ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- VectorFourier.fderiv_fourierIntegral
lemma fderiv_fourierIntegral
    [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : Measure V}
    (hf : Integrable f μ) (hf' : Integrable (fun v : V ↦ ‖v‖ * ‖f v‖) μ) :
    fderiv ℝ (fourierIntegral 𝐞 μ L.toLinearMap₂ f) =
      fourierIntegral 𝐞 μ L.toLinearMap₂ (fourierSMulRight L f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : MeasureTheory.Measure V} (hf : MeasureTheory.Integrable f μ) (hf' : MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) μ)
Goal: fderiv ℝ (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f) =
  VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (VectorFourier.fourierSMulRight L f)
-- VectorFourier.differentiable_fourierIntegral
lemma differentiable_fourierIntegral
    [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : Measure V}
    (hf : Integrable f μ) (hf' : Integrable (fun v : V ↦ ‖v‖ * ‖f v‖) μ) :
    Differentiable ℝ (fourierIntegral 𝐞 μ L.toLinearMap₂ f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : MeasureTheory.Measure V} (hf : MeasureTheory.Integrable f μ) (hf' : MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) μ)
Goal: Differentiable ℝ (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f)
-- VectorFourier.fourierIntegral_fderiv
theorem fourierIntegral_fderiv [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V]
    {μ : Measure V} [Measure.IsAddHaarMeasure μ]
    (hf : Integrable f μ) (h'f : Differentiable ℝ f) (hf' : Integrable (fderiv ℝ f) μ) :
    fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f)
      = fourierSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [μ.IsAddHaarMeasure] (hf : MeasureTheory.Integrable f μ) (h'f : Differentiable ℝ f) (hf' : MeasureTheory.Integrable (fderiv ℝ f) μ)
Goal: VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (fderiv ℝ f) =
  VectorFourier.fourierSMulRight (-L.flip) (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f)
-- VectorFourier.fourierPowSMulRight_apply
@[simp 1100] lemma fourierPowSMulRight_apply {f : V → E} {v : V} {n : ℕ} {m : Fin n → W} :
    fourierPowSMulRight L f v n m = (- (2 * π * I))^n • (∏ i, L v (m i)) • f v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} {v : V} {n : ℕ} {m : Fin n → W}
Goal: (VectorFourier.fourierPowSMulRight L f v n) m = (-(2 * ↑π * Complex.I)) ^ n • (∏ i : Fin n, (L v) (m i)) • f v
-- VectorFourier.fourierPowSMulRight_eq_comp
lemma fourierPowSMulRight_eq_comp {f : V → E} {v : V} {n : ℕ} :
    fourierPowSMulRight L f v n = (- (2 * π * I))^n • smulRightL ℝ (fun (_ : Fin n) ↦ W) E
      (compContinuousLinearMapLRight
        (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ) (fun _ ↦ L v)) (f v) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} {v : V} {n : ℕ}
Goal: VectorFourier.fourierPowSMulRight L f v n =
  (-(2 * ↑π * Complex.I)) ^ n •
    ((ContinuousMultilinearMap.smulRightL ℝ (fun x => W) E)
        ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v))
      (f v)
-- Continuous.fourierPowSMulRight
@[continuity, fun_prop]
lemma _root_.Continuous.fourierPowSMulRight {f : V → E} (hf : Continuous f) (n : ℕ) :
    Continuous (fun v ↦ fourierPowSMulRight L f v n) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} (hf : Continuous f) (n : ℕ)
Goal: Continuous fun v => VectorFourier.fourierPowSMulRight L f v n
-- ContDiff.fourierPowSMulRight
lemma _root_.ContDiff.fourierPowSMulRight {f : V → E} {k : ℕ∞} (hf : ContDiff ℝ k f) (n : ℕ) :
    ContDiff ℝ k (fun v ↦ fourierPowSMulRight L f v n) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} {k : ℕ∞} (hf : ContDiff ℝ k f) (n : ℕ)
