ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mem_unitary_of_spectrum_subset_circle
lemma mem_unitary_of_spectrum_subset_circle {u : A}
    [IsStarNormal u] (hu : spectrum ℂ u ⊆ circle) : u ∈ unitary A 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {u : A} [IsStarNormal u] (hu : spectrum ℂ u ⊆ ↑circle)
Goal: u ∈ unitary A
-- spectrum_subset_circle_of_mem_unitary
lemma spectrum_subset_circle_of_mem_unitary {u : A} (hu : u ∈ unitary A) :
    spectrum ℂ u ⊆ circle 
Arguments: {A : Type u_1} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A] [ContinuousFunctionalCalculus ℂ IsStarNormal] {u : A} (hu : u ∈ unitary A)
Goal: spectrum ℂ u ⊆ ↑circle
-- InnerProductSpaceable.parallelogram_identity
class InnerProductSpaceable : Prop where
  parallelogram_identity :
    ∀ x y : E, ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)
Arguments: {E : Type u_2} [NormedAddCommGroup E] [self : InnerProductSpaceable E] (x y : E)
Goal: ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)
-- InnerProductSpace.toInnerProductSpaceable
theorem InnerProductSpace.toInnerProductSpaceable [InnerProductSpace 𝕜 E] :
    InnerProductSpaceable E 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: InnerProductSpaceable E
-- InnerProductSpaceable.innerProp_neg_one
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: InnerProductSpaceable.innerProp' E ↑(-1)
-- Continuous.inner_
theorem _root_.Continuous.inner_ {f g : ℝ → E} (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => inner_ 𝕜 (f x) (g x) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : ℝ → E} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => inner_ 𝕜 (f x) (g x)
-- InnerProductSpaceable.inner_.norm_sq
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: ‖x‖ ^ 2 = RCLike.re (inner_ 𝕜 x x)
-- InnerProductSpaceable.inner_.conj_symm
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x y : E)
Goal: (starRingEnd 𝕜) (inner_ 𝕜 y x) = inner_ 𝕜 x y
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux1
private theorem add_left_aux1 (x y z : E) : ‖x + y + z‖ * ‖x + y + z‖ =
    (‖2 • x + y‖ * ‖2 • x + y‖ + ‖2 • z + y‖ * ‖2 • z + y‖) / 2 - ‖x - z‖ * ‖x - z‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpaceable E] (x y z : E)
Goal: ‖x + y + z‖ * ‖x + y + z‖ = (‖2 • x + y‖ * ‖2 • x + y‖ + ‖2 • z + y‖ * ‖2 • z + y‖) / 2 - ‖x - z‖ * ‖x - z‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux2
private theorem add_left_aux2 (x y z : E) : ‖x + y - z‖ * ‖x + y - z‖ =
    (‖2 • x + y‖ * ‖2 • x + y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 - ‖x + z‖ * ‖x + z‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpaceable E] (x y z : E)
Goal: ‖x + y - z‖ * ‖x + y - z‖ = (‖2 • x + y‖ * ‖2 • x + y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 - ‖x + z‖ * ‖x + z‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux2'
private theorem add_left_aux2' (x y z : E) :
    ‖x + y + z‖ * ‖x + y + z‖ - ‖x + y - z‖ * ‖x + y - z‖ =
    ‖x + z‖ * ‖x + z‖ - ‖x - z‖ * ‖x - z‖ +
    (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpaceable E] (x y z : E)
Goal: ‖x + y + z‖ * ‖x + y + z‖ - ‖x + y - z‖ * ‖x + y - z‖ =
  ‖x + z‖ * ‖x + z‖ - ‖x - z‖ * ‖x - z‖ + (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux3
private theorem add_left_aux3 (y z : E) :
    ‖2 • z + y‖ * ‖2 • z + y‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpaceable E] (y z : E)
Goal: ‖2 • z + y‖ * ‖2 • z + y‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux4
private theorem add_left_aux4 (y z : E) :
    ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpaceable E] (y z : E)
Goal: ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux4'
private theorem add_left_aux4' (y z : E) :
    (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 =
    ‖y + z‖ * ‖y + z‖ - ‖y - z‖ * ‖y - z‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpaceable E] (y z : E)
