ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>ι : Type u_4<VAR>M : Type u_5<VAR>M' : Type u_6<VAR>N : Type u_7<VAR>P : Type u_8<VAR>G : Type u_9<VAR>H : Type u_10<VAR>R : Type u_11<VAR>S : Type u_12<VAR>inst✝¹ : AddCommMonoid M<VAR>inst✝ : AddCommMonoid N<VAR>f : α × β →₀ M<VAR>g : α → β → M → N<VAR>hg₀ : ∀ (a : α) (b : β), g a b 0 = 0<VAR>hg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁<VAR>p : α × β<VAR>c : M<GOAL>((single p.1 (single p.2 c)).sum fun a f => f.sum (g a)) = g p.1 p.2 c
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>ι : Type u_4<VAR>M : Type u_5<VAR>M' : Type u_6<VAR>N : Type u_7<VAR>P : Type u_8<VAR>G : Type u_9<VAR>H : Type u_10<VAR>R : Type u_11<VAR>S : Type u_12<VAR>inst✝¹ : AddCommMonoid M<VAR>inst✝ : AddCommMonoid N<VAR>f : α × β →₀ M<VAR>g : α → β → M → N<VAR>hg₀ : ∀ (a : α) (b : β), g a b 0 = 0<VAR>hg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁<VAR>p : α × β<VAR>c : M<GOAL>(single p.2 c).sum (g p.1) = g p.1 p.2 c
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝¹ : LinearOrder α<VAR>inst✝ : LinearOrder β<VAR>f : α → β<VAR>s : Set α<VAR>a✝ a₁ a₂ b✝ b₁ b₂ c✝ d x a b c : α<VAR>h : (fun b => Ι b a) b = (fun b => Ι b a) c<GOAL>b = c
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝¹ : LinearOrder α<VAR>inst✝ : LinearOrder β<VAR>f : α → β<VAR>s : Set α<VAR>a✝ a₁ a₂ b✝ b₁ b₂ c✝ d x a b c : α<VAR>h : ∀ (x : α), x ∈ (fun b => Ι b a) b ↔ x ∈ (fun b => Ι b a) c<GOAL>b = c
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝¹ : LinearOrder α<VAR>inst✝ : LinearOrder β<VAR>f : α → β<VAR>s : Set α<VAR>a✝ a₁ a₂ b✝ b₁ b₂ c✝ d x a b c : α<VAR>h : ∀ (x : α), x ∈ (fun b => Ι b a) b ↔ x ∈ (fun b => Ι b a) c<VAR>ha : b ≤ a<GOAL>b = c
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝¹ : LinearOrder α<VAR>inst✝ : LinearOrder β<VAR>f : α → β<VAR>s : Set α<VAR>a✝ a₁ a₂ b✝ b₁ b₂ c✝ d x a b c : α<VAR>h : ∀ (x : α), x ∈ (fun b => Ι b a) b ↔ x ∈ (fun b => Ι b a) c<VAR>ha : a < b<GOAL>b = c
<VAR>α : Type u_1<VAR>inst✝ : LinearOrderedSemiring α<VAR>m n : Num<GOAL>↑m ≤ ↑n ↔ m ≤ n
<VAR>α : Type u_1<VAR>inst✝ : LinearOrderedSemiring α<VAR>m n : Num<GOAL>¬↑n < ↑m ↔ m ≤ n
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>δ : Type u_4<VAR>s s₁ s₂ : Set α<VAR>t t₁ t₂ : Set β<VAR>a : α<VAR>b : β<VAR>inst✝¹ : Nonempty α<VAR>inst✝ : Nonempty β<GOAL>s ×ˢ t = univ ↔ s = univ ∧ t = univ
<VAR>R : Type u<VAR>inst✝¹ : CommRing R<VAR>W' : Jacobian R<VAR>F : Type v<VAR>inst✝ : Field F<VAR>W : Jacobian F<VAR>P Q : Fin 3 → F<VAR>hPz : P z ≠ 0<VAR>hQz : Q z ≠ 0<VAR>hx : P x * Q z ^ 2 = Q x * P z ^ 2<VAR>hy : P y * Q z ^ 3 = Q y * P z ^ 3<GOAL>P ≈ Q
<VAR>R : Type u<VAR>inst✝¹ : CommRing R<VAR>W' : Jacobian R<VAR>F : Type v<VAR>inst✝ : Field F<VAR>W : Jacobian F<VAR>P Q : Fin 3 → F<VAR>hPz : P z ≠ 0<VAR>hQz : Q z ≠ 0<VAR>hx : P x * Q z ^ 2 = Q x * P z ^ 2<VAR>hy : P y * Q z ^ 3 = Q y * P z ^ 3<GOAL>(fun m => m • Q) (Units.mk0 (P z) hPz / Units.mk0 (Q z) hQz) = P
