ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- VectorFourier.fourierIntegral_continuousLinearMap_apply
theorem fourierIntegral_continuousLinearMap_apply
    {f : V → (F →L[ℝ] E)} {a : F} {w : W} (he : Continuous e) (hf : Integrable f μ) :
    fourierIntegral e μ L.toLinearMap₂ f w a =
      fourierIntegral e μ L.toLinearMap₂ (fun x ↦ f x a) w 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} {V : Type u_5} {W : Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace 𝕜 W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →L[𝕜] W →L[𝕜] 𝕜} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → F →L[ℝ] E} {a : F} {w : W} (he : Continuous ⇑e) (hf : MeasureTheory.Integrable f μ)
Goal: (VectorFourier.fourierIntegral e μ L.toLinearMap₂ f w) a =
  VectorFourier.fourierIntegral e μ L.toLinearMap₂ (fun x => (f x) a) w
-- VectorFourier.fourierIntegral_continuousMultilinearMap_apply
theorem fourierIntegral_continuousMultilinearMap_apply
    {f : V → (ContinuousMultilinearMap ℝ M E)} {m : (i : ι) → M i} {w : W} (he : Continuous e)
    (hf : Integrable f μ) :
    fourierIntegral e μ L.toLinearMap₂ f w m =
      fourierIntegral e μ L.toLinearMap₂ (fun x ↦ f x m) w 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {V : Type u_5} {W : Type u_6} [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace 𝕜 W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →L[𝕜] W →L[𝕜] 𝕜} [NormedAddCommGroup E] [NormedSpace ℂ E] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace ℝ (M i)] {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {w : W} (he : Continuous ⇑e) (hf : MeasureTheory.Integrable f μ)
Goal: (VectorFourier.fourierIntegral e μ L.toLinearMap₂ f w) m =
  VectorFourier.fourierIntegral e μ L.toLinearMap₂ (fun x => (f x) m) w
-- Fourier.fourierIntegral_def
theorem fourierIntegral_def (e : AddChar 𝕜 𝕊) (μ : Measure 𝕜) (f : 𝕜 → E) (w : 𝕜) :
    fourierIntegral e μ f w = ∫ v : 𝕜, e (-(v * w)) • f v ∂μ 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (w : 𝕜)
Goal: Fourier.fourierIntegral e μ f w = ∫ (v : 𝕜), e (-(v * w)) • f v ∂μ
-- Fourier.fourierIntegral_smul_const
theorem fourierIntegral_smul_const (e : AddChar 𝕜 𝕊) (μ : Measure 𝕜) (f : 𝕜 → E) (r : ℂ) :
    fourierIntegral e μ (r • f) = r • fourierIntegral e μ f 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (r : ℂ)
Goal: Fourier.fourierIntegral e μ (r • f) = r • Fourier.fourierIntegral e μ f
-- Fourier.norm_fourierIntegral_le_integral_norm
theorem norm_fourierIntegral_le_integral_norm (e : AddChar 𝕜 𝕊) (μ : Measure 𝕜)
    (f : 𝕜 → E) (w : 𝕜) : ‖fourierIntegral e μ f w‖ ≤ ∫ x : 𝕜, ‖f x‖ ∂μ 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (w : 𝕜)
Goal: ‖Fourier.fourierIntegral e μ f w‖ ≤ ∫ (x : 𝕜), ‖f x‖ ∂μ
-- Fourier.fourierIntegral_comp_add_right
theorem fourierIntegral_comp_add_right [MeasurableAdd 𝕜] (e : AddChar 𝕜 𝕊) (μ : Measure 𝕜)
    [μ.IsAddRightInvariant] (f : 𝕜 → E) (v₀ : 𝕜) :
    fourierIntegral e μ (f ∘ fun v ↦ v + v₀) = fun w ↦ e (v₀ * w) • fourierIntegral e μ f w 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [MeasurableAdd 𝕜] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure 𝕜) [μ.IsAddRightInvariant] (f : 𝕜 → E) (v₀ : 𝕜)
