n  · rintro ⟨y, ⟨z, z_in, hz⟩, hy⟩\n    use z, z_in\n    calc ψ.comp φ z = ψ (φ z) := rfl\n    _               = ψ y := by congr\n    _               = x := hy\n-- BOTH:\n\nend exercises",
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        "id": "open scoped Classical\n\n-- EXAMPLES:\n\nexample {G : Type*} [Group G] (G' : Subgroup G) : Nat.card G' ∣ Nat.card G :=\n  ⟨G'.index, mul_comm G'.index _ ▸ G'.index_mul_card.symm⟩\n\n-- BOTH:\nopen Subgroup\n\n-- EXAMPLES:\nexample {G : Type*} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime]\n    (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n :=\n  Sylow.exists_subgroup_card_pow_prime p hdvd",
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        "id": "lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} :\n    H = ⊥ ↔ Nat.card H = 1 := by\n  suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by\n    simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  constructor\n  · intro h\n    use 1, H.one_mem\n  · rintro ⟨y, -, hy'⟩ x hx\n    calc x = y := hy' x hx\n    _      = 1 := (hy' 1 H.one_mem).symm\n-- EXAMPLES:\n\n#check card_dvd_of_le\n-- BOTH:\n\nlemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G)\n    (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  have D₁ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card H := card_dvd_of_le inf_le_left\n  have D₂ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card K := card_dvd_of_le inf_le_right\n  exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂)",
        "en": "lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} :\n    H = ⊥ ↔ Nat.card H = 1 := by\n  suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by\n    simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  constructor\n  · intro h\n    use 1, H.one_mem\n  · rintro ⟨y, -, hy'⟩ x hx\n    calc x = y := hy' x hx\n    _      = 1 := (hy' 1 H.one_mem).symm\n-- EXAMPLES:\n\n#check card_dvd_of_le\n-- BOTH:\n\nlemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G)\n    (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  have D₁ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card H := card_dvd_of_le inf_le_left\n  have D₂ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card K := card_dvd_of_le inf_le_right\n  exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂)",
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        "id": "section\nvariable {G : Type*} [Group G] {H K : Subgroup G}\n\nopen MonoidHom\n\n#check Nat.card_pos -- Argumen tak kosong akan disimpulkan secara otomatis untuk subgrup\n#check Subgroup.index_eq_card\n#check Subgroup.index_mul_card\n#check Nat.eq_of_mul_eq_mul_right\n\nlemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) :\n    Nat.card (G ⧸ H) = Nat.card K := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  have := calc\n    Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]\n    _                             = Nat.card K * Nat.card H := by rw [h', mul_comm]\n\n  exact Nat.eq_of_mul_eq_mul_right Nat.card_pos this",
        "en": "section\nvariable {G : Type*} [Group G] {H K : Subgroup G}\n\nopen MonoidHom\n\n#check Nat.card_pos -- The nonempty argument will be automatically inferred for subgroups\n#check Subgroup.index_eq_card\n#check Subgroup.index_mul_card\n#check Nat.eq_of_mul_eq_mul_right\n\nlemma aux_card_eq [Finite G] (h' : Nat.card G = Nat.card H * Nat.card K) :\n    Nat.card (G ⧸ H) = Nat.card K := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  have := calc\n    Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]\n    _                             = Nat.card K * Nat.card H := by rw [h', mul_comm]\n\n  exact Nat.eq_of_mul_eq_mul_right Nat.card_pos this",
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        "id": "def iso₂ : G ≃* (G ⧸ K) × (G ⧸ H) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply MulEquiv.ofBijective <| (QuotientGroup.mk' K).prod (QuotientGroup.mk' H)\n  rw [Nat.bijective_iff_injective_and_card]\n  constructor\n  · rw [← ker_eq_bot_iff, ker_prod]\n    simp [h.symm.eq_bot]\n  · rw [Nat.card_prod]\n    rw [aux_card_eq h', aux_card_eq (mul_comm (Nat.card H) _▸ h'), h']",
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        "id": "#check MulEquiv.prodCongr\n\n-- BOTH:\ndef finalIso : G ≃* H × K :=\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  (iso₂ h h').trans ((iso₁ h.symm (mul_comm (Nat.card H) _ ▸ h')).prodCongr (iso₁ h h')).symm\n\nend\nend QuotientGroup",
        "en": "#check MulEquiv.prodCongr\n\n-- BOTH:\ndef finalIso : G ≃* H × K :=\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  (iso₂ h h').trans ((iso₁ h.symm (mul_comm (Nat.card H) _ ▸ h')).prodCongr (iso₁ h h')).symm\n\nend\nend QuotientGroup",
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        "en": "lemma eq_bot_iff_card {G : Type*} [Group G] {H : Subgroup G} :\n    H = ⊥ ↔ Nat.card H = 1 := by\n  suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by\n    simpa [eq_bot_iff_forall, Nat.card_eq_one_iff_exists]\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  constructor\n  · intro h\n    use 1, H.one_mem\n  · rintro ⟨y, -, hy'⟩ x hx\n    calc x = y := hy' x hx\n    _      = 1 := (hy' 1 H.one_mem).symm\n-- EXAMPLES:\n\n#check card_dvd_of_le\n-- BOTH:\n\nlemma inf_bot_of_coprime {G : Type*} [Group G] (H K : Subgroup G)\n    (h : (Nat.card H).Coprime (Nat.card K)) : H ⊓ K = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  have D₁ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card H := card_dvd_of_le inf_le_left\n  have D₂ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card K := card_dvd_of_le inf_le_right\n  exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂)",
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            "sha256": "1ec2114a25a0152be8da055255ac95a5c390bc074c873c9923811fbb915ba14d",
            "text": "  ext x\n  simp only [mem_map]\n  constructor\n  · rintro ⟨y, y_in, hy⟩\n    exact ⟨φ y, ⟨y, y_in, rfl⟩, hy⟩\n  · rintro ⟨y, ⟨z, z_in, hz⟩, hy⟩\n    use z, z_in\n    calc ψ.comp φ z = ψ (φ z) := rfl\n    _               = ψ y := by congr\n    _               = x := hy",
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            "text": "  constructor\n  · intro h\n    use 1, H.one_mem\n  · rintro ⟨y, -, hy'⟩ x hx\n    calc x = y := hy' x hx\n    _      = 1 := (hy' 1 H.one_mem).symm",
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            "text": "  constructor\n  · intro h\n    use 1, H.one_mem\n  · rintro ⟨y, -, hy'⟩ x hx\n    calc x = y := hy' x hx\n    _      = 1 := (hy' 1 H.one_mem).symm",
            "line_start": 459,
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        "en": "  constructor\n  · intro h\n    use 1, H.one_mem\n  · rintro ⟨y, -, hy'⟩ x hx\n    calc x = y := hy' x hx\n    _      = 1 := (hy' 1 H.one_mem).symm"
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            "text": "  have D₁ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card H := card_dvd_of_le inf_le_left\n  have D₂ : Nat.card (H ⊓ K : Subgroup G) ∣ Nat.card K := card_dvd_of_le inf_le_right\n  exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂)",
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            "line_start": 475,
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    },
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      "id": "urn:mil:unit:c09:s01:solution:012",
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      "title": {
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            "sha256": "c93fecd227dadc9a66b2bfd55b2549c124c459f3a73bd6bec90a27968ccb5c5e",
            "text": "    intro x y H\n    ext z\n    constructor\n    · rintro ⟨h, h_in, rfl⟩\n      use y*h*y⁻¹\n      constructor\n      · use h\n      · group\n    · rintro ⟨-, ⟨h, h_in, rfl⟩, rfl⟩\n      use h, h_in\n      group",
            "line_start": 744,
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            "sha256": "c93fecd227dadc9a66b2bfd55b2549c124c459f3a73bd6bec90a27968ccb5c5e",
            "text": "    intro x y H\n    ext z\n    constructor\n    · rintro ⟨h, h_in, rfl⟩\n      use y*h*y⁻¹\n      constructor\n      · use h\n      · group\n    · rintro ⟨-, ⟨h, h_in, rfl⟩, rfl⟩\n      use h, h_in\n      group",
            "line_start": 718,
            "file_sha256": "957619a8127552299ab3ba7f3d886f7c7711b999b665ee8b2c0ec9aebfcd6ca2",
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        "id": "    intro x y H\n    ext z\n    constructor\n    · rintro ⟨h, h_in, rfl⟩\n      use y*h*y⁻¹\n      constructor\n      · use h\n      · group\n    · rintro ⟨-, ⟨h, h_in, rfl⟩, rfl⟩\n      use h, h_in\n      group",
        "en": "    intro x y H\n    ext z\n    constructor\n    · rintro ⟨h, h_in, rfl⟩\n      use y*h*y⁻¹\n      constructor\n      · use h\n      · group\n    · rintro ⟨-, ⟨h, h_in, rfl⟩, rfl⟩\n      use h, h_in\n      group"
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      "supports": [],
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      "source_record": {
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        "concept_ids": [
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          "urn:mil:concept:group",
          "urn:mil:concept:subgroup"
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        "data": {
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          "exercise_id": "urn:mil:unit:c09:s01:instance:001",
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          "hole_rank": 2,
          "hole_ranks": [
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        "language": "zxx",
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        "order_index": 912,
        "parent_id": "urn:mil:unit:c09:s01",
        "prerequisite_ids": [],
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        "qa_event_ids": [
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        "record_id": "urn:mil:unit:c09:s01:solution:012",
        "resource_id": "urn:mil:resource:avigad:mathematics-in-lean-source",
        "rights_id": "mil.repo.code",
        "schema": "mil-backend-record",
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    },
    {
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      "title": {
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      },
      "reader_routes": {
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          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C09_Groups_and_Rings.html#monoid-dan-grup",
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      "source_locator": "MIL/C09_Groups_and_Rings/S01_Groups.lean#line-834",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
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      "concept_ids": [
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        "urn:mil:concept:lagrange-theorem",
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        "urn:mil:concept:subgroup"
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      "code": {
        "sha256": {
          "id": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e",
          "en": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e",
            "text": "  have := calc\n    Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]\n    _                             = Nat.card K * Nat.card H := by rw [h', mul_comm]\n\n  exact Nat.eq_of_mul_eq_mul_right Nat.card_pos this",
            "line_start": 863,
            "file_sha256": "053ba52d913cb7e549d669e0169179d5d000c44a8f3ff4419be4820876480786",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e",
            "text": "  have := calc\n    Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]\n    _                             = Nat.card K * Nat.card H := by rw [h', mul_comm]\n\n  exact Nat.eq_of_mul_eq_mul_right Nat.card_pos this",
            "line_start": 834,
            "file_sha256": "957619a8127552299ab3ba7f3d886f7c7711b999b665ee8b2c0ec9aebfcd6ca2",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have := calc\n    Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]\n    _                             = Nat.card K * Nat.card H := by rw [h', mul_comm]\n\n  exact Nat.eq_of_mul_eq_mul_right Nat.card_pos this",
        "en": "  have := calc\n    Nat.card (G ⧸ H) * Nat.card H = Nat.card G := by rw [← H.index_eq_card, H.index_mul_card]\n    _                             = Nat.card K * Nat.card H := by rw [h', mul_comm]\n\n  exact Nat.eq_of_mul_eq_mul_right Nat.card_pos this"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c09:s01"
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        "concept_ids": [
          "urn:mil:concept:group",
          "urn:mil:concept:lagrange-theorem",
          "urn:mil:concept:normal-subgroup",
          "urn:mil:concept:subgroup"
        ],
        "data": {
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          "hole_rank": 1,
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          "is_alternative": false,
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          "solution_source_line": 834,
          "solution_status": "verified_solution_present",
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          "source_text_sha256": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e",
          "source_unit_id": "urn:mil:unit:c09:s01:exercise:aux-card-eq",
          "statement_signature": "have",
          "target_is_exercise": true,
          "target_source_line": 829,
          "target_text_sha256": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e",
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          "verified_solution": true
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        ],
        "record_id": "urn:mil:unit:c09:s01:solution:013",
        "resource_id": "urn:mil:resource:avigad:mathematics-in-lean-source",
        "rights_id": "mil.repo.code",
        "schema": "mil-backend-record",
        "schema_version": "1.0.0",
        "source_local_id": "solution-013",
        "source_locator": "MIL/C09_Groups_and_Rings/S01_Groups.lean#line-834",
        "source_path": "MIL/C09_Groups_and_Rings/S01_Groups.lean",
        "source_record_id": "",
        "source_sha256": "eb3d2a6273e58f2519a0794730c5b33c32d9bb6f2e994ed51e6e48cb84b4c61e",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
    },
    {
      "id": "urn:mil:unit:c09:s01:solution:014",
      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c09",
      "section_id": "urn:mil:unit:c09:s01",
      "title": {
        "id": "solution-014",
        "en": "solution-014"
      },
      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C09_Groups_and_Rings.html#monoid-dan-grup",
          "anchor": "monoid-dan-grup",
          "title": "Monoid dan Grup",
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        "en": {
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          "anchor": "monoids-and-groups",
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          "code_equivalence_claimed": false
        }
      },
      "source_locator": "MIL/C09_Groups_and_Rings/S01_Groups.lean#line-858",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
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      ],
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        "urn:mil:concept:normal-subgroup",
        "urn:mil:concept:subgroup"
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      "code": {
        "sha256": {