Goal: ContDiff ℝ k fun v => VectorFourier.fourierPowSMulRight L f v n
-- VectorFourier.norm_fourierPowSMulRight_le
lemma norm_fourierPowSMulRight_le (f : V → E) (v : V) (n : ℕ) :
    ‖fourierPowSMulRight L f v n‖ ≤ (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (f : V → E) (v : V) (n : ℕ)
Goal: ‖VectorFourier.fourierPowSMulRight L f v n‖ ≤ (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖
-- VectorFourier.norm_iteratedFDeriv_fourierPowSMulRight
lemma norm_iteratedFDeriv_fourierPowSMulRight
    {f : V → E} {K : ℕ∞} {C : ℝ} (hf : ContDiff ℝ K f) {n : 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 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {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)
Goal: ‖iteratedFDeriv ℝ k (fun v => VectorFourier.fourierPowSMulRight L f v n) v‖ ≤
  (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- MeasureTheory.AEStronglyMeasurable.fourierPowSMulRight
lemma _root_.MeasureTheory.AEStronglyMeasurable.fourierPowSMulRight
    (hf : AEStronglyMeasurable f μ) (n : ℕ) :
    AEStronglyMeasurable (fun v ↦ fourierPowSMulRight L f v n) μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} (hf : MeasureTheory.AEStronglyMeasurable f μ) (n : ℕ)
Goal: MeasureTheory.AEStronglyMeasurable (fun v => VectorFourier.fourierPowSMulRight L f v n) μ
-- VectorFourier.integrable_fourierPowSMulRight
lemma integrable_fourierPowSMulRight {n : ℕ} (hf : Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) μ)
    (h'f : AEStronglyMeasurable f μ) : Integrable (fun v ↦ fourierPowSMulRight L f v n) μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {n : ℕ} (hf : MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ) (h'f : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.Integrable (fun v => VectorFourier.fourierPowSMulRight L f v n) μ
-- VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral
lemma hasFTaylorSeriesUpTo_fourierIntegral {N : ℕ∞}
    (hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖) μ)
    (h'f : AEStronglyMeasurable f μ) :
    HasFTaylorSeriesUpTo N (fourierIntegral 𝐞 μ L.toLinearMap₂ f)
      (fun w n ↦ fourierIntegral 𝐞 μ L.toLinearMap₂ (fun v ↦ fourierPowSMulRight L f v n) w) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {N : ℕ∞} (hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ) (h'f : MeasureTheory.AEStronglyMeasurable f μ)
Goal: HasFTaylorSeriesUpTo N (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f) fun w n =>
  VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (fun v => VectorFourier.fourierPowSMulRight L f v n) w
-- VectorFourier.contDiff_fourierIntegral
theorem contDiff_fourierIntegral {N : ℕ∞}
    (hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖) μ) :
    ContDiff ℝ N (fourierIntegral 𝐞 μ L.toLinearMap₂ f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {N : ℕ∞} (hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ)
Goal: ContDiff ℝ N (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f)
-- VectorFourier.iteratedFDeriv_fourierIntegral
lemma iteratedFDeriv_fourierIntegral {N : ℕ∞}
    (hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖) μ)
    (h'f : AEStronglyMeasurable f μ) {n : ℕ} (hn : n ≤ N) :
    iteratedFDeriv ℝ n (fourierIntegral 𝐞 μ L.toLinearMap₂ f) =
      fourierIntegral 𝐞 μ L.toLinearMap₂ (fun v ↦ fourierPowSMulRight L f v n) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {N : ℕ∞} (hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ) (h'f : MeasureTheory.AEStronglyMeasurable f μ) {n : ℕ} (hn : ↑n ≤ N)