Goal: Fourier.fourierIntegral e μ (f ∘ fun v => v + v₀) = fun w => e (v₀ * w) • Fourier.fourierIntegral e μ f w
-- Real.fourierChar_apply
theorem fourierChar_apply (x : ℝ) : 𝐞 x = Complex.exp (↑(2 * π * x) * Complex.I) 
Arguments: (x : ℝ)
Goal: ↑(Real.fourierChar x) = Complex.exp (↑(2 * π * x) * Complex.I)
-- Real.continuous_fourierChar
@[continuity]
theorem continuous_fourierChar : Continuous 𝐞 
Goal: Continuous ⇑Real.fourierChar
-- Real.vector_fourierIntegral_eq_integral_exp_smul
theorem vector_fourierIntegral_eq_integral_exp_smul {V : Type*} [AddCommGroup V] [Module ℝ V]
    [MeasurableSpace V] {W : Type*} [AddCommGroup W] [Module ℝ W] (L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ)
    (μ : Measure V) (f : V → E) (w : W) :
    VectorFourier.fourierIntegral fourierChar μ L f w =
      ∫ v : V, Complex.exp (↑(-2 * π * L v w) * Complex.I) • f v ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [AddCommGroup V] [Module ℝ V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module ℝ W] (L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ) (μ : MeasureTheory.Measure V) (f : V → E) (w : W)
Goal: VectorFourier.fourierIntegral Real.fourierChar μ L f w =
  ∫ (v : V), Complex.exp (↑(-2 * π * (L v) w) * Complex.I) • f v ∂μ
-- Real.fourierIntegral_convergent_iff'
@[simp]
theorem fourierIntegral_convergent_iff' {V W : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
    [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace V] [BorelSpace V] {μ : Measure V}
    {f : V → E} (L : V →L[ℝ] W →L[ℝ] ℝ) (w : W) :
    Integrable (fun v : V ↦ 𝐞 (- L v w) • f v) μ ↔ Integrable f 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] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {f : V → E} (L : V →L[ℝ] W →L[ℝ] ℝ) (w : W)
Goal: MeasureTheory.Integrable (fun v => Real.fourierChar (-(L v) w) • f v) μ ↔ MeasureTheory.Integrable f μ
-- Real.fourierIntegral_continuousLinearMap_apply'
theorem fourierIntegral_continuousLinearMap_apply'
    {f : V → (F →L[ℝ] E)} {a : F} {w : W} (hf : Integrable f μ) :
    VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f w a =
      VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (fun x ↦ f x a) w 
Arguments: {E : Type u_1} {F : Type u_3} {V : Type u_4} {W : Type u_5} [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace ℝ W] {μ : MeasureTheory.Measure V} {L : V →L[ℝ] W →L[ℝ] ℝ} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → F →L[ℝ] E} {a : F} {w : W} (hf : MeasureTheory.Integrable f μ)
Goal: (VectorFourier.fourierIntegral Real.fourierChar μ L.toLinearMap₂ f w) a =
  VectorFourier.fourierIntegral Real.fourierChar μ L.toLinearMap₂ (fun x => (f x) a) w
-- Real.fourierIntegral_continuousMultilinearMap_apply'
theorem fourierIntegral_continuousMultilinearMap_apply'
    {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {w : W} (hf : Integrable f μ) :
    VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f w m =
      VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (fun x ↦ f x m) w 
