ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Sites/Over.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
F : SheafOfTypes J
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.forget X).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : C
f₁ : W ⟶ (Over.forget X).obj Y₁
f₂ : W ⟶ (Over.forget X).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.forget X).map g₁ = f₂ ≫ (Over.forget X).map g₂
W' : Over X := Over.mk (f₁ ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁ ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂ ⋯

⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
-- Mathlib/FieldTheory/IntermediateField.lean
K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
S : IntermediateField K L
f : L →ₐ[K] L'
a₁✝ a₂✝ : IntermediateField K L
h : map f a₁✝ = map f a₂✝

⊢ a₁✝ = a₂✝
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Algebra ℚ R
I : Ideal R
hI : I ≠ ⊤
a b : ℕ
h_ab : ↑a = ↑b

⊢ a = b
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Algebra ℚ R
I : Ideal R
a b : ℕ
h_ab : ↑a = ↑b
hI : a ≠ b

⊢ I = ⊤
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Algebra ℚ R
I : Ideal R
a b : ℕ
h_ab : ↑a = ↑b
hI : a ≠ b

⊢ (algebraMap ℚ R) (↑a - ↑b) ∈ I
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Algebra ℚ R
I : Ideal R
a b : ℕ
h_ab : ↑a = ↑b
hI : a ≠ b

⊢ ↑a - ↑b ≠ 0
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X

⊢ AnalyticSet (f '' s)
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X

⊢ AnalyticSet (f '' s)
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
b' : BorelSpace X

⊢ AnalyticSet (f '' s)
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle✝ : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
b' : BorelSpace X
hle : borel X ≤ borel X

⊢ AnalyticSet (f '' s)
-- Mathlib/Probability/Independence/Conditional.lean
Ω : Type u_1
ι : Type u_2
β✝ : Type u_3
β' : Type u_4
m' mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
f✝ : Ω → β✝
g : Ω → β'
β : ι → Type u_5
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
h_indep : iCondIndepFun m' hm' m f μ
hf : ∀ (i : ι), Measurable (f i)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l

⊢ CondIndepFun m' hm' (fun a => (f i a, f j a)) (fun a => (f k a, f l a)) μ
-- Mathlib/LinearAlgebra/Eigenspace/Zero.lean
R : Type u_1
K : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : Field K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Finite R M
inst✝² : Module.Free R M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : Module.End K M

⊢ [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ker φ = ⊥, ∀ (m : M), φ m = 0 → m = 0].TFAE
-- Mathlib/LinearAlgebra/Eigenspace/Zero.lean
R : Type u_1
K : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : Field K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Finite R M
inst✝² : Module.Free R M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : Module.End K M
this : (List.map Not [φ.HasEigenvalue 0, (minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) = 0, LinearMap.det φ = 0, ⊥ < ker φ, ∃ m, m ≠ 0 ∧ φ m = 0]).TFAE

⊢ [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ker φ = ⊥, ∀ (m : M), φ m = 0 → m = 0].TFAE
-- Mathlib/LinearAlgebra/Eigenspace/Zero.lean
R : Type u_1
K : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : Field K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Finite R M
inst✝² : Module.Free R M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : Module.End K M
this : [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ¬⊥ < ker φ, ∀ (m : M), m ≠ 0 → φ m ≠ 0].TFAE

⊢ [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ker φ = ⊥, ∀ (m : M), φ m = 0 → m = 0].TFAE
-- Mathlib/LinearAlgebra/Eigenspace/Zero.lean
R : Type u_1
K : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : Field K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Finite R M
inst✝² : Module.Free R M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : Module.End K M
this : [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ¬⊥ < ker φ, ∀ (m : M), m ≠ 0 → φ m ≠ 0].TFAE
aux₁ : ∀ (m : M), m ≠ 0 → φ m ≠ 0 ↔ φ m = 0 → m = 0

⊢ [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ker φ = ⊥, ∀ (m : M), φ m = 0 → m = 0].TFAE
-- Mathlib/LinearAlgebra/Eigenspace/Zero.lean
R : Type u_1
K : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
inst✝⁶ : Field K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Finite R M
inst✝² : Module.Free R M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : Module.End K M
this : [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ¬⊥ < ker φ, ∀ (m : M), m ≠ 0 → φ m ≠ 0].TFAE
aux₁ : ∀ (m : M), m ≠ 0 → φ m ≠ 0 ↔ φ m = 0 → m = 0
aux₂ : ker φ = ⊥ ↔ ¬⊥ < ker φ

⊢ [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ker φ = ⊥, ∀ (m : M), φ m = 0 → m = 0].TFAE
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C

⊢ ‖iteratedFDeriv ℝ k (fun v => fourierPowSMulRight L f v n) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C

⊢ ‖iteratedFDeriv ℝ k (fun v => (-(2 * ↑π * I)) ^ n • ((smulRightL ℝ (fun x => W) E) ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v)) (f v)) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight

⊢ ‖iteratedFDeriv ℝ k (fun v => (-(2 * ↑π * I)) ^ n • ((smulRightL ℝ (fun x => W) E) ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v)) (f v)) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)