Goal: iteratedFDeriv ℝ n (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f) =
  VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ fun v => VectorFourier.fourierPowSMulRight L f v n
-- VectorFourier.fourierIntegral_iteratedFDeriv
theorem fourierIntegral_iteratedFDeriv [FiniteDimensional ℝ V]
    {μ : Measure V} [Measure.IsAddHaarMeasure μ] {N : ℕ∞} (hf : ContDiff ℝ N f)
    (h'f : ∀ (n : ℕ), n ≤ N → Integrable (iteratedFDeriv ℝ n f) μ) {n : ℕ} (hn : n ≤ N) :
    fourierIntegral 𝐞 μ L.toLinearMap₂ (iteratedFDeriv ℝ n f)
      = (fun w ↦ fourierPowSMulRight (-L.flip) (fourierIntegral 𝐞 μ L.toLinearMap₂ f) w n) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [μ.IsAddHaarMeasure] {N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedFDeriv ℝ n f) μ) {n : ℕ} (hn : ↑n ≤ N)
Goal: VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (iteratedFDeriv ℝ n f) = fun w =>
  VectorFourier.fourierPowSMulRight (-L.flip) (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f) w n
-- VectorFourier.fourierPowSMulRight_iteratedFDeriv_fourierIntegral
theorem fourierPowSMulRight_iteratedFDeriv_fourierIntegral [FiniteDimensional ℝ V]
    {μ : Measure V} [Measure.IsAddHaarMeasure μ] {K N : ℕ∞} (hf : ContDiff ℝ N f)
    (h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖) μ)
    {k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) {w : W} :
    fourierPowSMulRight (-L.flip)
      (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w n =
    fourierIntegral 𝐞 μ L.toLinearMap₂
      (iteratedFDeriv ℝ n (fun v ↦ fourierPowSMulRight L f v k)) w 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [μ.IsAddHaarMeasure] {K N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ) {k n : ℕ} (hk : ↑k ≤ K) (hn : ↑n ≤ N) {w : W}
Goal: VectorFourier.fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w
    n =
  VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂
    (iteratedFDeriv ℝ n fun v => VectorFourier.fourierPowSMulRight L f v k) w
-- VectorFourier.norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le
theorem norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le [FiniteDimensional ℝ V]
    {μ : Measure V} [Measure.IsAddHaarMeasure μ] {K N : ℕ∞} (hf : ContDiff ℝ N f)
    (h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖) μ)
    {k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) {w : W} :
    ‖fourierPowSMulRight (-L.flip)
      (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w n‖ ≤
    (2 * π) ^ k * (2 * k + 2) ^ n * ‖L‖ ^ k * ∑ p in Finset.range (k + 1) ×ˢ Finset.range (n + 1),
      ∫ v, ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [μ.IsAddHaarMeasure] {K N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ) {k n : ℕ} (hk : ↑k ≤ K) (hn : ↑n ≤ N) {w : W}
Goal: ‖VectorFourier.fourierPowSMulRight (-L.flip) (iteratedFDeriv ℝ k (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w
      n‖ ≤
  (2 * π) ^ k * (2 * ↑k + 2) ^ n * ‖L‖ ^ k *
    ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ
-- VectorFourier.pow_mul_norm_iteratedFDeriv_fourierIntegral_le
lemma pow_mul_norm_iteratedFDeriv_fourierIntegral_le [FiniteDimensional ℝ V]
    {μ : Measure V} [Measure.IsAddHaarMeasure μ] {K N : ℕ∞} (hf : ContDiff ℝ N f)
    (h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖) μ)
    {k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) (v : V) (w : W) :
    |L v w| ^ n * ‖(iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w‖ ≤
      ‖v‖ ^ n * (2 * π * ‖L‖) ^ k * (2 * k + 2) ^ n *
        ∑ p in Finset.range (k + 1) ×ˢ Finset.range (n + 1),