Arguments: {E : Type u_1} {ι : Type u_2} {V : Type u_4} {W : Type u_5} [Fintype ι] [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace ℝ W] {μ : MeasureTheory.Measure V} {L : V →L[ℝ] W →L[ℝ] ℝ} [NormedAddCommGroup E] [NormedSpace ℂ E] {M : ι → Type u_6} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace ℝ (M i)] {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {w : W} (hf : MeasureTheory.Integrable f μ)
Goal: (VectorFourier.fourierIntegral Real.fourierChar μ L.toLinearMap₂ f w) m =
  VectorFourier.fourierIntegral Real.fourierChar μ L.toLinearMap₂ (fun x => (f x) m) w
-- Real.fourierIntegral_eq
lemma fourierIntegral_eq (f : V → E) (w : V) :
    𝓕 f w = ∫ v, 𝐞 (-⟪v, w⟫) • 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] [FiniteDimensional ℝ V] (f : V → E) (w : V)
Goal: Real.fourierIntegral f w = ∫ (v : V), Real.fourierChar (-⟪v, w⟫_ℝ) • f v
-- Real.fourierIntegral_eq'
lemma fourierIntegral_eq' (f : V → E) (w : V) :
    𝓕 f w = ∫ v, Complex.exp ((↑(-2 * π * ⟪v, w⟫) * Complex.I)) • 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] [FiniteDimensional ℝ V] (f : V → E) (w : V)
Goal: Real.fourierIntegral f w = ∫ (v : V), Complex.exp (↑(-2 * π * ⟪v, w⟫_ℝ) * Complex.I) • f v
-- Real.fourierIntegralInv_eq
lemma fourierIntegralInv_eq (f : V → E) (w : V) :
    𝓕⁻ f w = ∫ v, 𝐞 ⟪v, w⟫ • 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] [FiniteDimensional ℝ V] (f : V → E) (w : V)
Goal: Real.fourierIntegralInv f w = ∫ (v : V), Real.fourierChar ⟪v, w⟫_ℝ • f v
-- Real.fourierIntegralInv_eq'
lemma fourierIntegralInv_eq' (f : V → E) (w : V) :
    𝓕⁻ f w = ∫ v, Complex.exp ((↑(2 * π * ⟪v, w⟫) * Complex.I)) • 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] [FiniteDimensional ℝ V] (f : V → E) (w : V)
Goal: Real.fourierIntegralInv f w = ∫ (v : V), Complex.exp (↑(2 * π * ⟪v, w⟫_ℝ) * Complex.I) • f v
-- Real.fourierIntegral_comp_linearIsometry
lemma fourierIntegral_comp_linearIsometry (A : W ≃ₗᵢ[ℝ] V) (f : V → E) (w : W) :
    𝓕 (f ∘ A) w = (𝓕 f) (A w) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {W : Type u_3} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [MeasurableSpace W] [BorelSpace W] [FiniteDimensional ℝ W] (A : W ≃ₗᵢ[ℝ] V) (f : V → E) (w : W)
Goal: Real.fourierIntegral (f ∘ ⇑A) w = Real.fourierIntegral f (A w)
-- Real.fourierIntegralInv_eq_fourierIntegral_neg
lemma fourierIntegralInv_eq_fourierIntegral_neg (f : V → E) (w : V) :
    𝓕⁻ f w = 𝓕 f (-w) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V)
Goal: Real.fourierIntegralInv f w = Real.fourierIntegral f (-w)
-- Real.fourierIntegralInv_eq_fourierIntegral_comp_neg
lemma fourierIntegralInv_eq_fourierIntegral_comp_neg (f : V → E) :
    𝓕⁻ f = 𝓕 (fun x ↦ f (-x)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E)
Goal: Real.fourierIntegralInv f = Real.fourierIntegral fun x => f (-x)
-- Real.fourierIntegralInv_comm
lemma fourierIntegralInv_comm (f : V → E) :
    𝓕 (𝓕⁻ f) = 𝓕⁻ (𝓕 f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E)
Goal: Real.fourierIntegral (Real.fourierIntegralInv f) = Real.fourierIntegralInv (Real.fourierIntegral f)
-- Real.fourierIntegralInv_comp_linearIsometry
