ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- fderiv_id'
@[simp]
theorem fderiv_id' : fderiv 𝕜 (fun x : E => x) x = ContinuousLinearMap.id 𝕜 E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E}
Goal: fderiv 𝕜 (fun x => x) x = ContinuousLinearMap.id 𝕜 E
-- fderivWithin_id
theorem fderivWithin_id (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 id s x = id 𝕜 E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 id s x = ContinuousLinearMap.id 𝕜 E
-- fderivWithin_id'
theorem fderivWithin_id' (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x : E => x) s x = ContinuousLinearMap.id 𝕜 E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => x) s x = ContinuousLinearMap.id 𝕜 E
-- hasStrictFDerivAt_const
@[fun_prop]
theorem hasStrictFDerivAt_const (c : F) (x : E) :
    HasStrictFDerivAt (fun _ => c) (0 : E →L[𝕜] F) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F) (x : E)
Goal: HasStrictFDerivAt (fun x => c) 0 x
-- hasFDerivAtFilter_const
theorem hasFDerivAtFilter_const (c : F) (x : E) (L : Filter E) :
    HasFDerivAtFilter (fun _ => c) (0 : E →L[𝕜] F) x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F) (x : E) (L : Filter E)
Goal: HasFDerivAtFilter (fun x => c) 0 x L
-- hasFDerivWithinAt_const
@[fun_prop]
theorem hasFDerivWithinAt_const (c : F) (x : E) (s : Set E) :
    HasFDerivWithinAt (fun _ => c) (0 : E →L[𝕜] F) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F) (x : E) (s : Set E)
Goal: HasFDerivWithinAt (fun x => c) 0 s x
-- hasFDerivAt_const
@[fun_prop]
theorem hasFDerivAt_const (c : F) (x : E) : HasFDerivAt (fun _ => c) (0 : E →L[𝕜] F) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F) (x : E)
Goal: HasFDerivAt (fun x => c) 0 x
-- differentiableAt_const
@[simp, fun_prop]
theorem differentiableAt_const (c : F) : DifferentiableAt 𝕜 (fun _ => c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} (c : F)
Goal: DifferentiableAt 𝕜 (fun x => c) x
-- differentiableWithinAt_const
@[fun_prop]
theorem differentiableWithinAt_const (c : F) : DifferentiableWithinAt 𝕜 (fun _ => c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} (c : F)
Goal: DifferentiableWithinAt 𝕜 (fun x => c) s x
-- fderiv_const_apply
theorem fderiv_const_apply (c : F) : fderiv 𝕜 (fun _ => c) x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} (c : F)
Goal: fderiv 𝕜 (fun x => c) x = 0
-- fderiv_const
@[simp]
theorem fderiv_const (c : F) : (fderiv 𝕜 fun _ : E => c) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F)
Goal: (fderiv 𝕜 fun x => c) = 0
-- fderivWithin_const_apply
theorem fderivWithin_const_apply (c : F) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun _ => c) s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} (c : F) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => c) s x = 0
-- differentiable_const
@[simp, fun_prop]
theorem differentiable_const (c : F) : Differentiable 𝕜 fun _ : E => c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F)
Goal: Differentiable 𝕜 fun x => c
-- differentiableOn_const
@[simp, fun_prop]
theorem differentiableOn_const (c : F) : DifferentiableOn 𝕜 (fun _ => c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} (c : F)
