ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/FieldTheory/RatFunc/Basic.lean
K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidWithZeroHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p q : K[X]
hq : q ≠ 0

⊢ (map φ hφ) ((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q) = (algebraMap R[X] (RatFunc R)) (φ p) / (algebraMap R[X] (RatFunc R)) (φ q)
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u_4}
h : o.IsLimit
a : Ordinal.{u_4}

⊢ a < o ↔ ∃ x < o, a < x
-- Mathlib/SetTheory/Ordinal/NaturalOps.lean
a b : Ordinal.{u_1}

⊢ succ a ⨳ b = a ⨳ b ♯ b
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1

⊢ f.eigenspace (-q.coeff 0 / q.leadingCoeff) = LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0))
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1

⊢ q.leadingCoeff ≠ 0
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1
h : q.leadingCoeff = 0

⊢ False
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : ⊥ = 1
h : q.leadingCoeff = 0

⊢ False
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1

⊢ LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0)) = LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0)))
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1

⊢ LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0)) = LinearMap.ker (eval₂ (algebraMap K (End K V)) f (q.leadingCoeff • X + C (q.coeff 0)))
-- Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
K : Type v
V : Type w
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
q : K[X]
hq : q.degree = 1

⊢ LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0))) = LinearMap.ker ((aeval f) q)
-- Mathlib/Analysis/Analytic/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
c : F
e : E

⊢ HasFPowerSeriesOnBall (fun x => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤
-- Mathlib/Analysis/Analytic/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
c : F
e y✝ : E
x✝ : y✝ ∈ EMetric.ball 0 ⊤
n : ℕ
hn : n ≠ 0

⊢ ((constFormalMultilinearSeries 𝕜 E c n) fun x => y✝) = 0
-- Mathlib/RingTheory/PowerSeries/Trunc.lean
R : Type u_1
inst✝ : Semiring R
n d : ℕ

⊢ (trunc (n + 2) X).coeff d = Polynomial.X.coeff d
-- Mathlib/RingTheory/PowerSeries/Trunc.lean
R : Type u_1
inst✝ : Semiring R
n d : ℕ

⊢ (if d < n + 2 then if d = 1 then 1 else 0 else 0) = Polynomial.X.coeff d
-- Mathlib/MeasureTheory/Group/FundamentalDomain.lean
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝¹⁰ : Group G
inst✝⁹ : Group H
inst✝⁸ : MulAction G α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MulAction H β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x => g • x) ⁻¹' A = A
this : ∫⁻ (x : α) in s, A.indicator 1 x ∂μ = ∫⁻ (x : α) in t, A.indicator 1 x ∂μ

⊢ μ (A ∩ s) = μ (A ∩ t)
-- Mathlib/Order/Interval/Multiset.lean
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
a✝ b✝ a b : α
inst✝ : DecidablePred fun x => x ≤ a
hab : a < b

⊢ filter (fun x => x ≤ a) (Ico a b) = {a}
-- Mathlib/Order/Interval/Multiset.lean
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
a✝ b✝ a b : α
inst✝ : DecidablePred fun x => x ≤ a
hab : a < b

⊢ {a}.val = {a}
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))

⊢ Tendsto μs F (𝓝 μ)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))

⊢ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0

⊢ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))

⊢ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᶠ[ae ↑(μs i)] ⇑f₀

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᶠ[ae ↑(μs i)] ⇑f₀
aux : ∫ (a : Ω), f₀ a ∂↑μ = (∫⁻ (a : Ω), ENNReal.ofReal (f₀ a) ∂↑μ).toReal

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᶠ[ae ↑(μs i)] ⇑f₀
aux : ∫ (a : Ω), f₀ a ∂↑μ = (∫⁻ (a : Ω), ENNReal.ofReal (f₀ a) ∂↑μ).toReal
auxs : ∀ (i : γ), ∫ (a : Ω), f₀ a ∂↑(μs i) = (∫⁻ (a : Ω), ENNReal.ofReal (f₀ a) ∂↑(μs i)).toReal

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
key : Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᶠ[ae ↑(μs i)] ⇑f₀
aux : ∫ (a : Ω), ↑(f a) ∂↑μ = (∫⁻ (a : Ω), ↑(f a) ∂↑μ).toReal
auxs : ∀ (i : γ), ∫ (a : Ω), ↑(f a) ∂↑(μs i) = (∫⁻ (a : Ω), ↑(f a) ∂↑(μs i)).toReal

⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s

⊢ ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s

⊢ AEMeasurable g ((μ.restrict s).withDensity f)
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s

⊢ AEMeasurable g ((μ.withDensity f).restrict s)
-- Mathlib/Topology/Algebra/FilterBasis.lean
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
U : Set G
hU : U ∈ B

⊢ U ∈ 𝓝 1
-- Mathlib/Topology/Algebra/FilterBasis.lean
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
U : Set G
hU : U ∈ B

⊢ ∃ i ∈ B, id i ⊆ U
-- Mathlib/Topology/UniformSpace/Equicontinuity.lean
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
inst✝ : Finite ι
F : ι → X → α
x₀ : X

⊢ EquicontinuousAt F x₀ ↔ ∀ (i : ι), ContinuousAt (F i) x₀
-- Mathlib/Probability/Cdf.lean
μ ν : Measure ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
h : cdf μ = cdf ν

⊢ μ = ν
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
x : α

⊢ round x = 0 ↔ x ∈ Ico (-(1 / 2)) (1 / 2)
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
x : α

⊢ x ∈ Ico (0 - 1 / 2) (1 - 1 / 2) ↔ x ∈ Ico (-(1 / 2)) (1 / 2)
-- Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : T2Space M
hi : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := (ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))