lemma fourierIntegralInv_comp_linearIsometry (A : W ≃ₗᵢ[ℝ] V) (f : V → E) (w : W) :
    𝓕⁻ (f ∘ A) w = (𝓕⁻ f) (A w) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {W : Type u_3} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [MeasurableSpace W] [BorelSpace W] [FiniteDimensional ℝ W] (A : W ≃ₗᵢ[ℝ] V) (f : V → E) (w : W)
Goal: Real.fourierIntegralInv (f ∘ ⇑A) w = Real.fourierIntegralInv f (A w)
-- Real.fourierIntegral_real_eq
theorem fourierIntegral_real_eq (f : ℝ → E) (w : ℝ) :
    fourierIntegral f w = ∫ v : ℝ, 𝐞 (-(v * w)) • f v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (w : ℝ)
Goal: Real.fourierIntegral f w = ∫ (v : ℝ), Real.fourierChar (-(v * w)) • f v
-- Real.fourierIntegral_real_eq_integral_exp_smul
theorem fourierIntegral_real_eq_integral_exp_smul (f : ℝ → E) (w : ℝ) :
    𝓕 f w = ∫ v : ℝ, Complex.exp (↑(-2 * π * v * w) * Complex.I) • f v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (w : ℝ)
Goal: Real.fourierIntegral f w = ∫ (v : ℝ), Complex.exp (↑(-2 * π * v * w) * Complex.I) • f v
-- Real.fourierIntegral_convergent_iff
@[simp] theorem fourierIntegral_convergent_iff {μ : Measure V} {f : V → E} (w : V) :
    Integrable (fun v : V ↦ 𝐞 (- ⟪v, w⟫) • f v) μ ↔ Integrable f μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {f : V → E} (w : V)
Goal: MeasureTheory.Integrable (fun v => Real.fourierChar (-⟪v, w⟫_ℝ) • f v) μ ↔ MeasureTheory.Integrable f μ
-- Real.fourierIntegral_continuousLinearMap_apply
theorem fourierIntegral_continuousLinearMap_apply
    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
    {f : V → (F →L[ℝ] E)} {a : F} {v : V} (hf : Integrable f) :
    𝓕 f v a = 𝓕 (fun x ↦ f x a) v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : V → F →L[ℝ] E} {a : F} {v : V} (hf : MeasureTheory.Integrable f MeasureTheory.volume)
Goal: (Real.fourierIntegral f v) a = Real.fourierIntegral (fun x => (f x) a) v
-- Real.fourierIntegral_continuousMultilinearMap_apply
theorem fourierIntegral_continuousMultilinearMap_apply {ι : Type*} [Fintype ι]
    {M : ι → Type*} [∀ i, NormedAddCommGroup (M i)] [∀ i, NormedSpace ℝ (M i)]
    {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {v : V} (hf : Integrable f) :
    𝓕 f v m = 𝓕 (fun x ↦ f x m) v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {ι : Type u_4} [Fintype ι] {M : ι → Type u_5} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace ℝ (M i)] {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {v : V} (hf : MeasureTheory.Integrable f MeasureTheory.volume)
Goal: (Real.fourierIntegral f v) m = Real.fourierIntegral (fun x => (f x) m) v
-- AddCircle.volume_eq_smul_haarAddCircle
theorem volume_eq_smul_haarAddCircle :
    (volume : Measure (AddCircle T)) = ENNReal.ofReal T • (@haarAddCircle T _) 
Arguments: {T : ℝ} [hT : Fact (0 < T)]
Goal: MeasureTheory.volume = ENNReal.ofReal T • AddCircle.haarAddCircle
-- fourier_apply
@[simp]
theorem fourier_apply {n : ℤ} {x : AddCircle T} : fourier n x = toCircle (n • x :) 
Arguments: {T : ℝ} {n : ℤ} {x : AddCircle T}
Goal: (fourier n) x = ↑(n • x).toCircle
-- fourier_coe_apply
theorem fourier_coe_apply {n : ℤ} {x : ℝ} :