Goal: (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 = ‖y + z‖ * ‖y + z‖ - ‖y - z‖ * ‖y - z‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux5
private theorem add_left_aux5 (x y z : E) :
    ‖(I : 𝕜) • (x + y) + z‖ * ‖(I : 𝕜) • (x + y) + z‖ =
    (‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖ +
    ‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖) / 2 -
    ‖(I : 𝕜) • x - z‖ * ‖(I : 𝕜) • x - z‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (x y z : E)
Goal: ‖RCLike.I • (x + y) + z‖ * ‖RCLike.I • (x + y) + z‖ =
  (‖RCLike.I • (2 • x + y)‖ * ‖RCLike.I • (2 • x + y)‖ + ‖RCLike.I • y + 2 • z‖ * ‖RCLike.I • y + 2 • z‖) / 2 -
    ‖RCLike.I • x - z‖ * ‖RCLike.I • x - z‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux6
private theorem add_left_aux6 (x y z : E) :
    ‖(I : 𝕜) • (x + y) - z‖ * ‖(I : 𝕜) • (x + y) - z‖ =
    (‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖ +
    ‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖) / 2 -
    ‖(I : 𝕜) • x + z‖ * ‖(I : 𝕜) • x + z‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (x y z : E)
Goal: ‖RCLike.I • (x + y) - z‖ * ‖RCLike.I • (x + y) - z‖ =
  (‖RCLike.I • (2 • x + y)‖ * ‖RCLike.I • (2 • x + y)‖ + ‖RCLike.I • y - 2 • z‖ * ‖RCLike.I • y - 2 • z‖) / 2 -
    ‖RCLike.I • x + z‖ * ‖RCLike.I • x + z‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux7
private theorem add_left_aux7 (y z : E) :
    ‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖ =
    2 * (‖(I : 𝕜) • y + z‖ * ‖(I : 𝕜) • y + z‖ + ‖z‖ * ‖z‖) - ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (y z : E)
Goal: ‖RCLike.I • y + 2 • z‖ * ‖RCLike.I • y + 2 • z‖ =
  2 * (‖RCLike.I • y + z‖ * ‖RCLike.I • y + z‖ + ‖z‖ * ‖z‖) - ‖RCLike.I • y‖ * ‖RCLike.I • y‖
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.add_left_aux8
private theorem add_left_aux8 (y z : E) :
    ‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖ =
    2 * (‖(I : 𝕜) • y - z‖ * ‖(I : 𝕜) • y - z‖ + ‖z‖ * ‖z‖) - ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (y z : E)
Goal: ‖RCLike.I • y - 2 • z‖ * ‖RCLike.I • y - 2 • z‖ =
  2 * (‖RCLike.I • y - z‖ * ‖RCLike.I • y - z‖ + ‖z‖ * ‖z‖) - ‖RCLike.I • y‖ * ‖RCLike.I • y‖
-- InnerProductSpaceable.add_left
theorem add_left (x y z : E) : inner_ 𝕜 (x + y) z = inner_ 𝕜 x z + inner_ 𝕜 y z 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (x y z : E)
Goal: inner_ 𝕜 (x + y) z = inner_ 𝕜 x z + inner_ 𝕜 y z
-- InnerProductSpaceable.nat
theorem nat (n : ℕ) (x y : E) : inner_ 𝕜 ((n : 𝕜) • x) y = (n : 𝕜) * inner_ 𝕜 x y 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (n : ℕ) (x y : E)
Goal: inner_ 𝕜 (↑n • x) y = ↑n * inner_ 𝕜 x y
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.nat_prop
private theorem nat_prop (r : ℕ) : innerProp' E (r : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (r : ℕ)
Goal: InnerProductSpaceable.innerProp' E ↑r
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.int_prop
private theorem int_prop (n : ℤ) : innerProp' E (n : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (n : ℤ)
Goal: InnerProductSpaceable.innerProp' E ↑n
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.rat_prop
private theorem rat_prop (r : ℚ) : innerProp' E (r : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (r : ℚ)
Goal: InnerProductSpaceable.innerProp' E ↑r
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.real_prop
private theorem real_prop (r : ℝ) : innerProp' E (r : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (r : ℝ)
Goal: InnerProductSpaceable.innerProp' E ↑r
-- _private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.I_prop
private theorem I_prop : innerProp' E (I : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E]
Goal: InnerProductSpaceable.innerProp' E RCLike.I
-- InnerProductSpaceable.innerProp