          ∫ v, ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [μ.IsAddHaarMeasure] {K N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ) {k n : ℕ} (hk : ↑k ≤ K) (hn : ↑n ≤ N) (v : V) (w : W)
Goal: |(L v) w| ^ n * ‖iteratedFDeriv ℝ k (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f) w‖ ≤
  ‖v‖ ^ n * (2 * π * ‖L‖) ^ k * (2 * ↑k + 2) ^ n *
    ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ
-- Real.hasFDerivAt_fourierIntegral
theorem hasFDerivAt_fourierIntegral
    (hf_int : Integrable f) (hvf_int : Integrable (fun v ↦ ‖v‖ * ‖f v‖)) (x : V) :
    HasFDerivAt (𝓕 f) (𝓕 (fourierSMulRight (innerSL ℝ) f) x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} (hf_int : MeasureTheory.Integrable f MeasureTheory.volume) (hvf_int : MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) MeasureTheory.volume) (x : V)
Goal: HasFDerivAt (𝓕 f) (𝓕 (VectorFourier.fourierSMulRight (innerSL ℝ) f) x) x
-- Real.fderiv_fourierIntegral
theorem fderiv_fourierIntegral
    (hf_int : Integrable f) (hvf_int : Integrable (fun v ↦ ‖v‖ * ‖f v‖)) :
    fderiv ℝ (𝓕 f) = 𝓕 (fourierSMulRight (innerSL ℝ) f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} (hf_int : MeasureTheory.Integrable f MeasureTheory.volume) (hvf_int : MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) MeasureTheory.volume)
Goal: fderiv ℝ (𝓕 f) = 𝓕 (VectorFourier.fourierSMulRight (innerSL ℝ) f)
-- Real.differentiable_fourierIntegral
theorem differentiable_fourierIntegral
    (hf_int : Integrable f) (hvf_int : Integrable (fun v ↦ ‖v‖ * ‖f v‖)) :
    Differentiable ℝ (𝓕 f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} (hf_int : MeasureTheory.Integrable f MeasureTheory.volume) (hvf_int : MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) MeasureTheory.volume)
Goal: Differentiable ℝ (𝓕 f)
-- Real.fourierIntegral_fderiv
theorem fourierIntegral_fderiv
    (hf : Integrable f) (h'f : Differentiable ℝ f) (hf' : Integrable (fderiv ℝ f)) :
    𝓕 (fderiv ℝ f) = fourierSMulRight (-innerSL ℝ) (𝓕 f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : Differentiable ℝ f) (hf' : MeasureTheory.Integrable (fderiv ℝ f) MeasureTheory.volume)
Goal: 𝓕 (fderiv ℝ f) = VectorFourier.fourierSMulRight (-innerSL ℝ) (𝓕 f)
-- Real.contDiff_fourierIntegral
theorem contDiff_fourierIntegral {N : ℕ∞}
    (hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖)) :
    ContDiff ℝ N (𝓕 f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} {N : ℕ∞} (hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) MeasureTheory.volume)
Goal: ContDiff ℝ N (𝓕 f)
-- Real.iteratedFDeriv_fourierIntegral
theorem iteratedFDeriv_fourierIntegral {N : ℕ∞}
    (hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖))
    (h'f : AEStronglyMeasurable f) {n : ℕ} (hn : n ≤ N) :
    iteratedFDeriv ℝ n (𝓕 f) = 𝓕 (fun v ↦ fourierPowSMulRight (innerSL ℝ) f v n) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} {N : ℕ∞} (hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) MeasureTheory.volume) (h'f : MeasureTheory.AEStronglyMeasurable f MeasureTheory.volume) {n : ℕ} (hn : ↑n ≤ N)
Goal: iteratedFDeriv ℝ n (𝓕 f) = 𝓕 fun v => VectorFourier.fourierPowSMulRight (innerSL ℝ) f v n
-- Real.fourierIntegral_iteratedFDeriv
theorem fourierIntegral_iteratedFDeriv {N : ℕ∞} (hf : ContDiff ℝ N f)
    (h'f : ∀ (n : ℕ), n ≤ N → Integrable (iteratedFDeriv ℝ n f)) {n : ℕ} (hn : n ≤ N) :
    𝓕 (iteratedFDeriv ℝ n f)
      = (fun w ↦ fourierPowSMulRight (-innerSL ℝ) (𝓕 f) w n) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} {N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedFDeriv ℝ n f) MeasureTheory.volume) {n : ℕ} (hn : ↑n ≤ N)