    fourier n (x : AddCircle T) = Complex.exp (2 * π * Complex.I * n * x / T) 
Arguments: {T : ℝ} {n : ℤ} {x : ℝ}
Goal: (fourier n) ↑x = Complex.exp (2 * ↑π * Complex.I * ↑n * ↑x / ↑T)
-- fourier_coe_apply'
@[simp]
theorem fourier_coe_apply' {n : ℤ} {x : ℝ} :
    toCircle (n • (x : AddCircle T) :) = Complex.exp (2 * π * Complex.I * n * x / T) 
Arguments: {T : ℝ} {n : ℤ} {x : ℝ}
Goal: ↑(AddCircle.toCircle (n • ↑x)) = Complex.exp (2 * ↑π * Complex.I * ↑n * ↑x / ↑T)
-- fourier_zero
theorem fourier_zero {x : AddCircle T} : fourier 0 x = 1 
Arguments: {T : ℝ} {x : AddCircle T}
Goal: (fourier 0) x = 1
-- fourier_zero'
@[simp]
theorem fourier_zero' {x : AddCircle T} : @toCircle T 0 = (1 : ℂ) 
Arguments: {T : ℝ} {x : AddCircle T}
Goal: ↑(AddCircle.toCircle 0) = 1
-- fourier_eval_zero
theorem fourier_eval_zero (n : ℤ) : fourier n (0 : AddCircle T) = 1 
Arguments: {T : ℝ} (n : ℤ)
Goal: (fourier n) 0 = 1
-- fourier_one
theorem fourier_one {x : AddCircle T} : fourier 1 x = toCircle x 
Arguments: {T : ℝ} {x : AddCircle T}
Goal: (fourier 1) x = ↑x.toCircle
-- fourier_neg
theorem fourier_neg {n : ℤ} {x : AddCircle T} : fourier (-n) x = conj (fourier n x) 
Arguments: {T : ℝ} {n : ℤ} {x : AddCircle T}
Goal: (fourier (-n)) x = (starRingEnd ℂ) ((fourier n) x)
-- fourier_neg'
@[simp]
theorem fourier_neg' {n : ℤ} {x : AddCircle T} : @toCircle T (-(n • x)) = conj (fourier n x) 
Arguments: {T : ℝ} {n : ℤ} {x : AddCircle T}
Goal: ↑(-(n • x)).toCircle = (starRingEnd ℂ) ((fourier n) x)
-- fourier_add
theorem fourier_add {m n : ℤ} {x : AddCircle T} : fourier (m+n) x = fourier m x * fourier n x 
Arguments: {T : ℝ} {m n : ℤ} {x : AddCircle T}
Goal: (fourier (m + n)) x = (fourier m) x * (fourier n) x
-- fourier_add'
@[simp]
theorem fourier_add' {m n : ℤ} {x : AddCircle T} :
    toCircle ((m + n) • x :) = fourier m x * fourier n x 
Arguments: {T : ℝ} {m n : ℤ} {x : AddCircle T}
Goal: ↑((m + n) • x).toCircle = (fourier m) x * (fourier n) x
-- fourier_norm
theorem fourier_norm [Fact (0 < T)] (n : ℤ) : ‖@fourier T n‖ = 1 
Arguments: {T : ℝ} [Fact (0 < T)] (n : ℤ)
Goal: ‖fourier n‖ = 1
-- fourier_add_half_inv_index
theorem fourier_add_half_inv_index {n : ℤ} (hn : n ≠ 0) (hT : 0 < T) (x : AddCircle T) :
    @fourier T n (x + ↑(T / 2 / n)) = -fourier n x 
Arguments: {T : ℝ} {n : ℤ} (hn : n ≠ 0) (hT : 0 < T) (x : AddCircle T)
Goal: (fourier n) (x + ↑(T / 2 / ↑n)) = -(fourier n) x
-- fourierSubalgebra_coe
theorem fourierSubalgebra_coe :
    Subalgebra.toSubmodule (@fourierSubalgebra T).toSubalgebra = span ℂ (range (@fourier T)) 
Arguments: {T : ℝ}
Goal: Subalgebra.toSubmodule fourierSubalgebra.toSubalgebra = Submodule.span ℂ (Set.range fourier)
-- fourierSubalgebra_separatesPoints
theorem fourierSubalgebra_separatesPoints : (@fourierSubalgebra T).SeparatesPoints 
Arguments: {T : ℝ} [hT : Fact (0 < T)]
Goal: fourierSubalgebra.SeparatesPoints
-- fourierSubalgebra_closure_eq_top
theorem fourierSubalgebra_closure_eq_top : (@fourierSubalgebra T).topologicalClosure = ⊤ 
Arguments: {T : ℝ} [hT : Fact (0 < T)]
Goal: fourierSubalgebra.topologicalClosure = ⊤
-- span_fourier_closure_eq_top