          "id": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5",
          "en": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5",
            "text": "  apply MulEquiv.ofBijective ((QuotientGroup.mk' H).restrict K)\n  rw [Nat.bijective_iff_injective_and_card]\n  constructor\n  · rw [← ker_eq_bot_iff, (QuotientGroup.mk' H).ker_restrict K]\n    simp [h]\n  · symm\n    exact aux_card_eq h'",
            "line_start": 888,
            "file_sha256": "053ba52d913cb7e549d669e0169179d5d000c44a8f3ff4419be4820876480786",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5",
            "text": "  apply MulEquiv.ofBijective ((QuotientGroup.mk' H).restrict K)\n  rw [Nat.bijective_iff_injective_and_card]\n  constructor\n  · rw [← ker_eq_bot_iff, (QuotientGroup.mk' H).ker_restrict K]\n    simp [h]\n  · symm\n    exact aux_card_eq h'",
            "line_start": 858,
            "file_sha256": "957619a8127552299ab3ba7f3d886f7c7711b999b665ee8b2c0ec9aebfcd6ca2",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  apply MulEquiv.ofBijective ((QuotientGroup.mk' H).restrict K)\n  rw [Nat.bijective_iff_injective_and_card]\n  constructor\n  · rw [← ker_eq_bot_iff, (QuotientGroup.mk' H).ker_restrict K]\n    simp [h]\n  · symm\n    exact aux_card_eq h'",
        "en": "  apply MulEquiv.ofBijective ((QuotientGroup.mk' H).restrict K)\n  rw [Nat.bijective_iff_injective_and_card]\n  constructor\n  · rw [← ker_eq_bot_iff, (QuotientGroup.mk' H).ker_restrict K]\n    simp [h]\n  · symm\n    exact aux_card_eq h'"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c09:s01"
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        "concept_ids": [
          "urn:mil:concept:group",
          "urn:mil:concept:normal-subgroup",
          "urn:mil:concept:subgroup"
        ],
        "data": {
          "alternative_rank": 1,
          "exercise_count": 1,
          "exercise_id": "urn:mil:unit:c09:s01:exercise:iso",
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          "hole_rank": 1,
          "hole_ranks": [
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          ],
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 858,
          "solution_status": "verified_solution_present",
          "solved_hole_count": 1,
          "source_text_sha256": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5",
          "source_unit_id": "urn:mil:unit:c09:s01:exercise:iso",
          "statement_signature": "apply MulEquiv.ofBijective ((QuotientGroup.mk' H).restrict K) rw [Nat.bijective_iff_injective_and_card] constructor · rw [← ker_eq_bot_iff, (QuotientGroup.mk' H).ker_restrict K] simp [h] · symm exact aux_card_eq h'",
          "target_is_exercise": true,
          "target_source_line": 854,
          "target_text_sha256": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5",
          "unit_type": "solution",
          "verified_solution": true
        },
        "edition_id": "urn:mil:edition:source:8e112cd63ff1bf2a1020ff88f22f77288e42b9a9",
        "entity_type": "unit",
        "language": "zxx",
        "locale": "und",
        "order_index": 914,
        "parent_id": "urn:mil:unit:c09:s01",
        "prerequisite_ids": [],
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        "qa_event_ids": [
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          "urn:mil:qa:topology:c09:s01"
        ],
        "record_id": "urn:mil:unit:c09:s01:solution:014",
        "resource_id": "urn:mil:resource:avigad:mathematics-in-lean-source",
        "rights_id": "mil.repo.code",
        "schema": "mil-backend-record",
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        "source_path": "MIL/C09_Groups_and_Rings/S01_Groups.lean",
        "source_record_id": "",
        "source_sha256": "078e1666127ad9907e528a096a971aa7d554bf96ea5893274ccbfa51d2181aa5",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
    },
    {
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      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c09",
      "section_id": "urn:mil:unit:c09:s01",
      "title": {
        "id": "solution-015",
        "en": "solution-015"
      },
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        "id": "lemma chineseMap_mk (I : ι → Ideal R) (x : R) :\n    chineseMap I (Quotient.mk _ x) = fun i : ι ↦ Ideal.Quotient.mk (I i) x :=\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rfl\n-- BOTH:\n\nlemma chineseMap_mk' (I : ι → Ideal R) (x : R) (i : ι) :\n    chineseMap I (mk _ x) i = mk (I i) x :=\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rfl",
        "en": "lemma chineseMap_mk (I : ι → Ideal R) (x : R) :\n    chineseMap I (Quotient.mk _ x) = fun i : ι ↦ Ideal.Quotient.mk (I i) x :=\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rfl\n-- BOTH:\n\nlemma chineseMap_mk' (I : ι → Ideal R) (x : R) (i : ι) :\n    chineseMap I (mk _ x) i = mk (I i) x :=\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rfl",
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        "id": "lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}\n    (hI : ∀ i j, i ≠ j → IsCoprime (I i) (I j)) : Surjective (chineseMap I) := by\n  classical\n  intro g\n  choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i)\n  have key : ∀ i, ∃ e : R, mk (I i) e = 1 ∧ ∀ j, j ≠ i → mk (I j) e = 0 := by\n    intro i\n    have hI' : ∀ j ∈ ({i} : Finset ι)ᶜ, IsCoprime (I i) (I j) := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      intros j hj\n      exact hI _ _ (by simpa [ne_comm, isCoprime_iff_add] using hj)\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rcases isCoprime_iff_exists.mp (isCoprime_Inf hI') with ⟨u, hu, e, he, hue⟩\n    replace he : ∀ j, j ≠ i → e ∈ I j := by simpa using he\n    refine ⟨e, ?_, ?_⟩\n    · simp [eq_sub_of_add_eq' hue, map_sub, eq_zero_iff_mem.mpr hu]\n    · exact fun j hj ↦ eq_zero_iff_mem.mpr (he j hj)\n-- BOTH:\n  choose e he using key\n  use mk _ (∑ i, f i * e i)\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext i\n  rw [chineseMap_mk', map_sum, Fintype.sum_eq_single i]\n  · simp [(he i).1, hf]\n  · intros j hj\n    simp [(he j).2 i hj.symm]",
        "en": "lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}\n    (hI : ∀ i j, i ≠ j → IsCoprime (I i) (I j)) : Surjective (chineseMap I) := by\n  classical\n  intro g\n  choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i)\n  have key : ∀ i, ∃ e : R, mk (I i) e = 1 ∧ ∀ j, j ≠ i → mk (I j) e = 0 := by\n    intro i\n    have hI' : ∀ j ∈ ({i} : Finset ι)ᶜ, IsCoprime (I i) (I j) := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      intros j hj\n      exact hI _ _ (by simpa [ne_comm, isCoprime_iff_add] using hj)\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rcases isCoprime_iff_exists.mp (isCoprime_Inf hI') with ⟨u, hu, e, he, hue⟩\n    replace he : ∀ j, j ≠ i → e ∈ I j := by simpa using he\n    refine ⟨e, ?_, ?_⟩\n    · simp [eq_sub_of_add_eq' hue, map_sub, eq_zero_iff_mem.mpr hu]\n    · exact fun j hj ↦ eq_zero_iff_mem.mpr (he j hj)\n-- BOTH:\n  choose e he using key\n  use mk _ (∑ i, f i * e i)\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext i\n  rw [chineseMap_mk', map_sum, Fintype.sum_eq_single i]\n  · simp [(he i).1, hf]\n  · intros j hj\n    simp [(he j).2 i hj.symm]",
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        "id": "noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R)\n    (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i :=\n  { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩,\n    chineseMap f with }",
        "en": "noncomputable def chineseIso [Fintype ι] (f : ι → Ideal R)\n    (hf : ∀ i j, i ≠ j → IsCoprime (f i) (f j)) : (R ⧸ ⨅ i, f i) ≃+* Π i, R ⧸ f i :=\n  { Equiv.ofBijective _ ⟨chineseMap_inj f, chineseMap_surj hf⟩,\n    chineseMap f with }",
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        "id": "example {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) :\n    (r + r') • a = r • a + r' • a :=\n  add_smul r r' a\n\nexample {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) :\n    (r * r') • a = r • r' • a :=\n  mul_smul r r' a",
        "en": "example {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) :\n    (r + r') • a = r • a + r' • a :=\n  add_smul r r' a\n\nexample {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (r r' : R) (a : A) :\n    (r * r') • a = r • r' • a :=\n  mul_smul r r' a",
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        "en": "example {R : Type*} [CommRing R] (r : R) : (X + C r) * (X - C r) = X ^ 2 - C (r ^ 2) := by\n  rw [C.map_pow]\n  ring",
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        "id": "open Complex Polynomial\n\nexample : aroots (X ^ 2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by\n  suffices roots (X ^ 2 + 1 : ℂ[X]) = {I, -I} by simpa [aroots_def]\n  have factored : (X ^ 2 + 1 : ℂ[X]) = (X - C I) * (X - C (-I)) := by\n    have key : (C I * C I : ℂ[X]) = -1 := by simp [← C_mul]\n    rw [C_neg]\n    linear_combination key\n  have p_ne_zero : (X - C I) * (X - C (-I)) ≠ 0 := by\n    intro H\n    apply_fun eval 0 at H\n    simp [eval] at H\n  simp only [factored, roots_mul p_ne_zero, roots_X_sub_C]\n  rfl\n\n-- Mathlib mengetahui teorema d'Alembert–Gauss: ``ℂ`` tertutup secara aljabar.\nexample : IsAlgClosed ℂ := inferInstance\n",
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        "en": "lemma chineseMap_surj [Fintype ι] {I : ι → Ideal R}\n    (hI : ∀ i j, i ≠ j → IsCoprime (I i) (I j)) : Surjective (chineseMap I) := by\n  classical\n  intro g\n  choose f hf using fun i ↦ Ideal.Quotient.mk_surjective (g i)\n  have key : ∀ i, ∃ e : R, mk (I i) e = 1 ∧ ∀ j, j ≠ i → mk (I j) e = 0 := by\n    intro i\n    have hI' : ∀ j ∈ ({i} : Finset ι)ᶜ, IsCoprime (I i) (I j) := by\n/- EXAMPLES:\n      sorry\nSOLUTIONS: -/\n      intros j hj\n      exact hI _ _ (by simpa [ne_comm, isCoprime_iff_add] using hj)\n/- EXAMPLES:\n    sorry\nSOLUTIONS: -/\n    rcases isCoprime_iff_exists.mp (isCoprime_Inf hI') with ⟨u, hu, e, he, hue⟩\n    replace he : ∀ j, j ≠ i → e ∈ I j := by simpa using he\n    refine ⟨e, ?_, ?_⟩\n    · simp [eq_sub_of_add_eq' hue, map_sub, eq_zero_iff_mem.mpr hu]\n    · exact fun j hj ↦ eq_zero_iff_mem.mpr (he j hj)\n-- BOTH:\n  choose e he using key\n  use mk _ (∑ i, f i * e i)\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  ext i\n  rw [chineseMap_mk', map_sum, Fintype.sum_eq_single i]\n  · simp [(he i).1, hf]\n  · intros j hj\n    simp [(he j).2 i hj.symm]",
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        "id": "#check Finset.mem_insert_of_mem\n#check Finset.mem_insert_self\n\n-- BOTH:\ntheorem isCoprime_Inf {I : Ideal R} {J : ι → Ideal R} {s : Finset ι}\n    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by\n  classical\n  simp_rw [isCoprime_iff_add] at *\n  induction s using Finset.induction with\n  | empty =>\n      simp\n  | @insert i s _ hs =>\n      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]\n      set K := ⨅ j ∈ s, J j\n      calc\n/- EXAMPLES:\n        1 = I + K                  := sorry\n        _ = I + K * (I + J i)      := sorry\n        _ = (1 + K) * I + K * J i  := sorry\n        _ ≤ I + K ⊓ J i            := sorry\nSOLUTIONS: -/\n        1 = I + K                  := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm\n        _ = I + K * (I + J i)      := by rw [hf i (Finset.mem_insert_self i s), mul_one]\n        _ = (1 + K) * I + K * J i  := by ring\n        _ ≤ I + K ⊓ J i            := by gcongr ; apply mul_le_left ; apply mul_le_inf\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "section families\nopen DirectSum\n\nvariable {ι : Type*} [DecidableEq ι]\n         (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]\n\n-- Sifat universal jumlah langsung merakit pemetaan dari suku-sukunya untuk\n-- membangun pemetaan dari jumlah langsung\nexample (φ : Π i, (V i →ₗ[K] W)) : (⨁ i, V i) →ₗ[K] W :=\n  DirectSum.toModule K ι W φ\n\n-- Sifat universal produk langsung merakit pemetaan menuju faktor-faktornya untuk\n-- membangun pemetaan menuju produk langsung\nexample (φ : Π i, (W →ₗ[K] V i)) : W →ₗ[K] (Π i, V i) :=\n  LinearMap.pi φ\n\n-- Pemetaan proyeksi dari produk\nexample (i : ι) : (Π j, V j) →ₗ[K] V i := LinearMap.proj i\n\n-- Pemetaan inklusi menuju jumlah\nexample (i : ι) : V i →ₗ[K] (⨁ i, V i) := DirectSum.lof K ι V i\n\n-- Pemetaan inklusi menuju produk\nexample (i : ι) : V i →ₗ[K] (Π i, V i) := LinearMap.single K V i\n\n-- Jika `ι` merupakan tipe hingga, terdapat isomorfisme antara jumlah dan produk.\nexample [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) :=\n  linearEquivFunOnFintype K ι V\n\nend families",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n  -- Pemetaan proyeksi pertama\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n  -- Pemetaan proyeksi kedua\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n  -- Sifat universal produk\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n  -- Pemetaan produk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n  -- Kita juga dapat menggabungkan pemetaan secara paralel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n  -- Hal ini cukup dilakukan dengan menggabungkan proyeksi dan sifat universal\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n  -- Pemetaan inklusi pertama\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n  -- Pemetaan inklusi kedua\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n  -- Sifat universal jumlah (alias koproduk)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen pertama\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n  -- Pemetaan koproduk bekerja sebagaimana diharapkan pada komponen kedua\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n  -- Pemetaan koproduk didefinisikan sebagaimana yang diharapkan\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
        "en": "\nsection binary_product\n\nvariable {W : Type*} [AddCommGroup W] [Module K W]\nvariable {U : Type*} [AddCommGroup U] [Module K U]\nvariable {T : Type*} [AddCommGroup T] [Module K T]\n\n-- First projection map\nexample : V × W →ₗ[K] V := LinearMap.fst K V W\n\n-- Second projection map\nexample : V × W →ₗ[K] W := LinearMap.snd K V W\n\n-- Universal property of the product\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ\n\n-- The product map does the expected thing, first component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl\n\n-- The product map does the expected thing, second component\nexample (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl\n\n-- We can also combine maps in parallel\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ\n\n-- This is simply done by combining the projections with the universal property\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :\n  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl\n\n-- First inclusion map\nexample : V →ₗ[K] V × W := LinearMap.inl K V W\n\n-- Second inclusion map\nexample : W →ₗ[K] V × W := LinearMap.inr K V W\n\n-- Universal property of the sum (aka coproduct)\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ\n\n-- The coproduct map does the expected thing, first component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=\n  LinearMap.coprod_inl φ ψ\n\n-- The coproduct map does the expected thing, second component\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=\n  LinearMap.coprod_inr φ ψ\n\n-- The coproduct map is defined in the expected way\nexample (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :\n    φ.coprod ψ (v, w) = φ v + ψ w :=\n  rfl\n\nend binary_product\n",