<VAR>a : ℕ<VAR>a1 : 1 < a<GOAL>az a * az a - ↑(Pell.d a1) * 1 * 1 = 1
<VAR>V : Type u_1<VAR>W : Type u_2<VAR>V₁ : Type u_3<VAR>V₂ : Type u_4<VAR>V₃ : Type u_5<VAR>inst✝⁴ : SeminormedAddCommGroup V<VAR>inst✝³ : SeminormedAddCommGroup W<VAR>inst✝² : SeminormedAddCommGroup V₁<VAR>inst✝¹ : SeminormedAddCommGroup V₂<VAR>inst✝ : SeminormedAddCommGroup V₃<VAR>f : NormedAddGroupHom V₁ V₂<VAR>g : NormedAddGroupHom V₂ V₃<GOAL>(f.comp (incl ⊤)).range = f.range
<VAR>V : Type u_1<VAR>W : Type u_2<VAR>V₁ : Type u_3<VAR>V₂ : Type u_4<VAR>V₃ : Type u_5<VAR>inst✝⁴ : SeminormedAddCommGroup V<VAR>inst✝³ : SeminormedAddCommGroup W<VAR>inst✝² : SeminormedAddCommGroup V₁<VAR>inst✝¹ : SeminormedAddCommGroup V₂<VAR>inst✝ : SeminormedAddCommGroup V₃<VAR>f : NormedAddGroupHom V₁ V₂<VAR>g : NormedAddGroupHom V₂ V₃<GOAL>f.toAddMonoidHom.range = f.range
<VAR>ι : Sort u_1<VAR>α : Type u<VAR>β : Type v<VAR>inst✝¹ : PseudoEMetricSpace α<VAR>inst✝ : PseudoEMetricSpace β<VAR>x y : α<VAR>s t u : Set α<VAR>Φ : α → β<VAR>fin : hausdorffEdist s t ≠ ⊤<GOAL>s = ∅ ∧ t = ∅ ∨ s.Nonempty ∧ t.Nonempty
<VAR>ι : Sort u_1<VAR>α : Type u<VAR>β : Type v<VAR>inst✝¹ : PseudoEMetricSpace α<VAR>inst✝ : PseudoEMetricSpace β<VAR>x y : α<VAR>s t u : Set α<VAR>Φ : α → β<VAR>fin : hausdorffEdist s t ≠ ⊤<VAR>hs : s = ∅<GOAL>s = ∅ ∧ t = ∅ ∨ s.Nonempty ∧ t.Nonempty
<VAR>ι : Sort u_1<VAR>α : Type u<VAR>β : Type v<VAR>inst✝¹ : PseudoEMetricSpace α<VAR>inst✝ : PseudoEMetricSpace β<VAR>x y : α<VAR>s t u : Set α<VAR>Φ : α → β<VAR>fin : hausdorffEdist s t ≠ ⊤<VAR>hs : s.Nonempty<GOAL>s = ∅ ∧ t = ∅ ∨ s.Nonempty ∧ t.Nonempty
<VAR>α : Type u<VAR>inst✝² : Group α<VAR>inst✝¹ : LE α<VAR>inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1<VAR>a b c d : α<GOAL>a * a⁻¹ ≤ a * b ↔ 1 ≤ a * b
<VAR>F : Type u_1<VAR>α✝ : Type u_2<VAR>β : Type u_3<VAR>R : Type u_4<VAR>inst✝² : Semiring α✝<VAR>inst✝¹ : Semiring β<VAR>a b : α✝<VAR>m n : ℕ<VAR>α : Type u_5<VAR>inst✝ : Semiring α<VAR>x : α<GOAL>(x ∈ Set.range fun x => 2 * x + 1) ↔ x ∈ {a | Odd a}
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>E : Type u_3<VAR>F : Type u_4<VAR>G : Type u_5<VAR>E' : Type u_6<VAR>F' : Type u_7<VAR>G' : Type u_8<VAR>E'' : Type u_9<VAR>F'' : Type u_10<VAR>G'' : Type u_11<VAR>E''' : Type u_12<VAR>R : Type u_13<VAR>R' : Type u_14<VAR>𝕜 : Type u_15<VAR>𝕜' : Type u_16<VAR>inst✝¹⁵ : Norm E<VAR>inst✝¹⁴ : Norm F<VAR>inst✝¹³ : Norm G<VAR>inst✝¹² : SeminormedAddCommGroup E'<VAR>inst✝¹¹ : SeminormedAddCommGroup F'<VAR>inst✝¹⁰ : SeminormedAddCommGroup G'<VAR>inst✝⁹ : NormedAddCommGroup E''<VAR>inst✝⁸ : NormedAddCommGroup F''<VAR>inst✝⁷ : NormedAddCommGroup G''<VAR>inst✝⁶ : SeminormedRing R<VAR>inst✝⁵ : SeminormedAddGroup E'''<VAR>inst✝⁴ : SeminormedRing R'<VAR>inst✝³ : NormedDivisionRing 𝕜<VAR>inst✝² : NormedDivisionRing 𝕜'<VAR>c c' c₁ c₂ : ℝ<VAR>f : α → E<VAR>g : α → F<VAR>k : α → G<VAR>f' : α → E'<VAR>g' : α → F'<VAR>k' : α → G'<VAR>f'' : α → E''<VAR>g'' : α → F''<VAR>k'' : α → G''<VAR>l l' : Filter α<VAR>inst✝¹ : One F<VAR>inst✝ : NormOneClass F<GOAL>(f =O[l] fun _x => 1) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => ‖f x‖