theorem innerProp (r : 𝕜) : innerProp' E r 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E] (r : 𝕜)
Goal: InnerProductSpaceable.innerProp' E r
-- nonempty_innerProductSpace
theorem nonempty_innerProductSpace : Nonempty (InnerProductSpace 𝕜 E) 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜] (E : Type u_2) [NormedAddCommGroup E] [NormedSpace 𝕜 E] [InnerProductSpaceable E]
Goal: Nonempty (InnerProductSpace 𝕜 E)
-- selfAdjoint.expUnitary_coe
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A] [CompleteSpace A] [StarModule ℂ A] (a : ↥(selfAdjoint A))
Goal: ↑(selfAdjoint.expUnitary a) = NormedSpace.exp ℂ (Complex.I • ↑a)
-- Commute.expUnitary_add
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
    expUnitary (a + b) = expUnitary a * expUnitary b 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A] [CompleteSpace A] [StarModule ℂ A] {a b : ↥(selfAdjoint A)} (h : Commute ↑a ↑b)
Goal: selfAdjoint.expUnitary (a + b) = selfAdjoint.expUnitary a * selfAdjoint.expUnitary b
-- Commute.expUnitary
theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
    Commute (expUnitary a) (expUnitary b) 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A] [CompleteSpace A] [StarModule ℂ A] {a b : ↥(selfAdjoint A)} (h : Commute ↑a ↑b)
Goal: Commute (selfAdjoint.expUnitary a) (selfAdjoint.expUnitary b)
-- ae_eq_zero_of_integral_smooth_smul_eq_zero
theorem ae_eq_zero_of_integral_smooth_smul_eq_zero (hf : LocallyIntegrable f μ)
    (h : ∀ g : M → ℝ, Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, f x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [MeasurableSpace M] [BorelSpace M] [SigmaCompactSpace M] [T2Space M] {f : M → F} {μ : MeasureTheory.Measure M} (hf : MeasureTheory.LocallyIntegrable f μ) (h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0)
Goal: ∀ᵐ (x : M) ∂μ, f x = 0
-- IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero'
nonrec theorem IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero' {U : Set M} (hU : IsOpen U)
    (hSig : IsSigmaCompact U) (hf : LocallyIntegrableOn f U μ)
    (h : ∀ g : M → ℝ,
      Smooth I 𝓘(ℝ) g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, x ∈ U → f x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [MeasurableSpace M] [BorelSpace M] [T2Space M] {f : M → F} {μ : MeasureTheory.Measure M} {U : Set M} (hU : IsOpen U) (hSig : IsSigmaCompact U) (hf : MeasureTheory.LocallyIntegrableOn f U μ) (h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → tsupport g ⊆ U → ∫ (x : M), g x • f x ∂μ = 0)
Goal: ∀ᵐ (x : M) ∂μ, x ∈ U → f x = 0
-- IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero
theorem IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero {U : Set M} (hU : IsOpen U)
    (hf : LocallyIntegrableOn f U μ)
    (h : ∀ g : M → ℝ,
      Smooth I 𝓘(ℝ) g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, x ∈ U → f x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [MeasurableSpace M] [BorelSpace M] [SigmaCompactSpace M] [T2Space M] {f : M → F} {μ : MeasureTheory.Measure M} {U : Set M} (hU : IsOpen U) (hf : MeasureTheory.LocallyIntegrableOn f U μ) (h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → tsupport g ⊆ U → ∫ (x : M), g x • f x ∂μ = 0)
Goal: ∀ᵐ (x : M) ∂μ, x ∈ U → f x = 0
-- ae_eq_of_integral_smooth_smul_eq
theorem ae_eq_of_integral_smooth_smul_eq
    (hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ),
      Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :
    ∀ᵐ x ∂μ, f x = f' x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [MeasurableSpace M] [BorelSpace M] [SigmaCompactSpace M] [T2Space M] {f f' : M → F} {μ : MeasureTheory.Measure M} (hf : MeasureTheory.LocallyIntegrable f μ) (hf' : MeasureTheory.LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = ∫ (x : M), g x • f' x ∂μ)
Goal: ∀ᵐ (x : M) ∂μ, f x = f' x
-- ae_eq_zero_of_integral_contDiff_smul_eq_zero