Goal: DifferentiableOn 𝕜 (fun x => c) s
-- hasFDerivWithinAt_singleton
@[fun_prop]
theorem hasFDerivWithinAt_singleton (f : E → F) (x : E) :
    HasFDerivWithinAt f (0 : E →L[𝕜] F) {x} x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) (x : E)
Goal: HasFDerivWithinAt f 0 {x} x
-- hasFDerivAt_of_subsingleton
@[fun_prop]
theorem hasFDerivAt_of_subsingleton [h : Subsingleton E] (f : E → F) (x : E) :
    HasFDerivAt f (0 : E →L[𝕜] F) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [h : Subsingleton E] (f : E → F) (x : E)
Goal: HasFDerivAt f 0 x
-- differentiableOn_empty
@[fun_prop]
theorem differentiableOn_empty : DifferentiableOn 𝕜 f ∅ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F}
Goal: DifferentiableOn 𝕜 f ∅
-- differentiableOn_singleton
@[fun_prop]
theorem differentiableOn_singleton : DifferentiableOn 𝕜 f {x} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E}
Goal: DifferentiableOn 𝕜 f {x}
-- Set.Subsingleton.differentiableOn
@[fun_prop]
theorem Set.Subsingleton.differentiableOn (hs : s.Subsingleton) : DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hs : s.Subsingleton)
Goal: DifferentiableOn 𝕜 f s
-- hasFDerivAt_zero_of_eventually_const
theorem hasFDerivAt_zero_of_eventually_const (c : F) (hf : f =ᶠ[𝓝 x] fun _ => c) :
    HasFDerivAt f (0 : E →L[𝕜] F) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (c : F) (hf : f =ᶠ[𝓝 x] fun x => c)
Goal: HasFDerivAt f 0 x
-- HasStrictFDerivAt.of_nmem_tsupport
theorem HasStrictFDerivAt.of_nmem_tsupport (h : x ∉ tsupport f) :
    HasStrictFDerivAt f (0 : E →L[𝕜] F) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : x ∉ tsupport f)
Goal: HasStrictFDerivAt f 0 x
-- HasFDerivAt.of_nmem_tsupport
theorem HasFDerivAt.of_nmem_tsupport (h : x ∉ tsupport f) :
    HasFDerivAt f (0 : E →L[𝕜] F) x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : x ∉ tsupport f)
Goal: HasFDerivAt f 0 x
-- HasFDerivWithinAt.of_not_mem_tsupport
theorem HasFDerivWithinAt.of_not_mem_tsupport {s : Set E} {x : E} (h : x ∉ tsupport f) :
    HasFDerivWithinAt f (0 : E →L[𝕜] F) s x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (h : x ∉ tsupport f)
Goal: HasFDerivWithinAt f 0 s x
-- fderiv_of_not_mem_tsupport
theorem fderiv_of_not_mem_tsupport (h : x ∉ tsupport f) : fderiv 𝕜 f x = 0 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : x ∉ tsupport f)
Goal: fderiv 𝕜 f x = 0
-- support_fderiv_subset
theorem support_fderiv_subset : support (fderiv 𝕜 f) ⊆ tsupport f 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F}
Goal: Function.support (fderiv 𝕜 f) ⊆ tsupport f
-- tsupport_fderiv_subset
theorem tsupport_fderiv_subset : tsupport (fderiv 𝕜 f) ⊆ tsupport f 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F}
Goal: tsupport (fderiv 𝕜 f) ⊆ tsupport f
-- HasCompactSupport.fderiv
protected theorem HasCompactSupport.fderiv (hf : HasCompactSupport f) :
    HasCompactSupport (fderiv 𝕜 f) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (hf : HasCompactSupport f)
Goal: HasCompactSupport (fderiv 𝕜 f)
-- HasCompactSupport.fderiv_apply
