ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hut' : UnifTight (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hut' : UnifTight (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hg : StronglyMeasurable (AEStronglyMeasurable.mk g ⋯)

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hut' : UnifTight (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hg : StronglyMeasurable (AEStronglyMeasurable.mk g ⋯)
hgg' : g =ᶠ[ae μ] AEStronglyMeasurable.mk g ⋯

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hut' : UnifTight (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hg : StronglyMeasurable (AEStronglyMeasurable.mk g ⋯)
hgg' : g =ᶠ[ae μ] AEStronglyMeasurable.mk g ⋯
hg'' : Memℒp (AEStronglyMeasurable.mk g ⋯) p μ

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hut' : UnifTight (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ
hg : StronglyMeasurable (AEStronglyMeasurable.mk g ⋯)
hgg' : g =ᶠ[ae μ] AEStronglyMeasurable.mk g ⋯
hg'' : Memℒp (AEStronglyMeasurable.mk g ⋯) p μ
haefg' : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g ⋯ x))

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
hg : StronglyMeasurable (AEStronglyMeasurable.mk g ⋯)
hgg' : g =ᶠ[ae μ] AEStronglyMeasurable.mk g ⋯
hg'' : Memℒp (AEStronglyMeasurable.mk g ⋯) p μ
haefg' : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g ⋯ x))
f' : ℕ → α → β := fun n => AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable f' p μ
hut' : UnifTight f' p μ

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
f' : ℕ → α → β := fun n => AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable f' p μ
hut' : UnifTight f' p μ
g' : α → β := AEStronglyMeasurable.mk g ⋯
hg : StronglyMeasurable g'
hgg' : g =ᶠ[ae μ] g'
hg'' : Memℒp g' p μ
haefg' : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (g' x))

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
f' : ℕ → α → β := fun n => AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable f' p μ
hut' : UnifTight f' p μ
g' : α → β := AEStronglyMeasurable.mk g ⋯
hg : StronglyMeasurable g'
hgg' : g =ᶠ[ae μ] g'
hg'' : Memℒp g' p μ
haefg' : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (g' x))
haefg : ∀ (n : ℕ), f n - g =ᶠ[ae μ] f' n - g'

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
f' : ℕ → α → β := fun n => AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable f' p μ
hut' : UnifTight f' p μ
g' : α → β := AEStronglyMeasurable.mk g ⋯
hg : StronglyMeasurable g'
hgg' : g =ᶠ[ae μ] g'
hg'' : Memℒp g' p μ
haefg' : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (g' x))
haefg : ∀ (n : ℕ), f n - g =ᶠ[ae μ] f' n - g'
hsnfg : ∀ (n : ℕ), eLpNorm (f n - g) p μ = eLpNorm (f' n - g') p μ

⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : Memℒp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯
f' : ℕ → α → β := fun n => AEStronglyMeasurable.mk (f n) ⋯
hui' : UnifIntegrable f' p μ
hut' : UnifTight f' p μ
g' : α → β := AEStronglyMeasurable.mk g ⋯
hg : StronglyMeasurable g'
hgg' : g =ᶠ[ae μ] g'
hg'' : Memℒp g' p μ
haefg' : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (g' x))
haefg : ∀ (n : ℕ), f n - g =ᶠ[ae μ] f' n - g'
hsnfg : ∀ (n : ℕ), eLpNorm (f n - g) p μ = eLpNorm (f' n - g') p μ

⊢ Tendsto (fun n => eLpNorm (f' n - g') p μ) atTop (𝓝 0)
-- Mathlib/Logic/Encodable/Basic.lean
α : Type u_1
inst✝ : Encodable α
a : α

⊢ (down a).up = a
-- Mathlib/Algebra/Ring/Centralizer.lean
M : Type u_1
S T : Set M
a : M
b : ?m.797
inst✝¹ : Mul M
inst✝ : HasDistribNeg M
ha : a ∈ S.centralizer
c : M
hc : c ∈ S

⊢ c * -a = -a * c
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : DifferentiableOn ℝ f s
hs : μ s = 0

⊢ μ (f '' s) = 0
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : DifferentiableOn ℝ f s
hs : μ s = 0