<VAR>G : Type u_1<VAR>inst✝⁸ : TopologicalSpace G<VAR>inst✝⁷ : Group G<VAR>inst✝⁶ : TopologicalGroup G<VAR>inst✝⁵ : MeasurableSpace G<VAR>inst✝⁴ : BorelSpace G<VAR>inst✝³ : LocallyCompactSpace G<VAR>μ' μ : Measure G<VAR>inst✝² : μ.IsHaarMeasure<VAR>inst✝¹ : IsFiniteMeasureOnCompacts μ'<VAR>inst✝ : μ'.IsMulLeftInvariant<VAR>s : Set G<VAR>hs : MeasurableSet s<VAR>h's : IsCompact (closure s)<VAR>ν : Measure G := haar<GOAL>μ' s = μ'.haarScalarFactor μ • μ s
<VAR>G : Type u_1<VAR>inst✝⁸ : TopologicalSpace G<VAR>inst✝⁷ : Group G<VAR>inst✝⁶ : TopologicalGroup G<VAR>inst✝⁵ : MeasurableSpace G<VAR>inst✝⁴ : BorelSpace G<VAR>inst✝³ : LocallyCompactSpace G<VAR>μ' μ : Measure G<VAR>inst✝² : μ.IsHaarMeasure<VAR>inst✝¹ : IsFiniteMeasureOnCompacts μ'<VAR>inst✝ : μ'.IsMulLeftInvariant<VAR>s : Set G<VAR>hs : MeasurableSet s<VAR>h's : IsCompact (closure s)<VAR>ν : Measure G := haar<VAR>A : μ' s = μ'.haarScalarFactor ν • ν s<GOAL>μ' s = μ'.haarScalarFactor μ • μ s
<VAR>G : Type u_1<VAR>inst✝⁸ : TopologicalSpace G<VAR>inst✝⁷ : Group G<VAR>inst✝⁶ : TopologicalGroup G<VAR>inst✝⁵ : MeasurableSpace G<VAR>inst✝⁴ : BorelSpace G<VAR>inst✝³ : LocallyCompactSpace G<VAR>μ' μ : Measure G<VAR>inst✝² : μ.IsHaarMeasure<VAR>inst✝¹ : IsFiniteMeasureOnCompacts μ'<VAR>inst✝ : μ'.IsMulLeftInvariant<VAR>s : Set G<VAR>hs : MeasurableSet s<VAR>h's : IsCompact (closure s)<VAR>ν : Measure G := haar<VAR>A : μ' s = μ'.haarScalarFactor ν • ν s<VAR>B : μ s = μ.haarScalarFactor ν • ν s<GOAL>μ' s = μ'.haarScalarFactor μ • μ s
<VAR>𝕜 : Type u_1<VAR>E : Type u_2<VAR>inst✝⁷ : RCLike 𝕜<VAR>inst✝⁶ : NormedAddCommGroup E<VAR>inst✝⁵ : InnerProductSpace 𝕜 E<VAR>ι : Type u_3<VAR>inst✝⁴ : LinearOrder ι<VAR>inst✝³ : LocallyFiniteOrderBot ι<VAR>inst✝² : IsWellOrder ι fun x x_1 => x < x_1<VAR>inst✝¹ : Fintype ι<VAR>inst✝ : FiniteDimensional 𝕜 E<VAR>h : finrank 𝕜 E = Fintype.card ι<VAR>f : ι → E<VAR>i j : ι<VAR>hij : i < j<GOAL>(gramSchmidtOrthonormalBasis h f).repr (f i) j = 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>δ : Type u_4<VAR>m : MeasurableSpace α<VAR>μ ν : Measure α<VAR>inst✝² : MeasurableSpace δ<VAR>inst✝¹ : NormedAddCommGroup β<VAR>inst✝ : NormedAddCommGroup γ<VAR>f : α → ℝ≥0∞<VAR>hfm : AEMeasurable f μ<VAR>hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤<GOAL>Memℒp (fun x => (f x).toReal) 1 μ