theorem span_fourier_closure_eq_top : (span ℂ (range <| @fourier T)).topologicalClosure = ⊤ 
Arguments: {T : ℝ} [hT : Fact (0 < T)]
Goal: (Submodule.span ℂ (Set.range fourier)).topologicalClosure = ⊤
-- coeFn_fourierLp
theorem coeFn_fourierLp (p : ℝ≥0∞) [Fact (1 ≤ p)] (n : ℤ) :
    @fourierLp T hT p _ n =ᵐ[haarAddCircle] fourier n 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (p : ℝ≥0∞) [Fact (1 ≤ p)] (n : ℤ)
Goal: ↑↑(fourierLp p n) =ᵐ[AddCircle.haarAddCircle] ⇑(fourier n)
-- span_fourierLp_closure_eq_top
theorem span_fourierLp_closure_eq_top {p : ℝ≥0∞} [Fact (1 ≤ p)] (hp : p ≠ ∞) :
    (span ℂ (range (@fourierLp T _ p _))).topologicalClosure = ⊤ 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {p : ℝ≥0∞} [Fact (1 ≤ p)] (hp : p ≠ ⊤)
Goal: (Submodule.span ℂ (Set.range (fourierLp p))).topologicalClosure = ⊤
-- orthonormal_fourier
theorem orthonormal_fourier : Orthonormal ℂ (@fourierLp T _ 2 _) 
Arguments: {T : ℝ} [hT : Fact (0 < T)]
Goal: Orthonormal ℂ (fourierLp 2)
-- fourierCoeff_eq_intervalIntegral
theorem fourierCoeff_eq_intervalIntegral (f : AddCircle T → E) (n : ℤ) (a : ℝ) :
    fourierCoeff f n = (1 / T) • ∫ x in a..a + T, @fourier T (-n) x • f x 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : AddCircle T → E) (n : ℤ) (a : ℝ)
Goal: fourierCoeff f n = (1 / T) • ∫ (x : ℝ) in a..a + T, (fourier (-n)) ↑x • f ↑x
-- fourierCoeff.const_smul
theorem fourierCoeff.const_smul (f : AddCircle T → E) (c : ℂ) (n : ℤ) :
    fourierCoeff (c • f :) n = c • fourierCoeff f n 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : AddCircle T → E) (c : ℂ) (n : ℤ)
Goal: fourierCoeff (c • f) n = c • fourierCoeff f n
-- fourierCoeff.const_mul
theorem fourierCoeff.const_mul (f : AddCircle T → ℂ) (c : ℂ) (n : ℤ) :
    fourierCoeff (fun x => c * f x) n = c * fourierCoeff f n 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (f : AddCircle T → ℂ) (c : ℂ) (n : ℤ)
Goal: fourierCoeff (fun x => c * f x) n = c * fourierCoeff f n
-- fourierCoeffOn_eq_integral
theorem fourierCoeffOn_eq_integral {a b : ℝ} (f : ℝ → E) (n : ℤ) (hab : a < b) :
    fourierCoeffOn hab f n =
      (1 / (b - a)) • ∫ x in a..b, fourier (-n) (x : AddCircle (b - a)) • f x 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} (f : ℝ → E) (n : ℤ) (hab : a < b)
Goal: fourierCoeffOn hab f n = (1 / (b - a)) • ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x • f x
-- fourierCoeffOn.const_smul
theorem fourierCoeffOn.const_smul {a b : ℝ} (f : ℝ → E) (c : ℂ) (n : ℤ) (hab : a < b) :
    fourierCoeffOn hab (c • f) n = c • fourierCoeffOn hab f n 
Arguments: {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} (f : ℝ → E) (c : ℂ) (n : ℤ) (hab : a < b)
Goal: fourierCoeffOn hab (c • f) n = c • fourierCoeffOn hab f n
-- fourierCoeffOn.const_mul
theorem fourierCoeffOn.const_mul {a b : ℝ} (f : ℝ → ℂ) (c : ℂ) (n : ℤ) (hab : a < b) :
    fourierCoeffOn hab (fun x => c * f x) n = c * fourierCoeffOn hab f n 
Arguments: {a b : ℝ} (f : ℝ → ℂ) (c : ℂ) (n : ℤ) (hab : a < b)
Goal: fourierCoeffOn hab (fun x => c * f x) n = c * fourierCoeffOn hab f n
-- fourierCoeff_liftIoc_eq