⊢ L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x) = mfderiv I I (↑(chartAt H (f.c (f.ind x hx)))) x
-- Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : T2Space M
hi : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := (ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))
this : HasMFDerivAt I 𝓘(ℝ, E) (⇑L ∘ ⇑f.embeddingPiTangent) x (L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))

⊢ L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x) = mfderiv I I (↑(chartAt H (f.c (f.ind x hx)))) x
-- Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : T2Space M
hi : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := (ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))
this : HasMFDerivAt I 𝓘(ℝ, E) (⇑L ∘ ⇑f.embeddingPiTangent) x (L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))

⊢ HasMFDerivAt I 𝓘(ℝ, E) (⇑L ∘ ⇑f.embeddingPiTangent) x (mfderiv I I (↑(chartAt H (f.c (f.ind x hx)))) x)
-- Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : T2Space M
hi : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := (ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))
this : HasMFDerivAt I 𝓘(ℝ, E) (⇑L ∘ ⇑f.embeddingPiTangent) x (L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))

⊢ ⇑L ∘ ⇑f.embeddingPiTangent =ᶠ[𝓝 x] ↑(extChartAt I (f.c (f.ind x hx)))
-- Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : T2Space M
hi : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := (ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))
this : HasMFDerivAt I 𝓘(ℝ, E) (⇑L ∘ ⇑f.embeddingPiTangent) x (L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))
y : M
hy : ↑(f.toFun (f.ind x hx)) y = 1 y

⊢ (⇑L ∘ ⇑f.embeddingPiTangent) y = ↑(extChartAt I (f.c (f.ind x hx))) y
-- Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : SmoothManifoldWithCorners I M
inst✝ : T2Space M
hi : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := (ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))
this : HasMFDerivAt I 𝓘(ℝ, E) (⇑L ∘ ⇑f.embeddingPiTangent) x (L.comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))
y : M
hy : ↑(f.toFun (f.ind x hx)) y = 1 y

⊢ ↑(f.toFun (f.ind x hx)) y • ↑(extChartAt I (f.c (f.ind x hx))) y = ↑(extChartAt I (f.c (f.ind x hx))) y
-- Mathlib/Topology/UniformSpace/Basic.lean
α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
inst✝¹ : UniformSpace α
ι' : Sort u_2
inst✝ : UniformSpace β
p : ι → Prop
s : ι → Set (α × α)
ha : (𝓤 α).HasBasis p s
q : ι' → Prop
t : ι' → Set (β × β)
hb : (𝓤 β).HasBasis q t
f : α → β
S : Set α

⊢ (∀ (ib : ι'), q ib → ∃ ia, p ia ∧ ∀ x ∈ s ia ∩ S ×ˢ S, (f x.1, f x.2) ∈ t ib) ↔ ∀ (i : ι'), q i → ∃ j, p j ∧ ∀ x ∈ S, ∀ y ∈ S, (x, y) ∈ s j → (f x, f y) ∈ t i
-- Mathlib/AlgebraicGeometry/Scheme.lean
R : CommRingCat
X✝ : Scheme
V✝ U✝ : X✝.Opens
f✝ g : ↑Γ(X✝, U✝)
X Y : Scheme
f : X ⟶ Y
U : X.Opens
V : Y.Opens
e : U ≤ f ⁻¹ᵁ V
s : ↑Γ(Y, V)

⊢ X.basicOpen ((Hom.appLE f V U e) s) = U ⊓ f ⁻¹ᵁ Y.basicOpen s
-- Mathlib/AlgebraicGeometry/Scheme.lean
R : CommRingCat
X✝ : Scheme
V✝ U✝ : X✝.Opens
f✝ g : ↑Γ(X✝, U✝)
X Y : Scheme
f : X ⟶ Y
U : X.Opens
V : Y.Opens
e : U ≤ f ⁻¹ᵁ V
s : ↑Γ(Y, V)

⊢ X.basicOpen ((X.presheaf.map (homOfLE e).op) ((Hom.app f V) s)) = U ⊓ X.basicOpen ((Hom.app f V) s)
-- Mathlib/CategoryTheory/SingleObj.lean
M G : Type u
inst✝³ : Monoid M
inst✝² : Group G
N : Type v
inst✝¹ : Monoid N
C : Type v
inst✝ : Category.{w, v} C
f : C → G
X✝ : C

⊢ { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)⁻¹ }.map (𝟙 X✝) = 𝟙 ({ obj := fun x => (), map := fun {x y} x_1 => f y * (f x)⁻¹ }.obj X✝)
-- Mathlib/CategoryTheory/SingleObj.lean
M G : Type u
inst✝³ : Monoid M
inst✝² : Group G
N : Type v
inst✝¹ : Monoid N
C : Type v
inst✝ : Category.{w, v} C
f : C → G
X✝ Y✝ Z✝ : C
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝

⊢ f Z✝ * (f X✝)⁻¹ = (f Y✝ * (f X✝)⁻¹) ≫ (f Z✝ * (f Y✝)⁻¹)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x

⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x

⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)

⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y

⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ

⊢ multiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = multiplicity 2 (x - y) + multiplicity 2 ↑(n + 1)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 n.succ).Dom

⊢ multiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = multiplicity 2 (x - y) + multiplicity 2 ↑(n + 1)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (n + 1)).Dom

⊢ multiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = multiplicity 2 (x - y) + multiplicity 2 ↑(n + 1)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (n + 1)).Dom
hpn : ¬2 ^ ((multiplicity 2 (n + 1)).get h + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ (multiplicity 2 (n + 1)).get h * k

⊢ multiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = multiplicity 2 (x - y) + multiplicity 2 ↑(n + 1)