ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- 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)
\u22a2 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
\u22a2 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)
\u22a2 (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)
\u22a2 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
\u22a2 ∀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
\u22a2 ∃ 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
\u22a2 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
\u22a2 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
\u22a2 ∀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
\u22a2 ∀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
\u22a2 ∀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
\u22a2 ∀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
\u22a2 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
\u22a2 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
\u22a2 ∀ (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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 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)
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 ∀ (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
\u22a2 ∀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
\u22a2 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
\u22a2 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))
\u22a2 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))
\u22a2 ∀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
\u22a2 ‖(sets n).indicator f x‖ ≤ ↑n
-- Mathlib/Analysis/Complex/Angle.lean
a x✝ y✝ : ℂ
x y : ℝ
\u22a2 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
\u22a2 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
\u22a2 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
\u22a2 (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
\u22a2 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
\u22a2 (μ.prod ν) (s ×ˢ t)ᶜ = 0
-- 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
\u22a2 (μ.prod ν) (sᶜ ×ˢ univ) = 0 ∧ (μ.prod ν) (univ ×ˢ tᶜ) = 0
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : OrderedAddCommGroup α
a b c : α
\u22a2 (fun x => x - a) ⁻¹' Icc b c = Icc (b + a) (c + a)
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{?u.235675}
x : α
\u22a2 #α = 2 ↔ ∃! y, y ≠ x
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{?u.235675}
x : α
\u22a2 (∃ x y, x ≠ y ∧ {x, y} = univ) ↔ ∃! y, y ≠ x
-- Mathlib/Topology/Algebra/Group/Basic.lean
G : Type w
H : Type x
α : Type u
β : Type v
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : TopologicalGroup G
s t✝ : Set G
hs : IsOpen s
t : Set G
\u22a2 s / closure t = s / t
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝³ : SeminormedGroup E
inst✝² : SeminormedGroup F
inst✝¹ : SeminormedGroup G
s✝ : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
inst✝ : FunLike 𝓕 E F
f : ι → κ → G
s : Set κ
l : Filter ι
\u22a2 TendstoUniformlyOn f 1 l s ↔ ∀ ε > 0, ∀ᶠ (i : ι) in l, ∀ x ∈ s, ‖f i x‖ < ε
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
a : ℝ≥0∞
\u22a2 ContinuousOn ((fun p => p.1 - p.2) ∘ fun x => (a, x)) {x | x ≠ ⊤}
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
a : ℝ≥0∞
\u22a2 MapsTo (fun y => (a, y)) {x | x ≠ ⊤} {p | p ≠ (⊤, ⊤)}
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
h : ⊤ ∈ {x | x ≠ ⊤}
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
s : Finset I
x y : ↑(π C fun x => x ∈ s)
\u22a2 (factors C s x).prod y = if y = x then 1 else 0
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α✝ : Type u_1
r : α✝ → α✝ → Prop
α β : Type u
s : Set α
t : (a : α) → a ∈ s → Set β
c : Cardinal.{u}
hc : c.IsRegular
hs : #↑s < c
\u22a2 #↑(⋃ a, ⋃ (h : a ∈ s), t a h) < c ↔ ∀ (a : α) (ha : a ∈ s), #↑(t a ha) < c
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
\u22a2 0 ≤ prehaar (↑K₀) U K
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
q : R[X]
toFinsupp✝ : R[ℕ]
\u22a2 X * { toFinsupp := toFinsupp✝ } = { toFinsupp := toFinsupp✝ } * X
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
q : R[X]
toFinsupp✝ : R[ℕ]
\u22a2 Finsupp.single 1 1 * toFinsupp✝ = toFinsupp✝ * Finsupp.single 1 1
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
q : R[X]
toFinsupp✝ : R[ℕ]
x✝ : ℕ
\u22a2 (Finsupp.single 1 1 * toFinsupp✝) x✝ = (toFinsupp✝ * Finsupp.single 1 1) x✝
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
\u22a2 o.cof = sInf {a | ∃ ι f, lsub f = o ∧ #ι = a}
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
\u22a2 ∀ b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a}, o.cof ≤ b
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
\u22a2 o.cof ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a}
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s s₁✝ s₂✝ t t₁ t₂ u v : Finset α
a✝ b : α
s₁ s₂ : Finset α
a : α
h : a ∈ s₂
x : α
\u22a2 x ∈ insert a s₁ ∩ s₂ ↔ x ∈ insert a (s₁ ∩ s₂)
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s s₁✝ s₂✝ t t₁ t₂ u v : Finset α
a✝ b : α
s₁ s₂ : Finset α
a : α
h : a ∈ s₂
x : α
this : x = a ∨ x ∈ s₂ ↔ x ∈ s₂
\u22a2 x ∈ insert a s₁ ∩ s₂ ↔ x ∈ insert a (s₁ ∩ s₂)
-- Mathlib/CategoryTheory/Sites/PreservesSheafification.lean
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝¹ : Category.{u_3, u_1} A
inst✝ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
\u22a2 J.PreservesSheafification F ↔ ∀ (P : Cᵒᵖ ⥤ A), IsIso (G₂.map (whiskerRight (adj₁.unit.app P) F))
-- Mathlib/CategoryTheory/Sites/PreservesSheafification.lean
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝¹ : Category.{u_3, u_1} A
inst✝ : Category.{u_4, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
\u22a2 J.PreservesSheafification F ↔ ∀ (P : Cᵒᵖ ⥤ A), J.W (whiskerRight (adj₁.unit.app P) F)
-- Mathlib/Data/Set/Basic.lean
α : Type u
β : Type v
γ : Type w
ι : Sort x
a b : α
s✝ s₁ s₂ t✝ t₁ t₂ u s t : Set α
h : s = t
x : α
\u22a2 x ∈ s ↔ x ∈ t
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G : MonCat
inst✝ : MonoidalCategory V
A B : Action V G
\u22a2 ((functorCategoryMonoidalEquivalence V G).μIso A B).inv.app PUnit.unit = 𝟙 (((functorCategoryMonoidalEquivalence V G).obj (A ⊗ B)).obj PUnit.unit)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G : MonCat
inst✝ : MonoidalCategory V
A B : Action V G
\u22a2 inv (((functorCategoryMonoidalEquivalence V G).μIso A B).app PUnit.unit).hom = 𝟙 (((functorCategoryMonoidalEquivalence V G).obj (A ⊗ B)).obj PUnit.unit)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G : MonCat
inst✝ : MonoidalCategory V
A B : Action V G
\u22a2 (((functorCategoryMonoidalEquivalence V G).μIso A B).app PUnit.unit).hom ≫ 𝟙 (((functorCategoryMonoidalEquivalence V G).obj (A ⊗ B)).obj PUnit.unit) = 𝟙 (((functorCategoryMonoidalEquivalence V G).obj A ⊗ (functorCategoryMonoidalEquivalence V G).obj B).obj PUnit.unit)