ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
c✝ : E
f : α → E
hf : AEStronglyMeasurable f μ
s✝ s : Set α
c : G
hμs : μ s ≠ 0
h : eLpNormEssSup (s.indicator fun x => c) μ < ↑‖c‖₊
h' : μ {a | ¬↑‖s.indicator (fun x => c) a‖₊ < ↑‖c‖₊} = 0

⊢ False
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
c✝ : E
f : α → E
hf : AEStronglyMeasurable f μ
s✝ s : Set α
c : G
hμs : μ s ≠ 0
h : eLpNormEssSup (s.indicator fun x => c) μ < ↑‖c‖₊
h' : μ {a | ↑‖c‖₊ ≤ ↑‖s.indicator (fun x => c) a‖₊} = 0

⊢ False
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
c✝ : E
f : α → E
hf : AEStronglyMeasurable f μ
s✝ s : Set α
c : G
hμs : μ s ≠ 0
h : eLpNormEssSup (s.indicator fun x => c) μ < ↑‖c‖₊
h' : μ {a | ↑‖c‖₊ ≤ ↑‖s.indicator (fun x => c) a‖₊} = 0
x : α
hx_mem : x ∈ s

⊢ x ∈ {a | ↑‖c‖₊ ≤ ↑‖s.indicator (fun x => c) a‖₊}
-- Mathlib/Data/List/Rotate.lean
α : Type u
l : List α
n : ℕ

⊢ [] = l.rotate n ↔ [] = l
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p q : MvPolynomial σ R
a : R
x : σ

⊢ degreeOf x (C a) = 0
-- Mathlib/Data/Finset/Interval.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder β
s t : Finset α
f : Finset α → β
inst✝ : DecidableEq α

⊢ StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f s < f (insert a s)
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b

⊢ IntegrableOn g' (Ioc a b) volume
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b

⊢ IntegrableOn g' (Ioo a b) volume
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))

⊢ IntegrableOn g' (Ioo a b) volume
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))

⊢ ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊ ≤ ENNReal.ofReal (g b - g a)
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊

⊢ False
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f✝ f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), ↑f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) < ⊤
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x)

⊢ False
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F✝ : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f✝ f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), ↑f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) < ⊤
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x)
F : ℝ → ℝ := NNReal.toReal ∘ ↑f

⊢ False
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F✝ : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f✝ f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), ↑f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) < ⊤
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x)
F : ℝ → ℝ := NNReal.toReal ∘ ↑f
intF : IntegrableOn F (Ioo a b) volume

⊢ False
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F✝ : Type u_4
A✝ : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f✝ f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), ↑f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) < ⊤
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x)
F : ℝ → ℝ := NNReal.toReal ∘ ↑f
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)

⊢ False
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F✝ : Type u_4
A✝ : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f✝ f' : ℝ → E
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), ↑f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) < ⊤
F : ℝ → ℝ := NNReal.toReal ∘ ↑f
hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(↑f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
B : ∫ (x : ℝ) in Ioo a b, F x ≤ g b - g a

⊢ False
-- Mathlib/Data/Matroid/Sum.lean
α : Type u
β : Type v
M✝ N✝ M : Matroid α
N : Matroid β
I X : Set (α ⊕ β)

⊢ (M.sum N).Basis I X ↔ M.Basis (Sum.inl ⁻¹' I) (Sum.inl ⁻¹' X) ∧ N.Basis (Sum.inr ⁻¹' I) (Sum.inr ⁻¹' X)
-- Mathlib/Data/Matroid/Sum.lean
α : Type u
β : Type v
M✝ N✝ M : Matroid α
N : Matroid β
I X : Set (α ⊕ β)