protected theorem HasCompactSupport.fderiv_apply (hf : HasCompactSupport f) (v : E) :
    HasCompactSupport (fderiv 𝕜 f · v) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (hf : HasCompactSupport f) (v : E)
Goal: HasCompactSupport fun x => (fderiv 𝕜 f x) v
-- Complex.hasSum_taylorSeries_on_ball
lemma hasSum_taylorSeries_on_ball :
    HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄ ⦃c : ℂ⦄ ⦃r : ℝ⦄ (hf : DifferentiableOn ℂ f (Metric.ball c r)) ⦃z : ℂ⦄ (hz : z ∈ Metric.ball c r)
Goal: HasSum (fun n => (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)
-- Complex.taylorSeries_eq_on_ball
lemma taylorSeries_eq_on_ball :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄ ⦃c : ℂ⦄ ⦃r : ℝ⦄ (hf : DifferentiableOn ℂ f (Metric.ball c r)) ⦃z : ℂ⦄ (hz : z ∈ Metric.ball c r)
Goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z
-- Complex.taylorSeries_eq_on_ball'
lemma taylorSeries_eq_on_ball' {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (Metric.ball c r)) :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z 
Arguments: ⦃c : ℂ⦄ ⦃r : ℝ⦄ ⦃z : ℂ⦄ (hz : z ∈ Metric.ball c r) {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (Metric.ball c r))
Goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z
-- Complex.hasSum_taylorSeries_on_emetric_ball
lemma hasSum_taylorSeries_on_emetric_ball :
    HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄ ⦃c : ℂ⦄ ⦃r : ENNReal⦄ (hf : DifferentiableOn ℂ f (EMetric.ball c r)) ⦃z : ℂ⦄ (hz : z ∈ EMetric.ball c r)
Goal: HasSum (fun n => (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)
-- Complex.taylorSeries_eq_on_emetric_ball
lemma taylorSeries_eq_on_emetric_ball :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄ ⦃c : ℂ⦄ ⦃r : ENNReal⦄ (hf : DifferentiableOn ℂ f (EMetric.ball c r)) ⦃z : ℂ⦄ (hz : z ∈ EMetric.ball c r)
Goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z
-- Complex.taylorSeries_eq_on_emetric_ball'
lemma taylorSeries_eq_on_emetric_ball' {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (EMetric.ball c r)) :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z 
Arguments: ⦃c : ℂ⦄ ⦃r : ENNReal⦄ ⦃z : ℂ⦄ (hz : z ∈ EMetric.ball c r) {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (EMetric.ball c r))
Goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z
-- Complex.hasSum_taylorSeries_of_entire
lemma hasSum_taylorSeries_of_entire :
    HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄ (hf : Differentiable ℂ f) (c z : ℂ)
Goal: HasSum (fun n => (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)
-- Complex.taylorSeries_eq_of_entire
lemma taylorSeries_eq_of_entire :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄ (hf : Differentiable ℂ f) (c z : ℂ)
Goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z
-- Complex.taylorSeries_eq_of_entire'
lemma taylorSeries_eq_of_entire' {f : ℂ → ℂ} (hf : Differentiable ℂ f) :
    ∑' n : ℕ, (n ! : ℂ)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z 
Arguments: (c z : ℂ) {f : ℂ → ℂ} (hf : Differentiable ℂ f)
Goal: ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z
-- Complex.angle_eq_abs_arg