⊢ ‖iteratedFDeriv ℝ k (fun v => (-(2 * ↑π * I)) ^ n • ((smulRightL ℝ (fun x => W) E) ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v)) (f v)) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)

⊢ ‖iteratedFDeriv ℝ k (fun v => (-(2 * ↑π * I)) ^ n • ((smulRightL ℝ (fun x => W) E) ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v)) (f v)) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y

⊢ ‖iteratedFDeriv ℝ k (fun v => (-(2 * ↑π * I)) ^ n • ((smulRightL ℝ (fun x => W) E) ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v)) (f v)) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y

⊢ ‖(-(2 * ↑π * I)) ^ n‖ * ‖iteratedFDeriv ℝ k (fun x => ((smulRightL ℝ (fun x => W) E) (T fun x_1 => L x, f x).1) (T fun x_1 => L x, f x).2) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y

⊢ (2 * π) ^ n * ‖iteratedFDeriv ℝ k (fun x => ((smulRightL ℝ (fun x => W) E) (T fun x_1 => L x)) (f x)) v‖ ≤ (2 * π) ^ n * ((2 * ↑n + 2) ^ k * (‖L‖ ^ n * C))
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y

⊢ ‖iteratedFDeriv ℝ k (fun x => ((smulRightL ℝ (fun x => W) E) (T fun x_1 => L x)) (f x)) v‖ ≤ (2 * ↑n + 2) ^ k * (‖L‖ ^ n * C)
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y

⊢ ∑ i ∈ Finset.range (k + 1), ↑(k.choose i) * ‖iteratedFDeriv ℝ i (fun y => T fun x => L y) v‖ * ‖iteratedFDeriv ℝ (k - i) f v‖ ≤ (2 * ↑n + 2) ^ k * (‖L‖ ^ n * C)
-- Mathlib/Data/Real/Basic.lean
x✝ y : ℝ
C : ℝ → Prop
h : ∀ (y : CauSeq ℚ abs), C (mk y)
x : Cauchy abs

⊢ C { cauchy := x }
-- Mathlib/Data/Real/Basic.lean
x✝ y : ℝ
C : ℝ → Prop
h : ∀ (y : CauSeq ℚ abs), C (mk y)
x : CauSeq ℚ abs

⊢ C { cauchy := Quot.mk Setoid.r x }
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t

⊢ MeasurableSet (g ⁻¹' t)
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}

⊢ MeasurableSet (g ⁻¹' t)
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}

⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x})
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}

⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x})
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}
this : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}

⊢ MeasurableSet (f ⁻¹' t ∩ {x | f x = g x})
-- Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝⁸ : CommSemiring R
inst✝⁷ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝⁶ : (i : ι) → AddCommMonoid (Nᵢ i)
inst✝⁵ : AddCommMonoid P
inst✝⁴ : (i : ι) → Module R (Mᵢ i)
inst✝³ : (i : ι) → Module R (Nᵢ i)
inst✝² : Module R P
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
Q : QuadraticMap R (Mᵢ i) P
m : (i : ι) → Mᵢ i

⊢ Q (m i) = (pi (Pi.single i Q)) m
-- Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝⁸ : CommSemiring R
inst✝⁷ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝⁶ : (i : ι) → AddCommMonoid (Nᵢ i)
inst✝⁵ : AddCommMonoid P
inst✝⁴ : (i : ι) → Module R (Mᵢ i)
inst✝³ : (i : ι) → Module R (Nᵢ i)
inst✝² : Module R P
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
Q : QuadraticMap R (Mᵢ i) P
m : (i : ι) → Mᵢ i
j : ι
hij : j ≠ i

⊢ (Pi.single i Q j) (m j) = 0
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e

⊢ ((splitting K).ι (unop A.fst).len ≫ (obj K).map A.e.op) ≫ (obj K).map θ = Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e

⊢ (splitting K).ι (unop A.fst).len ≫ (obj K).map (A.e.op ≫ θ) = Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e

⊢ Sigma.ι (summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ map K (A.e.op ≫ θ) = Termwise.mapMono K i ≫ Sigma.ι (summand K (op Δ'')) (Splitting.IndexSet.id (op Δ'')) ≫ map K (Splitting.IndexSet.mk e).e.op
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
γ' : Cochain L M 0

⊢ (γ.comp γ' ⋯).leftShift a n' hn' = (γ.leftShift a n' hn').comp γ' ⋯
-- Mathlib/Order/Interval/Set/Pi.lean
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i

⊢ update f i '' Ico a (f i) = Ico (update f i a) f
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
a : A

⊢ ℜ (z • a) = z.re • ℜ a - z.im • ℑ a
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
a : A
this : ℜ (z • a) = ℜ ((↑z.re + ↑z.im * I) • a)

⊢ ℜ (z • a) = z.re • ℜ a - z.im • ℑ a
-- Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
h : ((J.toPlus P).app (op X)) x = ((J.toPlus P).app (op X)) y

⊢ x = y