⊢ M.Basis ((fun a => a.down) '' (Sigma.mk false ⁻¹' ((fun s => Sum.elim (fun x => ⟨false, x⟩) (fun x => ⟨true, x⟩) s) '' ((fun a => Sum.map (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) a) '' I)))) ((fun a => a.down) '' (Sigma.mk false ⁻¹' ((fun s => Sum.elim (fun x => ⟨false, x⟩) (fun x => ⟨true, x⟩) s) '' ((fun a => Sum.map (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) a) '' X)))) ∧ N.Basis ((fun a => a.down) '' (Sigma.mk true ⁻¹' ((fun s => Sum.elim (fun x => ⟨false, x⟩) (fun x => ⟨true, x⟩) s) '' ((fun a => Sum.map (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) a) '' I)))) ((fun a => a.down) '' (Sigma.mk true ⁻¹' ((fun s => Sum.elim (fun x => ⟨false, x⟩) (fun x => ⟨true, x⟩) s) '' ((fun a => Sum.map (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) (⇑{ toFun := ULift.down, invFun := ULift.up, left_inv := ⋯, right_inv := ⋯ }.symm) a) '' X)))) ↔ M.Basis (Sum.inl ⁻¹' I) (Sum.inl ⁻¹' X) ∧ N.Basis (Sum.inr ⁻¹' I) (Sum.inr ⁻¹' X)
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDF f κ ν
a : α

⊢ ∀ᵐ (b : β) ∂ν a, Tendsto (f (a, b)) atTop (𝓝 1)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal

⊢ ∃ x, IsSelfAdjoint x ∧ QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal ∧ x * x = a
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal

⊢ QuasispectrumRestricts (cfcₙ Real.sqrt a) ⇑ContinuousMap.realToNNReal ∧ cfcₙ Real.sqrt a * cfcₙ Real.sqrt a = a
-- Mathlib/Analysis/Calculus/FDeriv/Add.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
hg : HasStrictFDerivAt g g' x

⊢ HasStrictFDerivAt (fun x => f x - g x) (f' - g') x
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : Adapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R

⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop atTop ↔ Tendsto (fun n => predictablePart f ℱ μ n ω) atTop atTop
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : Adapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atTop

⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop atTop ↔ Tendsto (fun n => predictablePart f ℱ μ n ω) atTop atTop
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : Adapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atTop
h₂ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atBot

⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop atTop ↔ Tendsto (fun n => predictablePart f ℱ μ n ω) atTop atTop
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : Adapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atTop
h₂ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atBot
h₃ : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), 0 ≤ (μ[f (n + 1) - f n|↑ℱ n]) ω

⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop atTop ↔ Tendsto (fun n => predictablePart f ℱ μ n ω) atTop atTop
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω✝ : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : Adapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atTop
h₂ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atBot
h₃ : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), 0 ≤ (μ[f (n + 1) - f n|↑ℱ n]) ω
ω : Ω
hω₁ : ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atTop
hω₂ : ¬Tendsto (fun n => martingalePart f ℱ μ n ω) atTop atBot
hω₃ : ∀ (n : ℕ), 0 ≤ (μ[f (n + 1) - f n|↑ℱ n]) ω
hω₄ : ∀ (n : ℕ), f n ω ≤ f (n + 1) ω

⊢ Tendsto (fun n => f n ω) atTop atTop ↔ Tendsto (fun n => predictablePart f ℱ μ n ω) atTop atTop
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
f : X → Y
g : Z → W
hf : IsOpenMap f
hg : IsOpenMap g

⊢ IsOpenMap fun p => (f p.1, g p.2)
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
f : X → Y
g : Z → W
hf : IsOpenMap f
hg : IsOpenMap g

⊢ ∀ (x : X × Z), 𝓝 (f x.1, g x.2) ≤ map (fun p => (f p.1, g p.2)) (𝓝 x)
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
f : X → Y
g : Z → W
hf : IsOpenMap f
hg : IsOpenMap g
a : X
b : Z

⊢ 𝓝 (f (a, b).1, g (a, b).2) ≤ map (fun p => (f p.1, g p.2)) (𝓝 (a, b))
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
f : X → Y
g : Z → W
hf : IsOpenMap f
hg : IsOpenMap g
a : X
b : Z

⊢ 𝓝 (f (a, b).1) ×ˢ 𝓝 (g (a, b).2) ≤ map f (𝓝 a) ×ˢ map g (𝓝 b)
-- Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean
C : Type u_1
ι : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
inst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)
i : ι

⊢ biprod.inr.f i ≫ biprod.snd.f i = 𝟙 (L.X i)
-- Mathlib/Data/Nat/Bits.lean
m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1