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>δ : Type u_4<VAR>m : MeasurableSpace α<VAR>μ ν : Measure α<VAR>inst✝² : MeasurableSpace δ<VAR>inst✝¹ : NormedAddCommGroup β<VAR>inst✝ : NormedAddCommGroup γ<VAR>f : α → ℝ≥0∞<VAR>hfm : AEMeasurable f μ<VAR>hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤<GOAL>AEStronglyMeasurable (fun x => (f x).toReal) μ ∧ ∫⁻ (x : α), ↑‖(f x).toReal‖₊ ∂μ < ⊤
<VAR>x y : ℂ<GOAL>cos (2 * x) = cos x ^ 2 - sin x ^ 2
<VAR>p : ℕ<VAR>hp : Prime p<VAR>b : ℕ<VAR>x✝ : log p 0 < b<GOAL>multiplicity p 0! = ↑(∑ i ∈ Ico 1 b, 0 / p ^ i)
<VAR>p : ℕ<VAR>hp : Prime p<VAR>n b : ℕ<VAR>hb : log p (n + 1) < b<GOAL>multiplicity p (n + 1)! = multiplicity p n ! + multiplicity p (n + 1)
<VAR>p : ℕ<VAR>hp : Prime p<VAR>n b : ℕ<VAR>hb : log p (n + 1) < b<GOAL>multiplicity p n ! + multiplicity p (n + 1) = ↑(∑ i ∈ Ico 1 b, n / p ^ i) + ↑(filter (fun i => p ^ i ∣ n + 1) (Ico 1 b)).card
<VAR>p : ℕ<VAR>hp : Prime p<VAR>n b : ℕ<VAR>hb : log p (n + 1) < b<GOAL>↑(∑ i ∈ Ico 1 b, n / p ^ i) + ↑(filter (fun i => p ^ i ∣ n + 1) (Ico 1 b)).card = ↑(∑ i ∈ Ico 1 b, (n / p ^ i + if p ^ i ∣ n + 1 then 1 else 0))
<VAR>p : ℕ<VAR>hp : Prime p<VAR>n b : ℕ<VAR>hb : log p (n + 1) < b<GOAL>↑(∑ i ∈ Ico 1 b, n / p ^ i) + ↑(filter (fun i => p ^ i ∣ n + 1) (Ico 1 b)).card = ↑(∑ x ∈ Ico 1 b, n / p ^ x + ↑(filter (fun x => p ^ x ∣ n + 1) (Ico 1 b)).card)
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<VAR>h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<VAR>h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop<VAR>A_meas : MeasurableSet A<GOAL>∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<VAR>h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop<VAR>A_meas : MeasurableSet A<GOAL>restrict s A ≤ restrict 0 A
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<VAR>h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop<VAR>A_meas : MeasurableSet A<VAR>hnn : ¬restrict s A ≤ restrict 0 A<GOAL>False
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<VAR>h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop<VAR>A_meas : MeasurableSet A<VAR>hnn : ¬∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ A → ↑s j ≤ ↑0 j<GOAL>False
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝³ : MeasurableSpace α<VAR>M : Type u_3<VAR>inst✝² : AddCommMonoid M<VAR>inst✝¹ : TopologicalSpace M<VAR>inst✝ : OrderedAddCommMonoid M<VAR>s : SignedMeasure α<VAR>i j : Set α<VAR>hi : ↑s i < 0<VAR>hi₁ : MeasurableSet i<VAR>h : ¬restrict s i ≤ restrict 0 i<VAR>hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l<VAR>bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)<VAR>hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)<VAR>h₂ : ↑s A ≤ ↑s i<VAR>h₃' : Summable fun n => 1 / (↑(bdd n) + 1)<VAR>h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop<VAR>h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop<VAR>A_meas : MeasurableSet A<VAR>E : Set α<VAR>hE₁ : MeasurableSet E<VAR>hE₂ : E ⊆ A<VAR>hE₃ : ↑0 E < ↑s E<VAR>k : ℕ<VAR>hk₁ : 1 ≤ bdd k<VAR>hk₂ : 1 / ↑(bdd k) < ↑s E<GOAL>False