lemma angle_eq_abs_arg (hx : x ≠ 0) (hy : y ≠ 0) : angle x y = |(x / y).arg| 
Arguments: {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: InnerProductGeometry.angle x y = |(x / y).arg|
-- Complex.angle_one_left
lemma angle_one_left (hy : y ≠ 0) : angle 1 y = |y.arg| 
Arguments: {y : ℂ} (hy : y ≠ 0)
Goal: InnerProductGeometry.angle 1 y = |y.arg|
-- Complex.angle_one_right
lemma angle_one_right (hx : x ≠ 0) : angle x 1 = |x.arg| 
Arguments: {x : ℂ} (hx : x ≠ 0)
Goal: InnerProductGeometry.angle x 1 = |x.arg|
-- Complex.angle_mul_left
@[simp] lemma angle_mul_left (ha : a ≠ 0) (x y : ℂ) : angle (a * x) (a * y) = angle x y 
Arguments: {a : ℂ} (ha : a ≠ 0) (x y : ℂ)
Goal: InnerProductGeometry.angle (a * x) (a * y) = InnerProductGeometry.angle x y
-- Complex.angle_mul_right
@[simp] lemma angle_mul_right (ha : a ≠ 0) (x y : ℂ) : angle (x * a) (y * a) = angle x y 
Arguments: {a : ℂ} (ha : a ≠ 0) (x y : ℂ)
Goal: InnerProductGeometry.angle (x * a) (y * a) = InnerProductGeometry.angle x y
-- Complex.angle_div_left_eq_angle_mul_right
lemma angle_div_left_eq_angle_mul_right (a x y : ℂ) : angle (x / a) y = angle x (y * a) 
Arguments: (a x y : ℂ)
Goal: InnerProductGeometry.angle (x / a) y = InnerProductGeometry.angle x (y * a)
-- Complex.angle_div_right_eq_angle_mul_left
lemma angle_div_right_eq_angle_mul_left (a x y : ℂ) : angle x (y / a) = angle (x * a) y 
Arguments: (a x y : ℂ)
Goal: InnerProductGeometry.angle x (y / a) = InnerProductGeometry.angle (x * a) y
-- Complex.angle_exp_exp
lemma angle_exp_exp (x y : ℝ) :
    angle (exp (x * I)) (exp (y * I)) = |toIocMod Real.two_pi_pos (-π) (x - y)| 
Arguments: (x y : ℝ)
Goal: InnerProductGeometry.angle (Complex.exp (↑x * Complex.I)) (Complex.exp (↑y * Complex.I)) =
  |toIocMod Real.two_pi_pos (-π) (x - y)|
-- Complex.angle_exp_one
lemma angle_exp_one (x : ℝ) : angle (exp (x * I)) 1 = |toIocMod Real.two_pi_pos (-π) x| 
Arguments: (x : ℝ)
Goal: InnerProductGeometry.angle (Complex.exp (↑x * Complex.I)) 1 = |toIocMod Real.two_pi_pos (-π) x|
-- Complex.norm_sub_mem_Icc_angle
lemma norm_sub_mem_Icc_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
    ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y) 
Arguments: {x y : ℂ} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1)
Goal: ‖x - y‖ ∈ Set.Icc (2 / π * InnerProductGeometry.angle x y) (InnerProductGeometry.angle x y)
-- Complex.norm_sub_le_angle
lemma norm_sub_le_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) : ‖x - y‖ ≤ angle x y 
Arguments: {x y : ℂ} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1)
Goal: ‖x - y‖ ≤ InnerProductGeometry.angle x y
-- Complex.mul_angle_le_norm_sub
lemma mul_angle_le_norm_sub (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) : 2 / π * angle x y ≤ ‖x - y‖ 
Arguments: {x y : ℂ} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1)
Goal: 2 / π * InnerProductGeometry.angle x y ≤ ‖x - y‖
-- Complex.angle_le_mul_norm_sub
lemma angle_le_mul_norm_sub (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) : angle x y ≤ π / 2 * ‖x - y‖ 
Arguments: {x y : ℂ} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1)