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        "id": "section families\nopen DirectSum\n\nvariable {ι : Type*} [DecidableEq ι]\n         (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]\n\n-- Sifat universal jumlah langsung merakit pemetaan dari suku-sukunya untuk\n-- membangun pemetaan dari jumlah langsung\nexample (φ : Π i, (V i →ₗ[K] W)) : (⨁ i, V i) →ₗ[K] W :=\n  DirectSum.toModule K ι W φ\n\n-- Sifat universal produk langsung merakit pemetaan menuju faktor-faktornya untuk\n-- membangun pemetaan menuju produk langsung\nexample (φ : Π i, (W →ₗ[K] V i)) : W →ₗ[K] (Π i, V i) :=\n  LinearMap.pi φ\n\n-- Pemetaan proyeksi dari produk\nexample (i : ι) : (Π j, V j) →ₗ[K] V i := LinearMap.proj i\n\n-- Pemetaan inklusi menuju jumlah\nexample (i : ι) : V i →ₗ[K] (⨁ i, V i) := DirectSum.lof K ι V i\n\n-- Pemetaan inklusi menuju produk\nexample (i : ι) : V i →ₗ[K] (Π i, V i) := LinearMap.single K V i\n\n-- Jika `ι` merupakan tipe hingga, terdapat isomorfisme antara jumlah dan produk.\nexample [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) :=\n  linearEquivFunOnFintype K ι V\n\nend families",
        "en": "section families\nopen DirectSum\n\nvariable {ι : Type*} [DecidableEq ι]\n         (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]\n\n-- The universal property of the direct sum assembles maps from the summands to build\n-- a map from the direct sum\nexample (φ : Π i, (V i →ₗ[K] W)) : (⨁ i, V i) →ₗ[K] W :=\n  DirectSum.toModule K ι W φ\n\n-- The universal property of the direct product assembles maps into the factors\n-- to build a map into the direct product\nexample (φ : Π i, (W →ₗ[K] V i)) : W →ₗ[K] (Π i, V i) :=\n  LinearMap.pi φ\n\n-- The projection maps from the product\nexample (i : ι) : (Π j, V j) →ₗ[K] V i := LinearMap.proj i\n\n-- The inclusion maps into the sum\nexample (i : ι) : V i →ₗ[K] (⨁ i, V i) := DirectSum.lof K ι V i\n\n-- The inclusion maps into the product\nexample (i : ι) : V i →ₗ[K] (Π i, V i) := LinearMap.single K V i\n\n-- In case `ι` is a finite type, there is an isomorphism between the sum and product.\nexample [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) :=\n  linearEquivFunOnFintype K ι V\n\nend families",
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        "en": "\n-- If two subspaces are in direct sum then they span the whole space.\nexample (U V : Submodule K V) (h : IsCompl U V) :\n  U ⊔ V = ⊤ := h.sup_eq_top\n\n-- If two subspaces are in direct sum then they intersect only at zero.\nexample (U V : Submodule K V) (h : IsCompl U V) :\n  U ⊓ V = ⊥ := h.inf_eq_bot\n\nsection\nopen DirectSum\nvariable {ι : Type*} [DecidableEq ι]\n\n-- If subspaces are in direct sum then they span the whole space.\nexample (U : ι → Submodule K V) (h : DirectSum.IsInternal U) :\n  ⨆ i, U i = ⊤ := h.submodule_iSup_eq_top\n\n-- If subspaces are in direct sum then they pairwise intersect only at zero.\nexample {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V) (h : DirectSum.IsInternal U)\n    {i j : ι} (hij : i ≠ j) : U i ⊓ U j = ⊥ :=\n  (h.submodule_iSupIndep.pairwiseDisjoint hij).eq_bot\n\n-- Those conditions characterize direct sums.\n#check DirectSum.isInternal_submodule_iff_iSupIndep_and_iSup_eq_top\n\n-- The relation with external direct sums: if a family of subspaces is\n-- in internal direct sum then the map from their external direct sum into `V`\n-- is a linear isomorphism.\nnoncomputable example {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V)\n    (h : DirectSum.IsInternal U) : (⨁ i, U i) ≃ₗ[K] V :=\n  LinearEquiv.ofBijective (coeLinearMap U) h\nend",
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        "en": "\n-- If two subspaces are in direct sum then they span the whole space.\nexample (U V : Submodule K V) (h : IsCompl U V) :\n  U ⊔ V = ⊤ := h.sup_eq_top\n\n-- If two subspaces are in direct sum then they intersect only at zero.\nexample (U V : Submodule K V) (h : IsCompl U V) :\n  U ⊓ V = ⊥ := h.inf_eq_bot\n\nsection\nopen DirectSum\nvariable {ι : Type*} [DecidableEq ι]\n\n-- If subspaces are in direct sum then they span the whole space.\nexample (U : ι → Submodule K V) (h : DirectSum.IsInternal U) :\n  ⨆ i, U i = ⊤ := h.submodule_iSup_eq_top\n\n-- If subspaces are in direct sum then they pairwise intersect only at zero.\nexample {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V) (h : DirectSum.IsInternal U)\n    {i j : ι} (hij : i ≠ j) : U i ⊓ U j = ⊥ :=\n  (h.submodule_iSupIndep.pairwiseDisjoint hij).eq_bot\n\n-- Those conditions characterize direct sums.\n#check DirectSum.isInternal_submodule_iff_iSupIndep_and_iSup_eq_top\n\n-- The relation with external direct sums: if a family of subspaces is\n-- in internal direct sum then the map from their external direct sum into `V`\n-- is a linear isomorphism.\nnoncomputable example {ι : Type*} [DecidableEq ι] (U : ι → Submodule K V)\n    (h : DirectSum.IsInternal U) : (⨁ i, U i) ≃ₗ[K] V :=\n  LinearEquiv.ofBijective (coeLinearMap U) h\nend",
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        "en": "\nexample (φ : End K V) (a : K) : φ.HasEigenvalue a ↔ φ.eigenspace a ≠ ⊥ :=\n  Iff.rfl\n\nexample (φ : End K V) (a : K) : φ.HasEigenvalue a ↔ ∃ v, φ.HasEigenvector a v  :=\n  ⟨End.HasEigenvalue.exists_hasEigenvector, fun ⟨_, hv⟩ ↦ φ.hasEigenvalue_of_hasEigenvector hv⟩\n\nexample (φ : End K V) : φ.Eigenvalues = {a // φ.HasEigenvalue a} :=\n  rfl\n\n-- Eigenvalue are roots of the minimal polynomial\nexample (φ : End K V) (a : K) : φ.HasEigenvalue a → (minpoly K φ).IsRoot a :=\n  φ.isRoot_of_hasEigenvalue\n\n-- In finite dimension, the converse is also true (we will discuss dimension below)\nexample [FiniteDimensional K V] (φ : End K V) (a : K) :\n    φ.HasEigenvalue a ↔ (minpoly K φ).IsRoot a :=\n  φ.hasEigenvalue_iff_isRoot\n\n-- Cayley-Hamilton\nexample [FiniteDimensional K V] (φ : End K V) : aeval φ φ.charpoly = 0 :=\n  φ.aeval_self_charpoly\n",
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        "id": "\n#check Submodule.eq_bot_iff\n#check Submodule.mem_inf\n#check LinearMap.mem_ker\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ ker (aeval φ Q) = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]\n-- BOTH:\n\n#check Submodule.add_mem_sup\n#check map_mul\n#check End.mul_apply\n#check LinearMap.ker_le_ker_comp\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) :\n    ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- atau alternatif di bawah:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- atau alternatif seperti di atas\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n",
        "en": "\n#check Submodule.eq_bot_iff\n#check Submodule.mem_inf\n#check LinearMap.mem_ker\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ ker (aeval φ Q) = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]\n-- BOTH:\n\n#check Submodule.add_mem_sup\n#check map_mul\n#check End.mul_apply\n#check LinearMap.ker_le_ker_comp\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) :\n    ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- or alternative below:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- or alternative as above\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n",
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        "id": "\n#check Submodule.eq_bot_iff\n#check Submodule.mem_inf\n#check LinearMap.mem_ker\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ ker (aeval φ Q) = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]\n-- BOTH:\n\n#check Submodule.add_mem_sup\n#check map_mul\n#check End.mul_apply\n#check LinearMap.ker_le_ker_comp\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) :\n    ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- atau alternatif di bawah:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- atau alternatif seperti di atas\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n",
        "en": "\n#check Submodule.eq_bot_iff\n#check Submodule.mem_inf\n#check LinearMap.mem_ker\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ ker (aeval φ Q) = ⊥ := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]\n-- BOTH:\n\n#check Submodule.add_mem_sup\n#check map_mul\n#check End.mul_apply\n#check LinearMap.ker_le_ker_comp\n\nexample (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) :\n    ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- or alternative below:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- or alternative as above\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n",
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            "text": "  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]",
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            "sha256": "50dd8fbbc37c9729b2f712e716d19665e8a1126241f9dda49a9ffc431bdeb357",
            "text": "  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]",
            "line_start": 63,
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        "id": "  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]",
        "en": "  rw [Submodule.eq_bot_iff]\n  rintro x hx\n  rw [Submodule.mem_inf, mem_ker, mem_ker] at hx\n  rcases h with ⟨U, V, hUV⟩\n  have := congr((aeval φ) $hUV.symm x)\n  simpa [hx]"
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            "state": "exact_hash_span",
            "sha256": "03781b14b79023004d592c623978e56dfd7001ceaa4c690be883882c40e636f4",
            "text": "  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- atau alternatif di bawah:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- atau alternatif seperti di atas\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]",
            "line_start": 84,
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            "method": "exact_contiguous_span"
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          "en": {
            "state": "exact_hash_span",
            "sha256": "42431d520242d72dfeb713b1e3df1c3d8ca04d0f5665867b8fc355ed407afd47",
            "text": "  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- or alternative below:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- or alternative as above\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]",
            "line_start": 81,
            "file_sha256": "209c2351df6bf0deb5cdf9a32a58fa5a7cd0ca41d87cbbbf03d548ba70f60a82",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- atau alternatif di bawah:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- atau alternatif seperti di atas\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]",
        "en": "  apply le_antisymm\n  · apply sup_le\n    · rw [mul_comm, map_mul]\n      apply ker_le_ker_comp -- or alternative below:\n      -- intro x hx\n      -- rw [mul_comm, mem_ker] at *\n      -- simp [hx]\n    · rw [map_mul]\n      apply ker_le_ker_comp -- or alternative as above\n  · intro x hx\n    rcases h with ⟨U, V, hUV⟩\n    have key : x = aeval φ (U*P) x + aeval φ (V*Q) x := by simpa using congr((aeval φ) $hUV.symm x)\n    rw [key, add_comm]\n    apply Submodule.add_mem_sup <;> rw [mem_ker] at *\n    · rw [← mul_apply, ← map_mul, show P*(V*Q) = V*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]\n    · rw [← mul_apply, ← map_mul, show Q*(U*P) = U*(P*Q) by ring, map_mul, mul_apply, hx,\n          map_zero]"
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        "id": "def Tendsto₁ {X Y : Type*} (f : X → Y) (F : Filter X) (G : Filter Y) :=\n  ∀ V ∈ G, f ⁻¹' V ∈ F",
        "en": "def Tendsto₁ {X Y : Type*} (f : X → Y) (F : Filter X) (G : Filter Y) :=\n  ∀ V ∈ G, f ⁻¹' V ∈ F",
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            "text": "example {X Y Z : Type*} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z}\n    (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H :=\n  calc\n    map (g ∘ f) F = map g (map f F) := by rw [map_map]\n    _ ≤ map g G := (map_mono hf)\n    _ ≤ H := hg",
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            "text": "example {X Y Z : Type*} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z}\n    (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H :=\n  calc\n    map (g ∘ f) F = map g (map f F) := by rw [map_map]\n    _ ≤ map g G := (map_mono hf)\n    _ ≤ H := hg",
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            "text": "example {X Y Z : Type*} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z}\n    (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by\n  intro V hV\n  rw [preimage_comp]\n  apply hf\n  apply hg\n  exact hV",
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            "text": "example {X Y Z : Type*} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z}\n    (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by\n  intro V hV\n  rw [preimage_comp]\n  apply hf\n  apply hg\n  exact hV",
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            "text": "example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔\n      Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) :=\n  calc\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl\n    _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq]\n    _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl\n    _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff\n    _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by\n      rw [← map_le_iff_le_comap, ← map_le_iff_le_comap]\n    _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by\n      rw [map_map, map_map]",
            "line_start": 324,
            "file_sha256": "29ed35bfa05a5ed79507eccf1f9a66ecf1ce05fc1541d56cd594b40216627b3f",
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            "sha256": "cc22bfd5962755380325373a5bcdc9cb375674b61fe369b1e620563528628030",
            "text": "example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔\n      Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) :=\n  calc\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl\n    _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq]\n    _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl\n    _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff\n    _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by\n      rw [← map_le_iff_le_comap, ← map_le_iff_le_comap]\n    _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by\n      rw [map_map, map_map]",
            "line_start": 281,
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        "en": "example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔\n      Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) :=\n  calc\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl\n    _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq]\n    _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl\n    _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff\n    _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by\n      rw [← map_le_iff_le_comap, ← map_le_iff_le_comap]\n    _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by\n      rw [map_map, map_map]"
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            "text": "example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔\n      Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by\n  rw [nhds_prod_eq]\n  unfold Tendsto SProd.sprod Filter.instSProd\n  rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map]",