theorem ae_eq_zero_of_integral_contDiff_smul_eq_zero (hf : LocallyIntegrable f μ)
    (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, f x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [MeasurableSpace E] [BorelSpace E] {f : E → F} {μ : MeasureTheory.Measure E} (hf : MeasureTheory.LocallyIntegrable f μ) (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → ∫ (x : E), g x • f x ∂μ = 0)
Goal: ∀ᵐ (x : E) ∂μ, f x = 0
-- ae_eq_of_integral_contDiff_smul_eq
theorem ae_eq_of_integral_contDiff_smul_eq
    (hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : E → ℝ),
      ContDiff ℝ ⊤ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :
    ∀ᵐ x ∂μ, f x = f' x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [MeasurableSpace E] [BorelSpace E] {f f' : E → F} {μ : MeasureTheory.Measure E} (hf : MeasureTheory.LocallyIntegrable f μ) (hf' : MeasureTheory.LocallyIntegrable f' μ) (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → ∫ (x : E), g x • f x ∂μ = ∫ (x : E), g x • f' x ∂μ)
Goal: ∀ᵐ (x : E) ∂μ, f x = f' x
-- IsOpen.ae_eq_zero_of_integral_contDiff_smul_eq_zero
theorem IsOpen.ae_eq_zero_of_integral_contDiff_smul_eq_zero {U : Set E} (hU : IsOpen U)
    (hf : LocallyIntegrableOn f U μ)
    (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → tsupport g ⊆ U →
        ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, x ∈ U → f x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [MeasurableSpace E] [BorelSpace E] {f : E → F} {μ : MeasureTheory.Measure E} {U : Set E} (hU : IsOpen U) (hf : MeasureTheory.LocallyIntegrableOn f U μ) (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → tsupport g ⊆ U → ∫ (x : E), g x • f x ∂μ = 0)
Goal: ∀ᵐ (x : E) ∂μ, x ∈ U → f x = 0
-- MellinConvergent.const_smul
theorem MellinConvergent.const_smul {f : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) {𝕜 : Type*}
    [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜) :
    MellinConvergent (fun t => c • f t) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜)
Goal: MellinConvergent (fun t => c • f t) s
-- MellinConvergent.cpow_smul
theorem MellinConvergent.cpow_smul {f : ℝ → E} {s a : ℂ} :
    MellinConvergent (fun t => (t : ℂ) ^ a • f t) s ↔ MellinConvergent f (s + a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {s a : ℂ}
Goal: MellinConvergent (fun t => ↑t ^ a • f t) s ↔ MellinConvergent f (s + a)
-- MellinConvergent.div_const
nonrec theorem MellinConvergent.div_const {f : ℝ → ℂ} {s : ℂ} (hf : MellinConvergent f s) (a : ℂ) :
    MellinConvergent (fun t => f t / a) s 
Arguments: {f : ℝ → ℂ} {s : ℂ} (hf : MellinConvergent f s) (a : ℂ)
Goal: MellinConvergent (fun t => f t / a) s
-- MellinConvergent.comp_mul_left
theorem MellinConvergent.comp_mul_left {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : 0 < a) :
    MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : 0 < a)
Goal: MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s
-- MellinConvergent.comp_rpow
theorem MellinConvergent.comp_rpow {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : a ≠ 0) :
    MellinConvergent (fun t => f (t ^ a)) s ↔ MellinConvergent f (s / a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : a ≠ 0)
Goal: MellinConvergent (fun t => f (t ^ a)) s ↔ MellinConvergent f (s / ↑a)
-- mellin_cpow_smul
theorem mellin_cpow_smul (f : ℝ → E) (s a : ℂ) :
    mellin (fun t => (t : ℂ) ^ a • f t) s = mellin f (s + a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (s a : ℂ)
Goal: mellin (fun t => ↑t ^ a • f t) s = mellin f (s + a)
-- mellin_const_smul
theorem mellin_const_smul (f : ℝ → E) (s : ℂ) {𝕜 : Type*} [NontriviallyNormedField 𝕜]
    [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜) :
    mellin (fun t => c • f t) s = c • mellin f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (s : ℂ) {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜)