⊢ shiftLeft' b m (n + 1 - (k + 1)) = shiftLeft' b m (n + 1) >>> (k + 1)
-- Mathlib/Data/Nat/Bits.lean
m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1

⊢ shiftLeft' b m (n - k) = bit b (shiftLeft' b m n) >>> 1 >>> k
-- Mathlib/Data/Nat/Bits.lean
m✝ n✝ : ℕ
b : Bool
m n k : ℕ
h : k + 1 ≤ n + 1

⊢ shiftLeft' b m n >>> k = (bit b (shiftLeft' b m n) / 2) >>> k
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)

⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
sets_prop : ∀ (n : ℕ), MeasurableSet (sets n) ∧ μ (sets n) < ⊤ ∧ ∀ x ∈ sets n, ‖f x‖ ≤ ↑n
h_univ : ⋃ i, sets i = Set.univ

⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
sets_prop : (∀ (x : ℕ), MeasurableSet (sets x)) ∧ (∀ (x : ℕ), μ (sets x) < ⊤) ∧ ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x

⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x

⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x

⊢ ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ sets n → (μ[f * g|m]) x = f x * (μ[g|m]) x
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
n : ℕ

⊢ ∀ᵐ (x : α) ∂μ.restrict (sets n), (μ[f * g|m]) x = f x * (μ[g|m]) x
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
n : ℕ

⊢ μ.restrict (sets n)[f * g|m] =ᶠ[ae (μ.restrict (sets n))] f * μ.restrict (sets n)[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
n : ℕ

⊢ μ.restrict (sets n)[(sets n).indicator f * g|m] =ᶠ[ae (μ.restrict (sets n))] (sets n).indicator f * μ.restrict (sets n)[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
n : ℕ
this : IsFiniteMeasure (μ.restrict (sets n))

⊢ μ.restrict (sets n)[(sets n).indicator f * g|m] =ᶠ[ae (μ.restrict (sets n))] (sets n).indicator f * μ.restrict (sets n)[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
n : ℕ
this : IsFiniteMeasure (μ.restrict (sets n))

⊢ ∀ᵐ (x : α) ∂μ.restrict (sets n), ‖(sets n).indicator f x‖ ≤ ↑n
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ⊤
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
n : ℕ
this : IsFiniteMeasure (μ.restrict (sets n))
x : α

⊢ ‖(sets n).indicator f x‖ ≤ ↑n
-- Mathlib/Analysis/Complex/Angle.lean
a x✝ y✝ : ℂ
x y : ℝ

⊢ angle (cexp (↑x * I)) (cexp (↑y * I)) = |toIocMod Real.two_pi_pos (-π) (x - y)|
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
a : α
l : List α
h : l ≠ []
ainl : a ∈ l

⊢ a ∈ cycle l h
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
a : α
l : List α
h : l ≠ []
ainl : a ∈ l

⊢ a ∈ l ++ₛ cycle l h
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁵ : Category.{v₁, u₁} J
K : Type u₂
inst✝⁴ : Category.{v₂, u₂} K
C : Type u
inst✝³ : Category.{v, u} C
F✝ : J ⥤ C
D : Type u'
inst✝² : Category.{v', u'} D
F : J ⥤ C
inst✝¹ : HasLimit F
G : C ⥤ D
inst✝ : HasLimit (F ⋙ G)
c : Cone F
j✝ : J

⊢ (G.map (lift F c) ≫ post F G) ≫ π (F ⋙ G) j✝ = lift (F ⋙ G) (G.mapCone c) ≫ π (F ⋙ G) j✝
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁵ : Category.{v₁, u₁} J
K : Type u₂
inst✝⁴ : Category.{v₂, u₂} K
C : Type u
inst✝³ : Category.{v, u} C
F✝ : J ⥤ C
D : Type u'
inst✝² : Category.{v', u'} D
F : J ⥤ C
inst✝¹ : HasLimit F
G : C ⥤ D
inst✝ : HasLimit (F ⋙ G)
c : Cone F
j✝ : J

⊢ G.map (c.π.app j✝) = (G.mapCone c).π.app j✝
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set α
t : Set β
s_ae_univ : μ sᶜ = 0
t_ae_univ : ν tᶜ = 0

⊢ (μ.prod ν) (s ×ˢ t)ᶜ = 0