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            "text": "example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔\n      Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by\n  rw [nhds_prod_eq]\n  unfold Tendsto SProd.sprod Filter.instSProd\n  rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map]",
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        "en": "example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :\n    Tendsto f atTop (𝓝 (x₀, y₀)) ↔\n      Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by\n  rw [nhds_prod_eq]\n  unfold Tendsto SProd.sprod Filter.instSProd\n  rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map]"
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            "text": "example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x))\n    (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M :=\n  mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)",
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        "id": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'\n",
        "en": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'\n",
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        "id": "example (u : ℕ → X) :\n    CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε :=\n  Metric.cauchySeq_iff\n\nexample (u : ℕ → X) :\n    CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε :=\n  Metric.cauchySeq_iff'\n\nexample [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) :\n    ∃ x, Tendsto u atTop (𝓝 x) :=\n  cauchySeq_tendsto_of_complete hu",
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        "id": "-- EXAMPLES:\ntheorem cauchySeq_of_le_geometric_two'αα {u : ℕ → X}\n    (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by\n  rw [Metric.cauchySeq_iff']\n  intro ε ε_pos\n  obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry\n  use N\n  intro n hn\n  obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn\n  calc\n    dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry\n    _ ≤ ∑ i  ∈ range k, dist (u (N + i)) (u (N + (i + 1))) := sorry\n    _ ≤ ∑ i  ∈ range k, (1 / 2 : ℝ) ^ (N + i) := sorry\n    _ = 1 / 2 ^ N * ∑ i  ∈ range k, (1 / 2 : ℝ) ^ i := sorry\n    _ ≤ 1 / 2 ^ N * 2 := sorry\n    _ < ε := sorry\n",
        "en": "-- EXAMPLES:\ntheorem cauchySeq_of_le_geometric_two'αα {u : ℕ → X}\n    (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by\n  rw [Metric.cauchySeq_iff']\n  intro ε ε_pos\n  obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry\n  use N\n  intro n hn\n  obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn\n  calc\n    dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry\n    _ ≤ ∑ i  ∈ range k, dist (u (N + i)) (u (N + (i + 1))) := sorry\n    _ ≤ ∑ i  ∈ range k, (1 / 2 : ℝ) ^ (N + i) := sorry\n    _ = 1 / 2 ^ N * ∑ i  ∈ range k, (1 / 2 : ℝ) ^ i := sorry\n    _ ≤ 1 / 2 ^ N * 2 := sorry\n    _ < ε := sorry\n",
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        "id": "open Metric\n\n-- EXAMPLES:\nexample [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :\n    Dense (⋂ n, f n) := by\n  let B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n\n  have Bpos : ∀ n, 0 < B n\n  sorry\n  /- Ubah asumsi kerapatan menjadi dua fungsi `center` dan `radius` yang mengaitkan\n    setiap n, x, δ, δpos dengan sebuah pusat dan jari-jari positif sedemikian sehingga\n    `closedBall center radius` tercakup dalam `f n` sekaligus `closedBall x δ`.\n    Kita juga dapat mensyaratkan `radius ≤ (1/2)^(n+1)` agar kelak memperoleh barisan Cauchy. -/\n  have :\n    ∀ (n : ℕ) (x : X),\n      ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n :=\n    by sorry\n  choose! center radius Hpos HB Hball using this\n  intro x\n  rw [mem_closure_iff_nhds_basis nhds_basis_closedBall]\n  intro ε εpos\n  /- `ε` bernilai positif. Kita harus menemukan sebuah titik dalam bola berjari-jari `ε`\n    di sekitar `x` yang termasuk dalam semua `f n`. Untuk itu, secara induktif kita\n    bangun barisan `F n = (c n, r n)` sedemikian sehingga bola tertutup\n    `closedBall (c n) (r n)` tercakup dalam bola sebelumnya sekaligus dalam `f n`,\n    dan `r n` cukup kecil untuk menjamin bahwa `c n` adalah barisan Cauchy.\n    Kemudian `c n` konvergen ke suatu limit yang termasuk dalam semua `f n`. -/\n  let F : ℕ → X × ℝ := fun n ↦\n    Nat.recOn n (Prod.mk x (min ε (B 0)))\n      fun n p ↦ Prod.mk (center n p.1 p.2) (radius n p.1 p.2)\n  let c : ℕ → X := fun n ↦ (F n).1\n  let r : ℕ → ℝ := fun n ↦ (F n).2\n  have rpos : ∀ n, 0 < r n := by sorry\n  have rB : ∀ n, r n ≤ B n := by sorry\n  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by\n    sorry\n  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry\n  have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist\n  -- karena barisan `c n` bersifat Cauchy dalam ruang lengkap, barisan itu konvergen ke limit `y`.\n  rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩\n  -- titik `y` ini akan menjadi titik yang dicari. Kita akan memeriksa bahwa titik tersebut termasuk\n  -- dalam semua `f n` dan dalam `ball x ε`.\n  use y\n  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry\n  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry\n  sorry",
        "en": "open Metric\n\n-- EXAMPLES:\nexample [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :\n    Dense (⋂ n, f n) := by\n  let B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n\n  have Bpos : ∀ n, 0 < B n\n  sorry\n  /- Translate the density assumption into two functions `center` and `radius` associating\n    to any n, x, δ, δpos a center and a positive radius such that\n    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.\n    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/\n  have :\n    ∀ (n : ℕ) (x : X),\n      ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n :=\n    by sorry\n  choose! center radius Hpos HB Hball using this\n  intro x\n  rw [mem_closure_iff_nhds_basis nhds_basis_closedBall]\n  intro ε εpos\n  /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x`\n    belonging to all `f n`. For this, we construct inductively a sequence\n    `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included\n    in the previous ball and in `f n`, and such that `r n` is small enough to ensure\n    that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs\n    to all the `f n`. -/\n  let F : ℕ → X × ℝ := fun n ↦\n    Nat.recOn n (Prod.mk x (min ε (B 0)))\n      fun n p ↦ Prod.mk (center n p.1 p.2) (radius n p.1 p.2)\n  let c : ℕ → X := fun n ↦ (F n).1\n  let r : ℕ → ℝ := fun n ↦ (F n).2\n  have rpos : ∀ n, 0 < r n := by sorry\n  have rB : ∀ n, r n ≤ B n := by sorry\n  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by\n    sorry\n  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry\n  have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist\n  -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.\n  rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩\n  -- this point `y` will be the desired point. We will check that it belongs to all\n  -- `f n` and to `ball x ε`.\n  use y\n  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry\n  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry\n  sorry",
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        "id": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'\n",
        "en": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'\n",
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        "id": "open Metric\n\n-- EXAMPLES:\nexample [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :\n    Dense (⋂ n, f n) := by\n  let B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n\n  have Bpos : ∀ n, 0 < B n\n  sorry\n  /- Ubah asumsi kerapatan menjadi dua fungsi `center` dan `radius` yang mengaitkan\n    setiap n, x, δ, δpos dengan sebuah pusat dan jari-jari positif sedemikian sehingga\n    `closedBall center radius` tercakup dalam `f n` sekaligus `closedBall x δ`.\n    Kita juga dapat mensyaratkan `radius ≤ (1/2)^(n+1)` agar kelak memperoleh barisan Cauchy. -/\n  have :\n    ∀ (n : ℕ) (x : X),\n      ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n :=\n    by sorry\n  choose! center radius Hpos HB Hball using this\n  intro x\n  rw [mem_closure_iff_nhds_basis nhds_basis_closedBall]\n  intro ε εpos\n  /- `ε` bernilai positif. Kita harus menemukan sebuah titik dalam bola berjari-jari `ε`\n    di sekitar `x` yang termasuk dalam semua `f n`. Untuk itu, secara induktif kita\n    bangun barisan `F n = (c n, r n)` sedemikian sehingga bola tertutup\n    `closedBall (c n) (r n)` tercakup dalam bola sebelumnya sekaligus dalam `f n`,\n    dan `r n` cukup kecil untuk menjamin bahwa `c n` adalah barisan Cauchy.\n    Kemudian `c n` konvergen ke suatu limit yang termasuk dalam semua `f n`. -/\n  let F : ℕ → X × ℝ := fun n ↦\n    Nat.recOn n (Prod.mk x (min ε (B 0)))\n      fun n p ↦ Prod.mk (center n p.1 p.2) (radius n p.1 p.2)\n  let c : ℕ → X := fun n ↦ (F n).1\n  let r : ℕ → ℝ := fun n ↦ (F n).2\n  have rpos : ∀ n, 0 < r n := by sorry\n  have rB : ∀ n, r n ≤ B n := by sorry\n  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by\n    sorry\n  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry\n  have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist\n  -- karena barisan `c n` bersifat Cauchy dalam ruang lengkap, barisan itu konvergen ke limit `y`.\n  rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩\n  -- titik `y` ini akan menjadi titik yang dicari. Kita akan memeriksa bahwa titik tersebut termasuk\n  -- dalam semua `f n` dan dalam `ball x ε`.\n  use y\n  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry\n  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry\n  sorry",
        "en": "open Metric\n\n-- EXAMPLES:\nexample [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :\n    Dense (⋂ n, f n) := by\n  let B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n\n  have Bpos : ∀ n, 0 < B n\n  sorry\n  /- Translate the density assumption into two functions `center` and `radius` associating\n    to any n, x, δ, δpos a center and a positive radius such that\n    `closedBall center radius` is included both in `f n` and in `closedBall x δ`.\n    We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/\n  have :\n    ∀ (n : ℕ) (x : X),\n      ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n :=\n    by sorry\n  choose! center radius Hpos HB Hball using this\n  intro x\n  rw [mem_closure_iff_nhds_basis nhds_basis_closedBall]\n  intro ε εpos\n  /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x`\n    belonging to all `f n`. For this, we construct inductively a sequence\n    `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included\n    in the previous ball and in `f n`, and such that `r n` is small enough to ensure\n    that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs\n    to all the `f n`. -/\n  let F : ℕ → X × ℝ := fun n ↦\n    Nat.recOn n (Prod.mk x (min ε (B 0)))\n      fun n p ↦ Prod.mk (center n p.1 p.2) (radius n p.1 p.2)\n  let c : ℕ → X := fun n ↦ (F n).1\n  let r : ℕ → ℝ := fun n ↦ (F n).2\n  have rpos : ∀ n, 0 < r n := by sorry\n  have rB : ∀ n, r n ≤ B n := by sorry\n  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by\n    sorry\n  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry\n  have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist\n  -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.\n  rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩\n  -- this point `y` will be the desired point. We will check that it belongs to all\n  -- `f n` and to `ball x ε`.\n  use y\n  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry\n  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry\n  sorry",
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            "sha256": "71582bad12813fb058bb037e1b0d00ec99579360926b1ee5b2338766190c03b4",
            "text": "example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) :\n    a ∈ closure s := by\n  rw [Metric.tendsto_atTop] at hu\n  rw [Metric.mem_closure_iff]\n  intro ε ε_pos\n  rcases hu ε ε_pos with ⟨N, hN⟩\n  refine ⟨u N, hs _, ?_⟩\n  rw [dist_comm]\n  exact hN N le_rfl",
            "line_start": 260,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
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          },
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            "state": "exact_hash_projection",
            "sha256": "71582bad12813fb058bb037e1b0d00ec99579360926b1ee5b2338766190c03b4",
            "text": "example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) :\n    a ∈ closure s := by\n  rw [Metric.tendsto_atTop] at hu\n  rw [Metric.mem_closure_iff]\n  intro ε ε_pos\n  rcases hu ε ε_pos with ⟨N, hN⟩\n  refine ⟨u N, hs _, ?_⟩\n  rw [dist_comm]\n  exact hN N le_rfl",
            "line_start": 223,
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        "id": "example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) :\n    a ∈ closure s := by\n  rw [Metric.tendsto_atTop] at hu\n  rw [Metric.mem_closure_iff]\n  intro ε ε_pos\n  rcases hu ε ε_pos with ⟨N, hN⟩\n  refine ⟨u N, hs _, ?_⟩\n  rw [dist_comm]\n  exact hN N le_rfl",
        "en": "example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) :\n    a ∈ closure s := by\n  rw [Metric.tendsto_atTop] at hu\n  rw [Metric.mem_closure_iff]\n  intro ε ε_pos\n  rcases hu ε ε_pos with ⟨N, hN⟩\n  refine ⟨u N, hs _, ?_⟩\n  rw [dist_comm]\n  exact hN N le_rfl"
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            "sha256": "f02ce4fb4f2951055395dffde7e40a4ca58ac4d092d3ab0ada7e67d7e5120702",
            "text": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'",
            "line_start": 404,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "native_conditional_first_marker_projection",
            "source_region": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'"
          },
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            "state": "exact_hash_projection",
            "sha256": "f02ce4fb4f2951055395dffde7e40a4ca58ac4d092d3ab0ada7e67d7e5120702",
            "text": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'",
            "line_start": 351,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "native_conditional_first_marker_projection",
            "source_region": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n/- EXAMPLES:\n  sorry\nSOLUTIONS: -/\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'"
          }
        },
        "id": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'",
        "en": "example {X : Type*} [MetricSpace X] [CompactSpace X]\n      {Y : Type*} [MetricSpace Y] {f : X → Y}\n    (hf : Continuous f) : UniformContinuous f := by\n  rw [Metric.uniformContinuous_iff]\n  intro ε ε_pos\n  let φ : X × X → ℝ := fun p ↦ dist (f p.1) (f p.2)\n  have φ_cont : Continuous φ := hf.fst'.dist hf.snd'\n  let K := { p : X × X | ε ≤ φ p }\n  have K_closed : IsClosed K := isClosed_le continuous_const φ_cont\n  have K_cpct : IsCompact K := K_closed.isCompact\n  rcases eq_empty_or_nonempty K with hK | hK\n  · use 1, by norm_num\n    intro x y _\n    have : (x, y) ∉ K := by simp [hK]\n    simpa [K] using this\n  · rcases K_cpct.exists_isMinOn hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩\n    use dist x₀ x₁\n    constructor\n    · change _ < _\n      rw [dist_pos]\n      intro h\n      have : ε ≤ 0 := by simpa [K, φ, *] using xx_in\n      linarith\n    · intro x x'\n      contrapose!\n      intro (hxx' : (x, x') ∈ K)\n      exact H hxx'"