Goal: mellin (fun t => c • f t) s = c • mellin f s
-- mellin_div_const
theorem mellin_div_const (f : ℝ → ℂ) (s a : ℂ) : mellin (fun t => f t / a) s = mellin f s / a 
Arguments: (f : ℝ → ℂ) (s a : ℂ)
Goal: mellin (fun t => f t / a) s = mellin f s / a
-- mellin_comp_rpow
theorem mellin_comp_rpow (f : ℝ → E) (s : ℂ) (a : ℝ) :
    mellin (fun t => f (t ^ a)) s = |a|⁻¹ • mellin f (s / a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (s : ℂ) (a : ℝ)
Goal: mellin (fun t => f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)
-- mellin_comp_mul_left
theorem mellin_comp_mul_left (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
    mellin (fun t => f (a * t)) s = (a : ℂ) ^ (-s) • mellin f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a)
Goal: mellin (fun t => f (a * t)) s = ↑a ^ (-s) • mellin f s
-- mellin_comp_mul_right
theorem mellin_comp_mul_right (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
    mellin (fun t => f (t * a)) s = (a : ℂ) ^ (-s) • mellin f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a)
Goal: mellin (fun t => f (t * a)) s = ↑a ^ (-s) • mellin f s
-- mellin_comp_inv
theorem mellin_comp_inv (f : ℝ → E) (s : ℂ) : mellin (fun t => f t⁻¹) s = mellin f (-s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (s : ℂ)
Goal: mellin (fun t => f t⁻¹) s = mellin f (-s)
-- hasMellin_add
theorem hasMellin_add {f g : ℝ → E} {s : ℂ} (hf : MellinConvergent f s)
    (hg : MellinConvergent g s) : HasMellin (fun t => f t + g t) s (mellin f s + mellin g s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) (hg : MellinConvergent g s)
Goal: HasMellin (fun t => f t + g t) s (mellin f s + mellin g s)
-- hasMellin_sub
theorem hasMellin_sub {f g : ℝ → E} {s : ℂ} (hf : MellinConvergent f s)
    (hg : MellinConvergent g s) : HasMellin (fun t => f t - g t) s (mellin f s - mellin g s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) (hg : MellinConvergent g s)
Goal: HasMellin (fun t => f t - g t) s (mellin f s - mellin g s)
-- mellin_convergent_iff_norm
theorem mellin_convergent_iff_norm [NormedSpace ℂ E] {f : ℝ → E} {T : Set ℝ} (hT : T ⊆ Ioi 0)
    (hT' : MeasurableSet T) (hfc : AEStronglyMeasurable f <| volume.restrict <| Ioi 0) {s : ℂ} :
    IntegrableOn (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) T ↔
      IntegrableOn (fun t : ℝ => t ^ (s.re - 1) * ‖f t‖) T 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {T : Set ℝ} (hT : T ⊆ Set.Ioi 0) (hT' : MeasurableSet T) (hfc : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.volume.restrict (Set.Ioi 0))) {s : ℂ}
Goal: MeasureTheory.IntegrableOn (fun t => ↑t ^ (s - 1) • f t) T MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn (fun t => t ^ (s.re - 1) * ‖f t‖) T MeasureTheory.volume
-- mellin_convergent_top_of_isBigO
theorem mellin_convergent_top_of_isBigO {f : ℝ → ℝ}
    (hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0)) {a s : ℝ}
    (hf : f =O[atTop] (· ^ (-a))) (hs : s < a) :
    ∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioi c) 
Arguments: {f : ℝ → ℝ} (hfc : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.volume.restrict (Set.Ioi 0))) {a s : ℝ} (hf : f =O[Filter.atTop] fun x => x ^ (-a)) (hs : s < a)
Goal: ∃ c, 0 < c ∧ MeasureTheory.IntegrableOn (fun t => t ^ (s - 1) * f t) (Set.Ioi c) MeasureTheory.volume
-- mellin_convergent_zero_of_isBigO
theorem mellin_convergent_zero_of_isBigO {b : ℝ} {f : ℝ → ℝ}
    (hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0))
    (hf :  f =O[𝓝[>] 0] (· ^ (-b))) {s : ℝ} (hs : b < s) :
    ∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioc 0 c) 
Arguments: {b : ℝ} {f : ℝ → ℝ} (hfc : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.volume.restrict (Set.Ioi 0))) (hf : f =O[𝓝[>] 0] fun x => x ^ (-b)) {s : ℝ} (hs : b < s)
Goal: ∃ c, 0 < c ∧ MeasureTheory.IntegrableOn (fun t => t ^ (s - 1) * f t) (Set.Ioc 0 c) MeasureTheory.volume