Goal: InnerProductGeometry.angle x y ≤ π / 2 * ‖x - y‖
-- Complex.convexHull_reProdIm
lemma convexHull_reProdIm (s t : Set ℝ) :
    convexHull ℝ (s ×ℂ t) = convexHull ℝ s ×ℂ convexHull ℝ t 
Arguments: (s t : Set ℝ)
Goal: (convexHull ℝ) (s ×ℂ t) = (convexHull ℝ) s ×ℂ (convexHull ℝ) t
-- Complex.starConvex_slitPlane
lemma starConvex_slitPlane {z : ℂ} (hz : 0 < z) : StarConvex ℝ z slitPlane 
Arguments: {z : ℂ} (hz : 0 < z)
Goal: StarConvex ℝ z Complex.slitPlane
-- Complex.starConvex_ofReal_slitPlane
lemma starConvex_ofReal_slitPlane {x : ℝ} (hx : 0 < x) : StarConvex ℝ ↑x slitPlane 
Arguments: {x : ℝ} (hx : 0 < x)
Goal: StarConvex ℝ (↑x) Complex.slitPlane
-- Complex.starConvex_one_slitPlane
lemma starConvex_one_slitPlane : StarConvex ℝ 1 slitPlane 
Goal: StarConvex ℝ 1 Complex.slitPlane
-- Real.hasDerivAt_fourierChar
lemma hasDerivAt_fourierChar (x : ℝ) : HasDerivAt (𝐞 · : ℝ → ℂ) (2 * π * I * 𝐞 x) x 
Arguments: (x : ℝ)
Goal: HasDerivAt (fun x => ↑(𝐞 x)) (2 * ↑π * Complex.I * ↑(𝐞 x)) x
-- Real.differentiable_fourierChar
lemma differentiable_fourierChar : Differentiable ℝ (𝐞 · : ℝ → ℂ) 
Goal: Differentiable ℝ fun x => ↑(𝐞 x)
-- Real.deriv_fourierChar
lemma deriv_fourierChar (x : ℝ) : deriv (𝐞 · : ℝ → ℂ) x = 2 * π * I * 𝐞 x 
Arguments: (x : ℝ)
Goal: deriv (fun x => ↑(𝐞 x)) x = 2 * ↑π * Complex.I * ↑(𝐞 x)
-- Real.hasFDerivAt_fourierChar_neg_bilinear_right
lemma hasFDerivAt_fourierChar_neg_bilinear_right (v : V) (w : W) :
    HasFDerivAt (fun w ↦ (𝐞 (-L v w) : ℂ))
      ((-2 * π * I * 𝐞 (-L v w)) • (ofRealCLM ∘L (L v))) w 
Arguments: {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (w : W)
Goal: HasFDerivAt (fun w => ↑(𝐞 (-(L v) w))) ((-2 * ↑π * Complex.I * ↑(𝐞 (-(L v) w))) • Complex.ofRealCLM.comp (L v)) w
-- Real.fderiv_fourierChar_neg_bilinear_right_apply
lemma fderiv_fourierChar_neg_bilinear_right_apply (v : V) (w y : W) :
    fderiv ℝ (fun w ↦ (𝐞 (-L v w) : ℂ)) w y = -2 * π * I * L v y * 𝐞 (-L v w) 
Arguments: {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (w y : W)
Goal: (fderiv ℝ (fun w => ↑(𝐞 (-(L v) w))) w) y = -2 * ↑π * Complex.I * ↑((L v) y) * ↑(𝐞 (-(L v) w))
-- Real.differentiable_fourierChar_neg_bilinear_right
lemma differentiable_fourierChar_neg_bilinear_right (v : V) :
    Differentiable ℝ (fun w ↦ (𝐞 (-L v w) : ℂ)) 
Arguments: {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V)
Goal: Differentiable ℝ fun w => ↑(𝐞 (-(L v) w))
-- Real.hasFDerivAt_fourierChar_neg_bilinear_left
lemma hasFDerivAt_fourierChar_neg_bilinear_left (v : V) (w : W) :
    HasFDerivAt (fun v ↦ (𝐞 (-L v w) : ℂ))
      ((-2 * π * I * 𝐞 (-L v w)) • (ofRealCLM ∘L (L.flip w))) v 
Arguments: {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (w : W)
Goal: HasFDerivAt (fun v => ↑(𝐞 (-(L v) w))) ((-2 * ↑π * Complex.I * ↑(𝐞 (-(L v) w))) • Complex.ofRealCLM.comp (L.flip w)) v
-- Real.fderiv_fourierChar_neg_bilinear_left_apply
lemma fderiv_fourierChar_neg_bilinear_left_apply (v y : V) (w : W) :
    fderiv ℝ (fun v ↦ (𝐞 (-L v w) : ℂ)) v y = -2 * π * I * L y w * 𝐞 (-L v w) 
Arguments: {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v y : V) (w : W)