theorem fourierCoeff_liftIoc_eq {a : ℝ} (f : ℝ → ℂ) (n : ℤ) :
    fourierCoeff (AddCircle.liftIoc T a f) n =
    fourierCoeffOn (lt_add_of_pos_right a hT.out) f n 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {a : ℝ} (f : ℝ → ℂ) (n : ℤ)
Goal: fourierCoeff (AddCircle.liftIoc T a f) n = fourierCoeffOn ⋯ f n
-- fourierCoeff_liftIco_eq
theorem fourierCoeff_liftIco_eq {a : ℝ} (f : ℝ → ℂ) (n : ℤ) :
    fourierCoeff (AddCircle.liftIco T a f) n =
    fourierCoeffOn (lt_add_of_pos_right a hT.out) f n 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {a : ℝ} (f : ℝ → ℂ) (n : ℤ)
Goal: fourierCoeff (AddCircle.liftIco T a f) n = fourierCoeffOn ⋯ f n
-- coe_fourierBasis
@[simp]
theorem coe_fourierBasis : ⇑(@fourierBasis T hT) = @fourierLp T hT 2 _ 
Arguments: {T : ℝ} [hT : Fact (0 < T)]
Goal: (fun i => fourierBasis.repr.symm (lp.single 2 i 1)) = fourierLp 2
-- fourierBasis_repr
theorem fourierBasis_repr (f : Lp ℂ 2 <| @haarAddCircle T hT) (i : ℤ) :
    fourierBasis.repr f i = fourierCoeff f i 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)) (i : ℤ)
Goal: ↑(fourierBasis.repr f) i = fourierCoeff (↑↑f) i
-- hasSum_fourier_series_L2
theorem hasSum_fourier_series_L2 (f : Lp ℂ 2 <| @haarAddCircle T hT) :
    HasSum (fun i => fourierCoeff f i • fourierLp 2 i) f 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle))
Goal: HasSum (fun i => fourierCoeff (↑↑f) i • fourierLp 2 i) f
-- tsum_sq_fourierCoeff
theorem tsum_sq_fourierCoeff (f : Lp ℂ 2 <| @haarAddCircle T hT) :
    ∑' i : ℤ, ‖fourierCoeff f i‖ ^ 2 = ∫ t : AddCircle T, ‖f t‖ ^ 2 ∂haarAddCircle 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle))
Goal: ∑' (i : ℤ), ‖fourierCoeff (↑↑f) i‖ ^ 2 = ∫ (t : AddCircle T), ‖↑↑f t‖ ^ 2 ∂AddCircle.haarAddCircle
-- fourierCoeff_toLp
theorem fourierCoeff_toLp (n : ℤ) :
    fourierCoeff (toLp (E := ℂ) 2 haarAddCircle ℂ f) n = fourierCoeff f n 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (f : C(AddCircle T, ℂ)) (n : ℤ)
Goal: fourierCoeff (↑↑((ContinuousMap.toLp 2 AddCircle.haarAddCircle ℂ) f)) n = fourierCoeff (⇑f) n
-- hasSum_fourier_series_of_summable
theorem hasSum_fourier_series_of_summable (h : Summable (fourierCoeff f)) :
    HasSum (fun i => fourierCoeff f i • fourier i) f 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {f : C(AddCircle T, ℂ)} (h : Summable (fourierCoeff ⇑f))
Goal: HasSum (fun i => fourierCoeff (⇑f) i • fourier i) f
-- has_pointwise_sum_fourier_series_of_summable
theorem has_pointwise_sum_fourier_series_of_summable (h : Summable (fourierCoeff f))
    (x : AddCircle T) : HasSum (fun i => fourierCoeff f i • fourier i x) (f x) 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {f : C(AddCircle T, ℂ)} (h : Summable (fourierCoeff ⇑f)) (x : AddCircle T)
Goal: HasSum (fun i => fourierCoeff (⇑f) i • (fourier i) x) (f x)
-- hasDerivAt_fourier
theorem hasDerivAt_fourier (n : ℤ) (x : ℝ) :
    HasDerivAt (fun y : ℝ => fourier n (y : AddCircle T))
      (2 * π * I * n / T * fourier n (x : AddCircle T)) x 
Arguments: (T : ℝ) (n : ℤ) (x : ℝ)
Goal: HasDerivAt (fun y => (fourier n) ↑y) (2 * ↑π * Complex.I * ↑n / ↑T * (fourier n) ↑x) x