Goal: 𝓕 (iteratedFDeriv ℝ n f) = fun w => VectorFourier.fourierPowSMulRight (-innerSL ℝ) (𝓕 f) w n
-- Real.pow_mul_norm_iteratedFDeriv_fourierIntegral_le
lemma pow_mul_norm_iteratedFDeriv_fourierIntegral_le [FiniteDimensional ℝ V]
    {K N : ℕ∞} (hf : ContDiff ℝ N f)
    (h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖))
    {k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) (w : V) :
    ‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖ ≤ (2 * π) ^ k * (2 * k + 2) ^ n *
      ∑ p in Finset.range (k + 1) ×ˢ Finset.range (n + 1),
        ∫ v, ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} [FiniteDimensional ℝ V] {K N : ℕ∞} (hf : ContDiff ℝ N f) (h'f :
  ∀ (k n : ℕ),
    ↑k ≤ K → ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) MeasureTheory.volume) {k n : ℕ} (hk : ↑k ≤ K) (hn : ↑n ≤ N) (w : V)
Goal: ‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖ ≤
  (2 * π) ^ k * (2 * ↑k + 2) ^ n *
    ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖
-- Real.hasDerivAt_fourierIntegral
lemma hasDerivAt_fourierIntegral
    {f : ℝ → E} (hf : Integrable f) (hf' : Integrable (fun x : ℝ ↦ x • f x)) (w : ℝ) :
    HasDerivAt (𝓕 f) (𝓕 (fun x : ℝ ↦ (-2 * π * I * x) • f x) w) w 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (hf' : MeasureTheory.Integrable (fun x => x • f x) MeasureTheory.volume) (w : ℝ)
Goal: HasDerivAt (𝓕 f) (𝓕 (fun x => (-2 * ↑π * Complex.I * ↑x) • f x) w) w
-- Real.deriv_fourierIntegral
theorem deriv_fourierIntegral
    {f : ℝ → E} (hf : Integrable f) (hf' : Integrable (fun x : ℝ ↦ x • f x)) :
    deriv (𝓕 f) = 𝓕 (fun x : ℝ ↦ (-2 * π * I * x) • f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (hf' : MeasureTheory.Integrable (fun x => x • f x) MeasureTheory.volume)
Goal: deriv (𝓕 f) = 𝓕 fun x => (-2 * ↑π * Complex.I * ↑x) • f x
-- Real.fourierIntegral_deriv
theorem fourierIntegral_deriv
    {f : ℝ → E} (hf : Integrable f) (h'f : Differentiable ℝ f) (hf' : Integrable (deriv f)) :
    𝓕 (deriv f) = fun (x : ℝ) ↦ (2 * π * I * x) • (𝓕 f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : Differentiable ℝ f) (hf' : MeasureTheory.Integrable (deriv f) MeasureTheory.volume)
Goal: 𝓕 (deriv f) = fun x => (2 * ↑π * Complex.I * ↑x) • 𝓕 f x
-- Real.iteratedDeriv_fourierIntegral
theorem iteratedDeriv_fourierIntegral {f : ℝ → E} {N : ℕ∞} {n : ℕ}
    (hf : ∀ (n : ℕ), n ≤ N → Integrable (fun x ↦ x^n • f x)) (hn : n ≤ N) :
    iteratedDeriv n (𝓕 f) = 𝓕 (fun x : ℝ ↦ (-2 * π * I * x) ^ n • f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {N : ℕ∞} {n : ℕ} (hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun x => x ^ n • f x) MeasureTheory.volume) (hn : ↑n ≤ N)
Goal: iteratedDeriv n (𝓕 f) = 𝓕 fun x => (-2 * ↑π * Complex.I * ↑x) ^ n • f x
-- Real.fourierIntegral_iteratedDeriv
theorem fourierIntegral_iteratedDeriv {f : ℝ → E} {N : ℕ∞} {n : ℕ} (hf : ContDiff ℝ N f)
    (h'f : ∀ (n : ℕ), n ≤ N → Integrable (iteratedDeriv n f)) (hn : n ≤ N) :
    𝓕 (iteratedDeriv n f) = fun (x : ℝ) ↦ (2 * π * I * x) ^ n • (𝓕 f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {N : ℕ∞} {n : ℕ} (hf : ContDiff ℝ N f) (h'f : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedDeriv n f) MeasureTheory.volume) (hn : ↑n ≤ N)
Goal: 𝓕 (iteratedDeriv n f) = fun x => (2 * ↑π * Complex.I * ↑x) ^ n • 𝓕 f x
-- mem_circle_iff_abs
@[simp]
theorem mem_circle_iff_abs {z : ℂ} : z ∈ circle ↔ abs z = 1 
Arguments: {z : ℂ}
Goal: z ∈ circle ↔ Complex.abs z = 1
-- circle_def
theorem circle_def : ↑circle = { z : ℂ | abs z = 1 } 
Goal: ↑circle = {z | Complex.abs z = 1}