      },
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            "text": "  obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by\n    have : Tendsto (fun N : ℕ ↦ (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by\n      rw [← zero_mul (2 : ℝ)]\n      apply Tendsto.mul\n      simp_rw [← one_div_pow (2 : ℝ)]\n      apply tendsto_pow_atTop_nhds_zero_of_lt_one <;> linarith\n      exact tendsto_const_nhds\n    rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩\n    exact ⟨N, by simpa using (hN N self_mem_Ici).2⟩",
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            "sha256": "546817779cc42fa242db344502c13d67a1a1542e02c7fa934cdc0a061fad238b",
            "text": "  obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by\n    have : Tendsto (fun N : ℕ ↦ (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by\n      rw [← zero_mul (2 : ℝ)]\n      apply Tendsto.mul\n      simp_rw [← one_div_pow (2 : ℝ)]\n      apply tendsto_pow_atTop_nhds_zero_of_lt_one <;> linarith\n      exact tendsto_const_nhds\n    rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩\n    exact ⟨N, by simpa using (hN N self_mem_Ici).2⟩",
            "line_start": 445,
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        "id": "  obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by\n    have : Tendsto (fun N : ℕ ↦ (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by\n      rw [← zero_mul (2 : ℝ)]\n      apply Tendsto.mul\n      simp_rw [← one_div_pow (2 : ℝ)]\n      apply tendsto_pow_atTop_nhds_zero_of_lt_one <;> linarith\n      exact tendsto_const_nhds\n    rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩\n    exact ⟨N, by simpa using (hN N self_mem_Ici).2⟩",
        "en": "  obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by\n    have : Tendsto (fun N : ℕ ↦ (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by\n      rw [← zero_mul (2 : ℝ)]\n      apply Tendsto.mul\n      simp_rw [← one_div_pow (2 : ℝ)]\n      apply tendsto_pow_atTop_nhds_zero_of_lt_one <;> linarith\n      exact tendsto_const_nhds\n    rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩\n    exact ⟨N, by simpa using (hN N self_mem_Ici).2⟩"
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            "text": "    dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero]",
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      "title": {
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        "en": "solution-011"
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      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
          "anchor": "ruang-metrik",
          "title": "Ruang metrik",
          "state": "enclosing_section",
          "page_sha256": "a743dc70f89f5023bd33ef496f1975bae518ac147656fe23b7bbe748a2c2ef75",
          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
        },
        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
          "page_sha256": "872f380ff4a2877b4ee9a9e5f63fdd5a1c98e5dc93bb667d732485d86e650e4e",
          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
        }
      },
      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-529",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
        "mil.repo.code"
      ],
      "concept_ids": [
        "urn:mil:concept:function-composition",
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      "code": {
        "sha256": {
          "id": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2",
          "en": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2",
            "text": "  have Bpos : ∀ n, 0 < B n := fun n ↦ pow_pos (by linarith) n",
            "line_start": 587,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2",
            "text": "  have Bpos : ∀ n, 0 < B n := fun n ↦ pow_pos (by linarith) n",
            "line_start": 529,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have Bpos : ∀ n, 0 < B n := fun n ↦ pow_pos (by linarith) n",
        "en": "  have Bpos : ∀ n, 0 < B n := fun n ↦ pow_pos (by linarith) n"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c11:s02"
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        "concept_ids": [
          "urn:mil:concept:function-composition",
          "urn:mil:concept:metric-space"
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        "data": {
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          "generated_learner_path": "MIL/C11_Topology/solutions/Solutions_S02_Metric_Spaces.lean",
          "hole_rank": 1,
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          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 529,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have Bpos : ∀ n, 0 < B n",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2",
          "unit_type": "solution",
          "verified_solution": true
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        "entity_type": "unit",
        "language": "zxx",
        "locale": "und",
        "order_index": 911,
        "parent_id": "urn:mil:unit:c11:s02",
        "prerequisite_ids": [],
        "provenance": [],
        "qa_event_ids": [
          "urn:mil:qa:code:c11:s02",
          "urn:mil:qa:language:c11:s02",
          "urn:mil:qa:topology:c11:s02"
        ],
        "record_id": "urn:mil:unit:c11:s02:solution:011",
        "resource_id": "urn:mil:resource:avigad:mathematics-in-lean-source",
        "rights_id": "mil.repo.code",
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        "schema_version": "1.0.0",
        "source_local_id": "solution-011",
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        "source_path": "MIL/C11_Topology/S02_Metric_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "f7f45c1c9ed3ec57436fcb8ef7a6b0c96540a6b6ee73153f089ab7f14b7c3ca2",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
    },
    {
      "id": "urn:mil:unit:c11:s02:solution:012",
      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c11",
      "section_id": "urn:mil:unit:c11:s02",
      "title": {
        "id": "solution-012",
        "en": "solution-012"
      },
      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
          "anchor": "ruang-metrik",
          "title": "Ruang metrik",
          "state": "enclosing_section",
          "page_sha256": "a743dc70f89f5023bd33ef496f1975bae518ac147656fe23b7bbe748a2c2ef75",
          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
        },
        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
          "page_sha256": "872f380ff4a2877b4ee9a9e5f63fdd5a1c98e5dc93bb667d732485d86e650e4e",
          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
        }
      },
      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-537",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
        "mil.repo.code"
      ],
      "concept_ids": [
        "urn:mil:concept:function-composition",
        "urn:mil:concept:metric-space"
      ],
      "code": {
        "sha256": {
          "id": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1",
          "en": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1",
            "text": "    intro n x δ δpos\n    have : x ∈ closure (f n) := hd n x\n    rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩\n    rw [dist_comm] at xy\n    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=\n      nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)\n    refine ⟨y, min (min (δ / 2) r) (B (n + 1)), ?_, ?_, fun z hz ↦ ⟨?_, ?_⟩⟩\n    show 0 < min (min (δ / 2) r) (B (n + 1))\n    exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1))\n    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)\n    exact min_le_right _ _\n    show z ∈ closedBall x δ\n    exact\n      calc\n        dist z x ≤ dist z y + dist y x := dist_triangle _ _ _\n        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le)\n        _ ≤ δ / 2 + δ / 2 := (add_le_add_left ((min_le_left _ _).trans (min_le_left _ _)) _)\n        _ = δ := add_halves δ\n\n    show z ∈ f n\n    exact\n      hr\n        (calc\n          dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz\n          _ ≤ r := (min_le_left _ _).trans (min_le_right _ _)\n          )",
            "line_start": 595,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1",
            "text": "    intro n x δ δpos\n    have : x ∈ closure (f n) := hd n x\n    rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩\n    rw [dist_comm] at xy\n    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=\n      nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)\n    refine ⟨y, min (min (δ / 2) r) (B (n + 1)), ?_, ?_, fun z hz ↦ ⟨?_, ?_⟩⟩\n    show 0 < min (min (δ / 2) r) (B (n + 1))\n    exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1))\n    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)\n    exact min_le_right _ _\n    show z ∈ closedBall x δ\n    exact\n      calc\n        dist z x ≤ dist z y + dist y x := dist_triangle _ _ _\n        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le)\n        _ ≤ δ / 2 + δ / 2 := (add_le_add_left ((min_le_left _ _).trans (min_le_left _ _)) _)\n        _ = δ := add_halves δ\n\n    show z ∈ f n\n    exact\n      hr\n        (calc\n          dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz\n          _ ≤ r := (min_le_left _ _).trans (min_le_right _ _)\n          )",
            "line_start": 537,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "    intro n x δ δpos\n    have : x ∈ closure (f n) := hd n x\n    rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩\n    rw [dist_comm] at xy\n    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=\n      nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)\n    refine ⟨y, min (min (δ / 2) r) (B (n + 1)), ?_, ?_, fun z hz ↦ ⟨?_, ?_⟩⟩\n    show 0 < min (min (δ / 2) r) (B (n + 1))\n    exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1))\n    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)\n    exact min_le_right _ _\n    show z ∈ closedBall x δ\n    exact\n      calc\n        dist z x ≤ dist z y + dist y x := dist_triangle _ _ _\n        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le)\n        _ ≤ δ / 2 + δ / 2 := (add_le_add_left ((min_le_left _ _).trans (min_le_left _ _)) _)\n        _ = δ := add_halves δ\n\n    show z ∈ f n\n    exact\n      hr\n        (calc\n          dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz\n          _ ≤ r := (min_le_left _ _).trans (min_le_right _ _)\n          )",
        "en": "    intro n x δ δpos\n    have : x ∈ closure (f n) := hd n x\n    rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩\n    rw [dist_comm] at xy\n    obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=\n      nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)\n    refine ⟨y, min (min (δ / 2) r) (B (n + 1)), ?_, ?_, fun z hz ↦ ⟨?_, ?_⟩⟩\n    show 0 < min (min (δ / 2) r) (B (n + 1))\n    exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1))\n    show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)\n    exact min_le_right _ _\n    show z ∈ closedBall x δ\n    exact\n      calc\n        dist z x ≤ dist z y + dist y x := dist_triangle _ _ _\n        _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le)\n        _ ≤ δ / 2 + δ / 2 := (add_le_add_left ((min_le_left _ _).trans (min_le_left _ _)) _)\n        _ = δ := add_halves δ\n\n    show z ∈ f n\n    exact\n      hr\n        (calc\n          dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz\n          _ ≤ r := (min_le_left _ _).trans (min_le_right _ _)\n          )"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c11:s02"
        ],
        "concept_ids": [
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          "urn:mil:concept:metric-space"
        ],
        "data": {
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          "generated_learner_path": "MIL/C11_Topology/solutions/Solutions_S02_Metric_Spaces.lean",
          "hole_rank": 2,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 537,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "intro n x δ δpos have : x ∈ closure (f n)",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1",
          "unit_type": "solution",
          "verified_solution": true
        },
        "edition_id": "urn:mil:edition:source:8e112cd63ff1bf2a1020ff88f22f77288e42b9a9",
        "entity_type": "unit",
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        "qa_event_ids": [
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          "urn:mil:qa:language:c11:s02",
          "urn:mil:qa:topology:c11:s02"
        ],
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        "source_path": "MIL/C11_Topology/S02_Metric_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "be4595ee55c30ffe989c0eba5745f5d4483ee446be4d49d8240c71070e0ec2d1",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
    },
    {
      "id": "urn:mil:unit:c11:s02:solution:013",
      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c11",
      "section_id": "urn:mil:unit:c11:s02",
      "title": {
        "id": "solution-013",
        "en": "solution-013"
      },
      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
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          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
        },
        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
          "page_sha256": "872f380ff4a2877b4ee9a9e5f63fdd5a1c98e5dc93bb667d732485d86e650e4e",
          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
        }
      },
      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-574",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
        "mil.repo.code"
      ],
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        "urn:mil:concept:function-composition",
        "urn:mil:concept:metric-space"
      ],
      "code": {
        "sha256": {
          "id": "fd0dfa5ff476e1b5042beb3bc82e9a9dd6a78a58626f4239db78ff44072e0d29",
          "en": "fd0dfa5ff476e1b5042beb3bc82e9a9dd6a78a58626f4239db78ff44072e0d29"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
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            "sha256": "fd0dfa5ff476e1b5042beb3bc82e9a9dd6a78a58626f4239db78ff44072e0d29",
            "text": "  have rpos : ∀ n, 0 < r n := by\n    intro n\n    induction' n with n hn\n    exact lt_min εpos (Bpos 0)\n    exact Hpos n (c n) (r n) hn",
            "line_start": 633,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "fd0dfa5ff476e1b5042beb3bc82e9a9dd6a78a58626f4239db78ff44072e0d29",
            "text": "  have rpos : ∀ n, 0 < r n := by\n    intro n\n    induction' n with n hn\n    exact lt_min εpos (Bpos 0)\n    exact Hpos n (c n) (r n) hn",
            "line_start": 574,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have rpos : ∀ n, 0 < r n := by\n    intro n\n    induction' n with n hn\n    exact lt_min εpos (Bpos 0)\n    exact Hpos n (c n) (r n) hn",
        "en": "  have rpos : ∀ n, 0 < r n := by\n    intro n\n    induction' n with n hn\n    exact lt_min εpos (Bpos 0)\n    exact Hpos n (c n) (r n) hn"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
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          "urn:mil:concept:metric-space"
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        "data": {