⊢ μ (f '' s) ≤ 0
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : DifferentiableOn ℝ f s
hs : μ s = 0
this : ∀ (A : E →L[ℝ] E), ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → μ (f '' t) ≤ ↑(|A.det|.toNNReal + 1) * μ t

⊢ μ (f '' s) ≤ 0
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : DifferentiableOn ℝ f s
hs : μ s = 0
δ : (E →L[ℝ] E) → ℝ≥0
hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → μ (f '' t) ≤ ↑(|A.det|.toNNReal + 1) * μ t

⊢ μ (f '' s) ≤ 0
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : DifferentiableOn ℝ f s
hs : μ s = 0
δ : (E →L[ℝ] E) → ℝ≥0
hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → μ (f '' t) ≤ ↑(|A.det|.toNNReal + 1) * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
left✝¹ : Pairwise (Disjoint on t)
left✝ : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))

⊢ μ (f '' s) ≤ 0
-- Mathlib/Algebra/Lie/Quotient.lean
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N N' : LieSubmodule R L M
I J : LieIdeal R L

⊢ (mk' N).range = ⊤
-- Mathlib/ModelTheory/Semantics.lean
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
φ✝ ψ : L.Formula α
v✝ : α → M
φ : L.Formula (Fin n)
v : Empty → M
x : Fin n → M

⊢ (BoundedFormula.relabel Sum.inr φ).Realize v x ↔ φ.Realize x
-- Mathlib/Algebra/Order/Group/Basic.lean
α : Type u_1
M : Type u_2
R : Type u_3
inst✝ : OrderedCommGroup α
m✝ n✝ : ℤ
a b : α
ha : 1 < a
m n : ℤ
h : m < n

⊢ a ^ m * a ^ (n - m) = a ^ n
-- Mathlib/Tactic/NormNum/Result.lean
α : Type u
u : Level
n : ℤ
d : ℕ
inst✝ : DivisionRing α
inv : Invertible ↑d

⊢ ↑n * ⅟↑d = Rat.rawCast n d
-- Mathlib/RingTheory/Polynomial/Dickson.lean
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
k : ℕ
a : R
inst✝ : Invertible 2

⊢ C ⅟2 * 2 = 1
-- Mathlib/Order/Interval/Set/Pi.lean
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → Preorder (α i)
x y✝ y : (i : ι) → α i

⊢ (y ∈ univ.pi fun i => Ici (x i)) ↔ y ∈ Ici x
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{?u.77613, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ

⊢ (↑(fst φ)).comp (inl φ) ⋯ + (snd φ).comp (Cochain.ofHom (inr φ)) ⋯ = Cochain.ofHom (𝟙 (mappingCone φ))
-- 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✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : SMulInvariantMeasure G α ν
inst✝ : MeasurableSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
vol_s : ν s = 0

⊢ QuotientMeasureEqMeasurePreimage ν 0
-- 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✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : SMulInvariantMeasure G α ν
inst✝ : MeasurableSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
vol_s : ν s = 0

⊢ 0 = Measure.map (Quotient.mk α_mod_G) (ν.restrict s)
-- 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✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : SMulInvariantMeasure G α ν
inst✝ : MeasurableSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
vol_s : ν s = 0
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U

⊢ 0 U = (Measure.map (Quotient.mk α_mod_G) (ν.restrict s)) U
-- 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✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : SMulInvariantMeasure G α ν
inst✝ : MeasurableSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
vol_s : ν s = 0
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U

⊢ 0 = (Measure.map (Quotient.mk α_mod_G) (ν.restrict s)) U
-- 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✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : SMulInvariantMeasure G α ν
inst✝ : MeasurableSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
vol_s : ν s = 0
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U

⊢ (Measure.map (Quotient.mk α_mod_G) (ν.restrict s)) U = ν (Quotient.mk α_mod_G ⁻¹' U ∩ s)
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m✝ : α → β
m' : β → γ
s : Set α
t : Set β
f : Filter β
m : α → β

⊢ (comap m f).NeBot ↔ ∀ t ∈ f, ∃ a, m a ∈ t
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m✝ : α → β
m' : β → γ
s : Set α
t : Set β
f : Filter β
m : α → β