Goal: (fderiv ℝ (fun v => ↑(𝐞 (-(L v) w))) v) y = -2 * ↑π * Complex.I * ↑((L y) w) * ↑(𝐞 (-(L v) w))
-- Real.differentiable_fourierChar_neg_bilinear_left
lemma differentiable_fourierChar_neg_bilinear_left (w : W) :
    Differentiable ℝ (fun v ↦ (𝐞 (-L v w) : ℂ)) 
Arguments: {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (w : W)
Goal: Differentiable ℝ fun v => ↑(𝐞 (-(L v) w))
-- VectorFourier.fourierSMulRight_apply
@[simp] lemma fourierSMulRight_apply (v : V) (w : W) :
    fourierSMulRight L f v w = -(2 * π * I) • L v w • 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) (w : W)
Goal: (VectorFourier.fourierSMulRight L f v) w = -(2 * ↑π * Complex.I) • (L v) w • f v
-- VectorFourier.hasFDerivAt_fourierChar_smul
lemma hasFDerivAt_fourierChar_smul (v : V) (w : W) :
    HasFDerivAt (fun w' ↦ 𝐞 (-L v w') • f v) (𝐞 (-L v w) • fourierSMulRight L f v) 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) (v : V) (w : W)
Goal: HasFDerivAt (fun w' => 𝐞 (-(L v) w') • f v) (𝐞 (-(L v) w) • VectorFourier.fourierSMulRight L f v) w
-- VectorFourier.norm_fourierSMulRight
lemma norm_fourierSMulRight (L : V →L[ℝ] W →L[ℝ] ℝ) (f : V → E) (v : V) :
    ‖fourierSMulRight L f v‖ = (2 * π) * ‖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)
Goal: ‖VectorFourier.fourierSMulRight L f v‖ = 2 * π * ‖L v‖ * ‖f v‖
-- VectorFourier.norm_fourierSMulRight_le
lemma norm_fourierSMulRight_le (L : V →L[ℝ] W →L[ℝ] ℝ) (f : V → E) (v : V) :
    ‖fourierSMulRight L f v‖ ≤ 2 * π * ‖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)
Goal: ‖VectorFourier.fourierSMulRight L f v‖ ≤ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
-- MeasureTheory.AEStronglyMeasurable.fourierSMulRight
lemma _root_.MeasureTheory.AEStronglyMeasurable.fourierSMulRight
    [SecondCountableTopologyEither V (W →L[ℝ] ℝ)] [MeasurableSpace V] [BorelSpace V]
    {L : V →L[ℝ] W →L[ℝ] ℝ} {f : V → E} {μ : Measure V}
    (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun v ↦ fourierSMulRight L 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] [SecondCountableTopologyEither V (W →L[ℝ] ℝ)] [MeasurableSpace V] [BorelSpace V] {L : V →L[ℝ] W →L[ℝ] ℝ} {f : V → E} {μ : MeasureTheory.Measure V} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun v => VectorFourier.fourierSMulRight L f v) μ
-- VectorFourier.hasFDerivAt_fourierIntegral
theorem hasFDerivAt_fourierIntegral
    [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : Measure V}
    (hf : Integrable f μ) (hf' : Integrable (fun v : V ↦ ‖v‖ * ‖f v‖) μ) (w : W) :
    HasFDerivAt (fourierIntegral 𝐞 μ L.toLinearMap₂ f)
      (fourierIntegral 𝐞 μ L.toLinearMap₂ (fourierSMulRight L f) w) 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} [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : MeasureTheory.Measure V} (hf : MeasureTheory.Integrable f μ) (hf' : MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) μ) (w : W)
Goal: HasFDerivAt (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ f)
  (VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (VectorFourier.fourierSMulRight L f) w) w