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          "generated_learner_path": "MIL/C11_Topology/solutions/Solutions_S02_Metric_Spaces.lean",
          "hole_rank": 3,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 574,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "fd0dfa5ff476e1b5042beb3bc82e9a9dd6a78a58626f4239db78ff44072e0d29",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have rpos : ∀ n, 0 < r n",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "fd0dfa5ff476e1b5042beb3bc82e9a9dd6a78a58626f4239db78ff44072e0d29",
          "unit_type": "solution",
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      "section_id": "urn:mil:unit:c11:s02",
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        "en": "solution-014"
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          "anchor": "ruang-metrik",
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      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
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          "en": "264ebc037f18d115133b58950827c6b8a8ddb8235b5f133a2920b59ae54fdafa"
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            "state": "exact_hash_span",
            "sha256": "264ebc037f18d115133b58950827c6b8a8ddb8235b5f133a2920b59ae54fdafa",
            "text": "  have rB : ∀ n, r n ≤ B n := by\n    intro n\n    induction' n with n hn\n    exact min_le_right _ _\n    exact HB n (c n) (r n) (rpos n)",
            "line_start": 638,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
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            "state": "exact_hash_span",
            "sha256": "264ebc037f18d115133b58950827c6b8a8ddb8235b5f133a2920b59ae54fdafa",
            "text": "  have rB : ∀ n, r n ≤ B n := by\n    intro n\n    induction' n with n hn\n    exact min_le_right _ _\n    exact HB n (c n) (r n) (rpos n)",
            "line_start": 579,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have rB : ∀ n, r n ≤ B n := by\n    intro n\n    induction' n with n hn\n    exact min_le_right _ _\n    exact HB n (c n) (r n) (rpos n)",
        "en": "  have rB : ∀ n, r n ≤ B n := by\n    intro n\n    induction' n with n hn\n    exact min_le_right _ _\n    exact HB n (c n) (r n) (rpos n)"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
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          "urn:mil:concept:metric-space"
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        "data": {
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          "source_text_sha256": "264ebc037f18d115133b58950827c6b8a8ddb8235b5f133a2920b59ae54fdafa",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have rB : ∀ n, r n ≤ B n",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "264ebc037f18d115133b58950827c6b8a8ddb8235b5f133a2920b59ae54fdafa",
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        "source_sha256": "264ebc037f18d115133b58950827c6b8a8ddb8235b5f133a2920b59ae54fdafa",
        "status": "active",
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        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
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      }
    },
    {
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      "section_id": "urn:mil:unit:c11:s02",
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        "en": "solution-015"
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          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
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          "title": "Ruang metrik",
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      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
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            "sha256": "040d5173c8cca86e6df4f3ea6e6e23d32e30586fd6275638f749d8c84f8e859b",
            "text": "  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n ↦\n    Hball n (c n) (r n) (rpos n)",
            "line_start": 643,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
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          "en": {
            "state": "exact_hash_span",
            "sha256": "040d5173c8cca86e6df4f3ea6e6e23d32e30586fd6275638f749d8c84f8e859b",
            "text": "  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n ↦\n    Hball n (c n) (r n) (rpos n)",
            "line_start": 584,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n ↦\n    Hball n (c n) (r n) (rpos n)",
        "en": "  have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n ↦\n    Hball n (c n) (r n) (rpos n)"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
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        "concept_ids": [
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          "urn:mil:concept:metric-space"
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        "data": {
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          "exercise_id": "urn:mil:unit:c11:s02:exercise:005",
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          "hole_rank": 5,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 584,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "040d5173c8cca86e6df4f3ea6e6e23d32e30586fd6275638f749d8c84f8e859b",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "040d5173c8cca86e6df4f3ea6e6e23d32e30586fd6275638f749d8c84f8e859b",
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        "source_path": "MIL/C11_Topology/S02_Metric_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "040d5173c8cca86e6df4f3ea6e6e23d32e30586fd6275638f749d8c84f8e859b",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
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    },
    {
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      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c11",
      "section_id": "urn:mil:unit:c11:s02",
      "title": {
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        "en": "solution-016"
      },
      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
          "anchor": "ruang-metrik",
          "title": "Ruang metrik",
          "state": "enclosing_section",
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          "code_equivalence_claimed": false
        },
        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
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          "code_equivalence_claimed": false
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      },
      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-586",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
        "mil.repo.code"
      ],
      "concept_ids": [
        "urn:mil:concept:function-composition",
        "urn:mil:concept:metric-space"
      ],
      "code": {
        "sha256": {
          "id": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96",
          "en": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96",
            "text": "  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by\n    intro n\n    rw [dist_comm]\n    have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) :=\n      mem_closedBall_self (rpos <| n + 1).le\n    have I :=\n      calc\n        closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=\n          (incl n).trans Set.inter_subset_left\n        _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)\n\n    exact I A",
            "line_start": 645,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96",
            "text": "  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by\n    intro n\n    rw [dist_comm]\n    have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) :=\n      mem_closedBall_self (rpos <| n + 1).le\n    have I :=\n      calc\n        closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=\n          (incl n).trans Set.inter_subset_left\n        _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)\n\n    exact I A",
            "line_start": 586,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by\n    intro n\n    rw [dist_comm]\n    have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) :=\n      mem_closedBall_self (rpos <| n + 1).le\n    have I :=\n      calc\n        closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=\n          (incl n).trans Set.inter_subset_left\n        _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)\n\n    exact I A",
        "en": "  have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by\n    intro n\n    rw [dist_comm]\n    have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) :=\n      mem_closedBall_self (rpos <| n + 1).le\n    have I :=\n      calc\n        closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=\n          (incl n).trans Set.inter_subset_left\n        _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)\n\n    exact I A"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c11:s02"
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        ],
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          "exercise_id": "urn:mil:unit:c11:s02:exercise:005",
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          "hole_rank": 6,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 586,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96",
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          "verified_solution": true
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        "source_path": "MIL/C11_Topology/S02_Metric_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "90dbe0d12fa3854f1737d14603059bc03b5bb2e33ec0cbaecfa7c432c15ddb96",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
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    {
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      "kind": "solution",
      "practice": false,
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          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
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        },
        "en": {
          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
          "page_sha256": "872f380ff4a2877b4ee9a9e5f63fdd5a1c98e5dc93bb667d732485d86e650e4e",
          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
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      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-604",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
        "mil.repo.code"
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        "urn:mil:concept:function-composition",
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          "id": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5",
          "en": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5",
            "text": "  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by\n    intro n\n    refine Nat.le_induction ?_ fun m hnm h ↦ ?_\n    · exact Subset.rfl\n    · exact (incl m).trans (Set.inter_subset_left.trans h)",
            "line_start": 663,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5",
            "text": "  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by\n    intro n\n    refine Nat.le_induction ?_ fun m hnm h ↦ ?_\n    · exact Subset.rfl\n    · exact (incl m).trans (Set.inter_subset_left.trans h)",
            "line_start": 604,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by\n    intro n\n    refine Nat.le_induction ?_ fun m hnm h ↦ ?_\n    · exact Subset.rfl\n    · exact (incl m).trans (Set.inter_subset_left.trans h)",
        "en": "  have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by\n    intro n\n    refine Nat.le_induction ?_ fun m hnm h ↦ ?_\n    · exact Subset.rfl\n    · exact (incl m).trans (Set.inter_subset_left.trans h)"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c11:s02"
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        "concept_ids": [
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          "urn:mil:concept:metric-space"
        ],
        "data": {
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          "exercise_id": "urn:mil:unit:c11:s02:exercise:005",
          "generated_learner_path": "MIL/C11_Topology/solutions/Solutions_S02_Metric_Spaces.lean",
          "hole_rank": 7,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 604,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n)",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5",
          "unit_type": "solution",
          "verified_solution": true
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        "source_path": "MIL/C11_Topology/S02_Metric_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "ef75428e7cc6a28322fab8f040ceba7af37169991a56b5a03cc7482d14190bb5",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
    },
    {
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      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c11",
      "section_id": "urn:mil:unit:c11:s02",
      "title": {
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        "en": "solution-018"
      },
      "reader_routes": {
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          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
          "anchor": "ruang-metrik",
          "title": "Ruang metrik",
          "state": "enclosing_section",
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          "bound_unit_id": "urn:mil:unit:c11:s02",
          "code_equivalence_claimed": false
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          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
          "page_sha256": "872f380ff4a2877b4ee9a9e5f63fdd5a1c98e5dc93bb667d732485d86e650e4e",
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          "code_equivalence_claimed": false
        }
      },
      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-609",
      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
      "rights_ids": [
        "mil.repo.code"
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      "code": {
        "sha256": {
          "id": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97",
          "en": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97",
            "text": "  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by\n    intro n\n    refine isClosed_closedBall.mem_of_tendsto ylim ?_\n    refine (Filter.eventually_ge_atTop n).mono fun m hm ↦ ?_\n    exact I n m hm (mem_closedBall_self (rpos _).le)",
            "line_start": 668,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
          },
          "en": {
            "state": "exact_hash_span",
            "sha256": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97",
            "text": "  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by\n    intro n\n    refine isClosed_closedBall.mem_of_tendsto ylim ?_\n    refine (Filter.eventually_ge_atTop n).mono fun m hm ↦ ?_\n    exact I n m hm (mem_closedBall_self (rpos _).le)",
            "line_start": 609,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by\n    intro n\n    refine isClosed_closedBall.mem_of_tendsto ylim ?_\n    refine (Filter.eventually_ge_atTop n).mono fun m hm ↦ ?_\n    exact I n m hm (mem_closedBall_self (rpos _).le)",
        "en": "  have yball : ∀ n, y ∈ closedBall (c n) (r n) := by\n    intro n\n    refine isClosed_closedBall.mem_of_tendsto ylim ?_\n    refine (Filter.eventually_ge_atTop n).mono fun m hm ↦ ?_\n    exact I n m hm (mem_closedBall_self (rpos _).le)"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
        "build_event_ids": [