⊢ (∀ (s : Set α), ∀ x ∈ f, m ⁻¹' x ⊆ s → s.Nonempty) ↔ ∀ t ∈ f, ∃ a, m a ∈ t
-- Mathlib/Data/Fintype/Card.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl

⊢ ∃ o, map length o = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise Disjoint o
-- Mathlib/Data/Fintype/Card.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α

⊢ ∃ o, map length o = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise Disjoint o
-- Mathlib/Data/Fintype/Card.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α
l : List (List (Fin (Fintype.card α))) := pmap klift' c.ranges hc'_lt

⊢ ∃ o, map length o = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise Disjoint o
-- Mathlib/Data/Fintype/Card.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α
l : List (List (Fin (Fintype.card α))) := pmap klift' c.ranges hc'_lt
hl : ∀ (a : List ℕ) (ha : a ∈ c.ranges), map (⇑Fin.valEmbedding) (klift' a ⋯) = a

⊢ ∃ o, map length o = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise Disjoint o
-- Mathlib/Data/Fintype/Card.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α
l : List (List (Fin (Fintype.card α))) := pmap klift' c.ranges hc'_lt
hl : ∀ (a : List ℕ) (ha : a ∈ c.ranges), map (⇑Fin.valEmbedding) (klift' a ⋯) = a

⊢ map length (map (map ⇑(Fintype.equivFin α).symm) l) = c ∧ (∀ s ∈ map (map ⇑(Fintype.equivFin α).symm) l, s.Nodup) ∧ Pairwise Disjoint (map (map ⇑(Fintype.equivFin α).symm) l)
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝ : DecidableEq α
s : Multiset α
a b : α

⊢ count a {b} = if a = b then 1 else 0
-- Mathlib/Data/Set/Pointwise/ListOfFn.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : Monoid α
s t : Set α
a✝ : α
m n✝ : ℕ
a : α
n : ℕ

⊢ a ∈ s ^ n ↔ ∃ f, (List.ofFn fun i => ↑(f i)).prod = a
-- Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H : Type u_4
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_6
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
inst✝⁵ : SmoothManifoldWithCorners I M
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : SmoothManifoldWithCorners I' M'
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : TangentBundle I H

⊢ (chartAt (ModelProd H E) p).source = (TotalSpace.toProd H E).toPartialEquiv.source
-- Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H : Type u_4
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_6
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
inst✝⁵ : SmoothManifoldWithCorners I M
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
inst✝² : SmoothManifoldWithCorners I' M'
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : TangentBundle I H

⊢ ({ toFun := fun p_1 => (p_1.proj, (tangentBundleCore I H).toFiberBundleCore.coordChange ((tangentBundleCore I H).toFiberBundleCore.indexAt p_1.proj) (achart H p.proj) p_1.proj p_1.snd), invFun := fun p_1 => { proj := p_1.1, snd := (tangentBundleCore I H).toFiberBundleCore.coordChange (achart H p.proj) ((tangentBundleCore I H).toFiberBundleCore.indexAt p_1.1) p_1.1 p_1.2 }, source := (tangentBundleCore I H).toFiberBundleCore.proj ⁻¹' (↑(achart H p.proj)).source, target := (↑(achart H p.proj)).source ×ˢ univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯, continuousOn_toFun := ⋯, continuousOn_invFun := ⋯ } ≫ₕ (chartAt H p.proj).prod (PartialHomeomorph.refl E)).source = (TotalSpace.toProd H E).toPartialEquiv.source
-- Mathlib/FieldTheory/RatFunc/Basic.lean
K : Type u
inst✝ : Field K
p q : K[X]
hq : q ≠ 0

⊢ C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q) ∣ p
-- Mathlib/Topology/Algebra/Module/StrongTopology.lean
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this : UniformSpace F := TopologicalAddGroup.toUniformSpace F

⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂
-- Mathlib/Topology/Algebra/Module/StrongTopology.lean
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalAddGroup F
h : 𝔖₂ ⊆ 𝔖₁
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F

⊢ instTopologicalSpace σ F 𝔖₁ ≤ instTopologicalSpace σ F 𝔖₂