          "urn:mil:qa:build:c11:s02"
        ],
        "concept_ids": [
          "urn:mil:concept:function-composition",
          "urn:mil:concept:metric-space"
        ],
        "data": {
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          "exercise_id": "urn:mil:unit:c11:s02:exercise:005",
          "generated_learner_path": "MIL/C11_Topology/solutions/Solutions_S02_Metric_Spaces.lean",
          "hole_rank": 8,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 609,
          "solution_status": "verified_solution_present",
          "source_text_sha256": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "have yball : ∀ n, y ∈ closedBall (c n) (r n)",
          "target_is_exercise": true,
          "target_source_line": 481,
          "target_text_sha256": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97",
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          "verified_solution": true
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        "parent_id": "urn:mil:unit:c11:s02",
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        "qa_event_ids": [
          "urn:mil:qa:code:c11:s02",
          "urn:mil:qa:language:c11:s02",
          "urn:mil:qa:topology:c11:s02"
        ],
        "record_id": "urn:mil:unit:c11:s02:solution:018",
        "resource_id": "urn:mil:resource:avigad:mathematics-in-lean-source",
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        "schema": "mil-backend-record",
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        "source_local_id": "solution-018",
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        "source_path": "MIL/C11_Topology/S02_Metric_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "125d07f5e6a6402bdb42f8c3ab33d8fa54f099d2b2ca011c6b36bf9dfefb3c97",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
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    },
    {
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      "kind": "solution",
      "practice": false,
      "chapter_id": "urn:mil:unit:c11",
      "section_id": "urn:mil:unit:c11:s02",
      "title": {
        "id": "solution-019",
        "en": "solution-019"
      },
      "reader_routes": {
        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C11_Topology.html#ruang-metrik",
          "anchor": "ruang-metrik",
          "title": "Ruang metrik",
          "state": "enclosing_section",
          "page_sha256": "a743dc70f89f5023bd33ef496f1975bae518ac147656fe23b7bbe748a2c2ef75",
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          "url": "https://kokunoyumeto.github.io/program-matematika-indonesia/en/courses/D110/C11_Topology.html#metric-spaces",
          "anchor": "metric-spaces",
          "title": "Metric spaces",
          "state": "enclosing_section",
          "page_sha256": "872f380ff4a2877b4ee9a9e5f63fdd5a1c98e5dc93bb667d732485d86e650e4e",
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      "source_locator": "MIL/C11_Topology/S02_Metric_Spaces.lean#line-614",
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      ],
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          "id": "2f2e774a6e7310b35d46bead7e0cc8dd29d07165e29206a5923f46fa07e7d16d",
          "en": "2f2e774a6e7310b35d46bead7e0cc8dd29d07165e29206a5923f46fa07e7d16d"
        },
        "scope": "source_solution_or_support_fragment",
        "witness": {
          "id": {
            "state": "exact_hash_span",
            "sha256": "2f2e774a6e7310b35d46bead7e0cc8dd29d07165e29206a5923f46fa07e7d16d",
            "text": "  constructor\n  · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter]\n    intro n\n    have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n :=\n      Subset.trans (incl n) Set.inter_subset_right\n    exact this (yball (n + 1))\n  calc\n    dist y x ≤ r 0 := yball 0\n    _ ≤ ε := min_le_left _ _",
            "line_start": 673,
            "file_sha256": "bce3b259923178888601d755159b743eb42abb61678f57b481713aaa5bcb8ff1",
            "method": "exact_contiguous_span"
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          "en": {
            "state": "exact_hash_span",
            "sha256": "2f2e774a6e7310b35d46bead7e0cc8dd29d07165e29206a5923f46fa07e7d16d",
            "text": "  constructor\n  · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter]\n    intro n\n    have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n :=\n      Subset.trans (incl n) Set.inter_subset_right\n    exact this (yball (n + 1))\n  calc\n    dist y x ≤ r 0 := yball 0\n    _ ≤ ε := min_le_left _ _",
            "line_start": 614,
            "file_sha256": "958c9a7c9c79865e3ca18a17f67be9447fad657d2fd23c7d2403c942788679ef",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  constructor\n  · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter]\n    intro n\n    have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n :=\n      Subset.trans (incl n) Set.inter_subset_right\n    exact this (yball (n + 1))\n  calc\n    dist y x ≤ r 0 := yball 0\n    _ ≤ ε := min_le_left _ _",
        "en": "  constructor\n  · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter]\n    intro n\n    have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n :=\n      Subset.trans (incl n) Set.inter_subset_right\n    exact this (yball (n + 1))\n  calc\n    dist y x ≤ r 0 := yball 0\n    _ ≤ ε := min_le_left _ _"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
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          "urn:mil:concept:metric-space"
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          "hole_rank": 9,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
          "solution_source_line": 614,
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          "source_text_sha256": "2f2e774a6e7310b35d46bead7e0cc8dd29d07165e29206a5923f46fa07e7d16d",
          "source_unit_id": "urn:mil:unit:c11:s02:exercise:005",
          "statement_signature": "constructor · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] intro n have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n",
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          "target_source_line": 481,
          "target_text_sha256": "2f2e774a6e7310b35d46bead7e0cc8dd29d07165e29206a5923f46fa07e7d16d",
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        ],
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        "translation_state": "structurally_verified",
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      }
    },
    {
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      "kind": "section",
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      "title": {
        "id": "Ruang topologi",
        "en": "Topological spaces"
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        "urn:mil:concept:set",
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        "id": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n/- EXAMPLES:\n  sorry\n\n#check HasBasis.tendsto_right_iff\n\nSOLUTIONS: -/\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
        "en": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n/- EXAMPLES:\n  sorry\n\n#check HasBasis.tendsto_right_iff\n\nSOLUTIONS: -/\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
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        "id": "-- EXAMPLES:\nexample [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) :\n    IsCompact (f '' s) := by\n  intro F F_ne F_le\n  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry\n  have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry\n  have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left\n  sorry",
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        "id": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n/- EXAMPLES:\n  sorry\n\n#check HasBasis.tendsto_right_iff\n\nSOLUTIONS: -/\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
        "en": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n/- EXAMPLES:\n  sorry\n\n#check HasBasis.tendsto_right_iff\n\nSOLUTIONS: -/\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
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        "id": "-- EXAMPLES:\nexample [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) :\n    IsCompact (f '' s) := by\n  intro F F_ne F_le\n  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry\n  have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry\n  have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left\n  sorry",
        "en": "-- EXAMPLES:\nexample [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) :\n    IsCompact (f '' s) := by\n  intro F F_ne F_le\n  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry\n  have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry\n  have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left\n  sorry",
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            "text": "theorem aux {X Y A : Type*} [TopologicalSpace X] {c : A → X}\n      {f : A → Y} {x : X} {F : Filter Y}\n      (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) :\n    ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by\n  simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in",
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        "en": "theorem aux {X Y A : Type*} [TopologicalSpace X] {c : A → X}\n      {f : A → Y} {x : X} {F : Filter Y}\n      (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) :\n    ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by\n  simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in"
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          "title": "Ruang topologi",
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          "anchor": "topological-spaces",
          "title": "Topological spaces",
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      "code": {
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          "en": "00490d76edf42d1dd3710af38e89bfc24898d06e931fdc8d6f0cf6011dc4b2bf"
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        "witness": {
          "id": {
            "state": "exact_hash_projection",
            "sha256": "00490d76edf42d1dd3710af38e89bfc24898d06e931fdc8d6f0cf6011dc4b2bf",
            "text": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
            "line_start": 442,
            "file_sha256": "febb5b074f41594c550aefb7aab7a7cef1b2b208900d75084c23344d8fc9e728",
            "method": "native_conditional_first_marker_projection",
            "source_region": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n/- EXAMPLES:\n  sorry\n\n#check HasBasis.tendsto_right_iff\n\nSOLUTIONS: -/\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt"
          },
          "en": {
            "state": "exact_hash_projection",
            "sha256": "00490d76edf42d1dd3710af38e89bfc24898d06e931fdc8d6f0cf6011dc4b2bf",
            "text": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
            "line_start": 385,
            "file_sha256": "59796ae1a75844151234dc8ef44c86f9a376781ee4fb222bf1224bae87a973e6",
            "method": "native_conditional_first_marker_projection",
            "source_region": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n/- EXAMPLES:\n  sorry\n\n#check HasBasis.tendsto_right_iff\n\nSOLUTIONS: -/\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt"
          }
        },
        "id": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt",
        "en": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X}\n    (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)\n    (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) :\n    ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by\n  choose φ hφ using hf\n  use φ\n  constructor\n  · rw [continuous_iff_continuousAt]\n    intro x\n    suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by\n      simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff]\n    intro V' V'_in V'_closed\n    obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in\n    suffices : ∀ y ∈ V, φ y ∈ V'\n    exact mem_of_superset V_in this\n    intro y y_in\n    have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in\n    haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y\n    apply V'_closed.mem_of_tendsto (hφ y)\n    exact mem_of_superset (preimage_mem_comap hVx) hV\n  · intro a\n    have lim : Tendsto f (𝓝 a) (𝓝 (φ a)) := by simpa [nhds_induced] using hφ a\n    exact tendsto_nhds_unique lim f_cont.continuousAt"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
      "source_record": {
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        "concept_ids": [],
        "data": {
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          "hole_rank": 1,
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          "source_unit_id": "urn:mil:unit:c11:s03:exercise:003",
          "statement_signature": "example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) (𝓝 x)) (𝓝 c)) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a",
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            "state": "exact_hash_span",
            "sha256": "c9213256380ce7b47b7ec0ec93920fa7f98c90187914cb89878c1d7659efc487",
            "text": "  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal]",
            "line_start": 567,
            "file_sha256": "febb5b074f41594c550aefb7aab7a7cef1b2b208900d75084c23344d8fc9e728",
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            "state": "exact_hash_span",
            "sha256": "c9213256380ce7b47b7ec0ec93920fa7f98c90187914cb89878c1d7659efc487",
            "text": "  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal]",
            "line_start": 502,
            "file_sha256": "59796ae1a75844151234dc8ef44c86f9a376781ee4fb222bf1224bae87a973e6",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal]",
        "en": "  have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal]"
      },
      "supports": [],
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        "data": {
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          "source_unit_id": "urn:mil:unit:c11:s03:exercise:004",
          "statement_signature": "have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F",
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        "id": "variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]\n  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]\n\nopen Metric\n\n-- EXAMPLES:\nexample {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) :\n    ∃ C', ∀ i, ‖g i‖ ≤ C' := by\n  -- barisan himpunan bagian berisi `x : E` dengan norma `‖g i x‖` yang dibatasi oleh `n`\n  let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n }\n  -- setiap himpunan ini tertutup\n  have hc : ∀ n : ℕ, IsClosed (e n)\n  sorry\n  -- gabungannya adalah seluruh ruang; di sinilah kita menggunakan `h`\n  have hU : (⋃ n : ℕ, e n) = univ\n  sorry\n  /- terapkan teorema kategori Baire untuk menyimpulkan bahwa, bagi suatu `m : ℕ`,\n       interior `e m` memuat suatu `x` -/\n  obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry\n  obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry\n  obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry\n  -- tunjukkan bahwa norma setiap elemen bola dibatasi oleh `m` setelah diterapkan sembarang `g i`\n  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m\n  sorry\n  have εk_pos : 0 < ε / ‖k‖ := sorry\n  refine ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.opNorm_le_of_shell ε_pos ?_ hk ?_⟩\n  sorry\n  sorry",
        "en": "variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]\n  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]\n\nopen Metric\n\n-- EXAMPLES:\nexample {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) :\n    ∃ C', ∀ i, ‖g i‖ ≤ C' := by\n  -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n`\n  let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n }\n  -- each of these sets is closed\n  have hc : ∀ n : ℕ, IsClosed (e n)\n  sorry\n  -- the union is the entire space; this is where we use `h`\n  have hU : (⋃ n : ℕ, e n) = univ\n  sorry\n  /- apply the Baire category theorem to conclude that for some `m : ℕ`,\n       the interior of `e m` contains some `x` -/\n  obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry\n  obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry\n  obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry\n  -- show all elements in the ball have norm bounded by `m` after applying any `g i`\n  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m\n  sorry\n  have εk_pos : 0 < ε / ‖k‖ := sorry\n  refine ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.opNorm_le_of_shell ε_pos ?_ hk ?_⟩\n  sorry\n  sorry",
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        "en": "variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]\n  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]\n\nopen Metric\n\n-- EXAMPLES:\nexample {ι : Type*} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) :\n    ∃ C', ∀ i, ‖g i‖ ≤ C' := by\n  -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n`\n  let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n }\n  -- each of these sets is closed\n  have hc : ∀ n : ℕ, IsClosed (e n)\n  sorry\n  -- the union is the entire space; this is where we use `h`\n  have hU : (⋃ n : ℕ, e n) = univ\n  sorry\n  /- apply the Baire category theorem to conclude that for some `m : ℕ`,\n       the interior of `e m` contains some `x` -/\n  obtain ⟨m, x, hx⟩ : ∃ m, ∃ x, x ∈ interior (e m) := sorry\n  obtain ⟨ε, ε_pos, hε⟩ : ∃ ε > 0, ball x ε ⊆ interior (e m) := sorry\n  obtain ⟨k, hk⟩ : ∃ k : 𝕜, 1 < ‖k‖ := sorry\n  -- show all elements in the ball have norm bounded by `m` after applying any `g i`\n  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m\n  sorry\n  have εk_pos : 0 < ε / ‖k‖ := sorry\n  refine ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.opNorm_le_of_shell ε_pos ?_ hk ?_⟩\n  sorry\n  sorry",
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        "urn:mil:unit:c12:s02:solution:004",
        "urn:mil:unit:c12:s02:solution:005",
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        "urn:mil:unit:c12:s02:solution:007",
        "urn:mil:unit:c12:s02:solution:008",
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        "en": "solution-001"
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          "anchor": "kalkulus-diferensial-pada-ruang-bernorma",
          "title": "Kalkulus Diferensial pada Ruang Bernorma",
          "state": "enclosing_section",
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          "anchor": "differential-calculus-in-normed-spaces",
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            "sha256": "4267e6c292c39b07f0545033c4dcab86f17687233d48ec9a77bb62d41c33647d",
            "text": "  have hc : ∀ n : ℕ, IsClosed (e n) := fun i ↦\n    isClosed_iInter fun i ↦ isClosed_le (g i).cont.norm continuous_const",
            "line_start": 213,
            "file_sha256": "7ed9d3342a2e48cdd2e1b28cfc7d3c58579b7d60fd09defa9650d86c8878c979",
            "method": "exact_contiguous_span"
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          "en": {
            "state": "exact_hash_span",
            "sha256": "4267e6c292c39b07f0545033c4dcab86f17687233d48ec9a77bb62d41c33647d",
            "text": "  have hc : ∀ n : ℕ, IsClosed (e n) := fun i ↦\n    isClosed_iInter fun i ↦ isClosed_le (g i).cont.norm continuous_const",
            "line_start": 214,
            "file_sha256": "9250f2b53f384e3477baa327fbcc7a8f239c3a02d7e5dd97cb9be2405816af63",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have hc : ∀ n : ℕ, IsClosed (e n) := fun i ↦\n    isClosed_iInter fun i ↦ isClosed_le (g i).cont.norm continuous_const",
        "en": "  have hc : ∀ n : ℕ, IsClosed (e n) := fun i ↦\n    isClosed_iInter fun i ↦ isClosed_le (g i).cont.norm continuous_const"
      },
      "supports": [],
      "supporting": [],
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          "hole_rank": 1,
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          "source_unit_id": "urn:mil:unit:c12:s02:exercise:001",
          "statement_signature": "have hc : ∀ n : ℕ, IsClosed (e n)",
          "target_is_exercise": true,
          "target_source_line": 184,
          "target_text_sha256": "4267e6c292c39b07f0545033c4dcab86f17687233d48ec9a77bb62d41c33647d",
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        "en": "solution-002"
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          "anchor": "kalkulus-diferensial-pada-ruang-bernorma",
          "title": "Kalkulus Diferensial pada Ruang Bernorma",
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        },
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            "sha256": "868517e4e58e5a8beb41e46214a0568d7d2e99c2564415389477c0e9d8b5f11a",
            "text": "  have hU : (⋃ n : ℕ, e n) = univ := by\n    refine eq_univ_of_forall fun x ↦ ?_\n    rcases h x with ⟨C, hC⟩\n    obtain ⟨m, hm⟩ := exists_nat_ge C\n    exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i ↦ le_trans (hC i) hm⟩",
            "line_start": 216,
            "file_sha256": "7ed9d3342a2e48cdd2e1b28cfc7d3c58579b7d60fd09defa9650d86c8878c979",
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            "sha256": "868517e4e58e5a8beb41e46214a0568d7d2e99c2564415389477c0e9d8b5f11a",
            "text": "  have hU : (⋃ n : ℕ, e n) = univ := by\n    refine eq_univ_of_forall fun x ↦ ?_\n    rcases h x with ⟨C, hC⟩\n    obtain ⟨m, hm⟩ := exists_nat_ge C\n    exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i ↦ le_trans (hC i) hm⟩",
            "line_start": 217,
            "file_sha256": "9250f2b53f384e3477baa327fbcc7a8f239c3a02d7e5dd97cb9be2405816af63",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  have hU : (⋃ n : ℕ, e n) = univ := by\n    refine eq_univ_of_forall fun x ↦ ?_\n    rcases h x with ⟨C, hC⟩\n    obtain ⟨m, hm⟩ := exists_nat_ge C\n    exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i ↦ le_trans (hC i) hm⟩",
        "en": "  have hU : (⋃ n : ℕ, e n) = univ := by\n    refine eq_univ_of_forall fun x ↦ ?_\n    rcases h x with ⟨C, hC⟩\n    obtain ⟨m, hm⟩ := exists_nat_ge C\n    exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i ↦ le_trans (hC i) hm⟩"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
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        ],
        "concept_ids": [
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          "urn:mil:concept:topological-space"
        ],
        "data": {
          "alternative_rank": 1,
          "exercise_count": 1,
          "exercise_id": "urn:mil:unit:c12:s02:exercise:001",
          "generated_learner_path": "MIL/C12_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean",
          "hole_rank": 2,
          "is_alternative": false,
          "matching_method": "source_region_and_line_plan",
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          "source_text_sha256": "868517e4e58e5a8beb41e46214a0568d7d2e99c2564415389477c0e9d8b5f11a",
          "source_unit_id": "urn:mil:unit:c12:s02:exercise:001",
          "statement_signature": "have hU : (⋃ n : ℕ, e n) = univ",
          "target_is_exercise": true,
          "target_source_line": 184,
          "target_text_sha256": "868517e4e58e5a8beb41e46214a0568d7d2e99c2564415389477c0e9d8b5f11a",
          "unit_type": "solution",
          "verified_solution": true
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        "qa_event_ids": [
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          "urn:mil:qa:language:c12:s02",
          "urn:mil:qa:topology:c12:s02"
        ],
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        "source_path": "MIL/C12_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean",
        "source_record_id": "",
        "source_sha256": "868517e4e58e5a8beb41e46214a0568d7d2e99c2564415389477c0e9d8b5f11a",
        "status": "active",
        "supersedes_id": "",
        "timestamp": "2026-08-20T21:30:02+02:00",
        "translation_state": "structurally_verified",
        "workflow_id": "mil-id-corpus-production-v1"
      }
    },
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      "practice": false,
      "chapter_id": "urn:mil:unit:c12",
      "section_id": "urn:mil:unit:c12:s02",
      "title": {
        "id": "solution-003",
        "en": "solution-003"
      },
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        "id": {
          "url": "https://kokunoyumeto.github.io/mathematics-in-lean-id/C12_Differential_Calculus.html#kalkulus-diferensial-pada-ruang-bernorma",
          "anchor": "kalkulus-diferensial-pada-ruang-bernorma",
          "title": "Kalkulus Diferensial pada Ruang Bernorma",
          "state": "enclosing_section",
          "page_sha256": "116fb94c26ef936c0e5d62d8c02d0bf9ccad6599a50be18cdab0e7937984a703",
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          "anchor": "differential-calculus-in-normed-spaces",
          "title": "Differential Calculus in Normed Spaces",
          "state": "enclosing_section",
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      "source_locator_scope": "original_authoring_witness_not_current_target_lines",
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        },
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            "state": "exact_hash_span",
            "sha256": "b4562587e511670f4526892786e11ddd9d09e86422018293b8f09f17d5c1292a",
            "text": "  obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU",
            "line_start": 223,
            "file_sha256": "7ed9d3342a2e48cdd2e1b28cfc7d3c58579b7d60fd09defa9650d86c8878c979",
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            "state": "exact_hash_span",
            "sha256": "b4562587e511670f4526892786e11ddd9d09e86422018293b8f09f17d5c1292a",
            "text": "  obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU",
            "line_start": 224,
            "file_sha256": "9250f2b53f384e3477baa327fbcc7a8f239c3a02d7e5dd97cb9be2405816af63",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU",
        "en": "  obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU"
      },
      "supports": [],
      "supporting": [],
      "companion": false,
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        "concept_ids": [
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          "title": "Kalkulus Diferensial pada Ruang Bernorma",
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            "sha256": "dda78eab48a5ad5ca6ef829c6b7f2b7da1d73980876ab2f1059cbe5a760e3cd4",
            "text": "  obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx",
            "line_start": 224,
            "file_sha256": "7ed9d3342a2e48cdd2e1b28cfc7d3c58579b7d60fd09defa9650d86c8878c979",
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            "sha256": "dda78eab48a5ad5ca6ef829c6b7f2b7da1d73980876ab2f1059cbe5a760e3cd4",
            "text": "  obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx",
            "line_start": 225,
            "file_sha256": "9250f2b53f384e3477baa327fbcc7a8f239c3a02d7e5dd97cb9be2405816af63",
            "method": "exact_contiguous_span"
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        },
        "id": "  obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx",
        "en": "  obtain ⟨ε, ε_pos, hε : ball x ε ⊆ interior (e m)⟩ := isOpen_iff.mp isOpen_interior x hx"
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          "title": "Kalkulus Diferensial pada Ruang Bernorma",
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            "sha256": "4423b2ba3465b7eb02034413214b0c75c363685543d758e4f973c856d23569a5",
            "text": "  obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜",
            "line_start": 225,
            "file_sha256": "7ed9d3342a2e48cdd2e1b28cfc7d3c58579b7d60fd09defa9650d86c8878c979",
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            "state": "exact_hash_span",
            "sha256": "4423b2ba3465b7eb02034413214b0c75c363685543d758e4f973c856d23569a5",
            "text": "  obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜",
            "line_start": 226,
            "file_sha256": "9250f2b53f384e3477baa327fbcc7a8f239c3a02d7e5dd97cb9be2405816af63",
            "method": "exact_contiguous_span"
          }
        },
        "id": "  obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜",
        "en": "  obtain ⟨k : 𝕜, hk : 1 < ‖k‖⟩ := NormedField.exists_one_lt_norm 𝕜"
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      "supporting": [],
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          "anchor": "kalkulus-diferensial-pada-ruang-bernorma",
          "title": "Kalkulus Diferensial pada Ruang Bernorma",
          "state": "enclosing_section",
          "page_sha256": "116fb94c26ef936c0e5d62d8c02d0bf9ccad6599a50be18cdab0e7937984a703",
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      "source_locator": "MIL/C12_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean#line-228",
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            "state": "exact_hash_span",
            "sha256": "80811f468edf8b272a3369696ee222a7a3e45324564cbc1e40d16c9a64020cfb",
            "text": "  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by\n    intro z hz i\n    replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i\n    apply interior_subset hz",
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            "sha256": "80811f468edf8b272a3369696ee222a7a3e45324564cbc1e40d16c9a64020cfb",
            "text": "  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by\n    intro z hz i\n    replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i\n    apply interior_subset hz",
            "line_start": 228,
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            "method": "exact_contiguous_span"
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        },
        "id": "  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by\n    intro z hz i\n    replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i\n    apply interior_subset hz",
        "en": "  have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m := by\n    intro z hz i\n    replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i\n    apply interior_subset hz"
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          "source_text_sha256": "80811f468edf8b272a3369696ee222a7a3e45324564cbc1e40d16c9a64020cfb",
          "source_unit_id": "urn:mil:unit:c12:s02:exercise:001",
          "statement_signature": "have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m",
          "target_is_exercise": true,
          "target_source_line": 184,
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