ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- measurableSet_eball
@[measurability]
theorem measurableSet_eball : MeasurableSet (EMetric.ball x ε) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} {ε : ℝ≥0∞}
Goal: MeasurableSet (EMetric.ball x ε)
-- measurable_edist_right
@[measurability, fun_prop]
theorem measurable_edist_right : Measurable (edist x) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α}
Goal: Measurable (edist x)
-- measurable_edist_left
@[measurability, fun_prop]
theorem measurable_edist_left : Measurable fun y => edist y x 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α}
Goal: Measurable fun y => edist y x
-- measurable_infEdist
@[measurability]
theorem measurable_infEdist {s : Set α} : Measurable fun x => infEdist x s 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α}
Goal: Measurable fun x => EMetric.infEdist x s
-- Measurable.infEdist
@[measurability, fun_prop]
theorem Measurable.infEdist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infEdist (f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α}
Goal: Measurable fun x => EMetric.infEdist (f x) s
-- tendsto_measure_cthickening
theorem tendsto_measure_cthickening {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) :
    Tendsto (fun r => μ (cthickening r s)) (𝓝 0) (𝓝 (μ (closure s))) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R > 0, μ (Metric.cthickening R s) ≠ ⊤)
Goal: Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ (closure s)))
-- tendsto_measure_cthickening_of_isClosed
theorem tendsto_measure_cthickening_of_isClosed {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) (h's : IsClosed s) :
    Tendsto (fun r => μ (cthickening r s)) (𝓝 0) (𝓝 (μ s)) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R > 0, μ (Metric.cthickening R s) ≠ ⊤) (h's : IsClosed s)
Goal: Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ s))
-- tendsto_measure_thickening
theorem tendsto_measure_thickening {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (thickening R s) ≠ ∞) :
    Tendsto (fun r => μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ (closure s))) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R > 0, μ (Metric.thickening R s) ≠ ⊤)
Goal: Filter.Tendsto (fun r => μ (Metric.thickening r s)) (𝓝[>] 0) (𝓝 (μ (closure s)))
-- tendsto_measure_thickening_of_isClosed
theorem tendsto_measure_thickening_of_isClosed {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (thickening R s) ≠ ∞) (h's : IsClosed s) :
    Tendsto (fun r => μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ s)) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R > 0, μ (Metric.thickening R s) ≠ ⊤) (h's : IsClosed s)
Goal: Filter.Tendsto (fun r => μ (Metric.thickening r s)) (𝓝[>] 0) (𝓝 (μ s))
-- measurable_edist
@[measurability]
theorem measurable_edist : Measurable fun p : α × α => edist p.1 p.2 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α]
Goal: Measurable fun p => edist p.1 p.2
-- Measurable.edist
@[measurability, fun_prop]
theorem Measurable.edist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => edist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun b => edist (f b) (g b)
-- AEMeasurable.edist
@[measurability, fun_prop]
theorem AEMeasurable.edist {f g : β → α} {μ : Measure β} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : AEMeasurable (fun a => edist (f a) (g a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => edist (f a) (g a)) μ
-- tendsto_measure_cthickening_of_isCompact
theorem tendsto_measure_cthickening_of_isCompact [MetricSpace α] [MeasurableSpace α]
    [OpensMeasurableSpace α] [ProperSpace α] {μ : Measure α} [IsFiniteMeasureOnCompacts μ]
    {s : Set α} (hs : IsCompact s) :
    Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ s)) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [ProperSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {s : Set α} (hs : IsCompact s)
Goal: Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ s))
-- exists_borelSpace_of_countablyGenerated_of_separatesPoints
theorem exists_borelSpace_of_countablyGenerated_of_separatesPoints (α : Type*)
    [m : MeasurableSpace α] [CountablyGenerated α] [SeparatesPoints α] :
    ∃ τ : TopologicalSpace α, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α 
Arguments: (α : Type u_5) [m : MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] [MeasurableSpace.SeparatesPoints α]
Goal: ∃ τ, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α
-- exists_opensMeasurableSpace_of_countablySeparated
theorem exists_opensMeasurableSpace_of_countablySeparated (α : Type*)
    [m : MeasurableSpace α] [CountablySeparated α] :
    ∃ τ : TopologicalSpace α, SecondCountableTopology α ∧ T4Space α ∧ OpensMeasurableSpace α 
Arguments: (α : Type u_5) [m : MeasurableSpace α] [MeasurableSpace.CountablySeparated α]
Goal: ∃ τ, SecondCountableTopology α ∧ T4Space α ∧ OpensMeasurableSpace α
-- measurable_norm
@[fun_prop, measurability]
theorem measurable_norm : Measurable (norm : α → ℝ) 
Arguments: {α : Type u_1} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α]
Goal: Measurable norm
-- Measurable.norm
@[measurability, fun_prop]
theorem Measurable.norm {f : β → α} (hf : Measurable f) : Measurable fun a => norm (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f)
Goal: Measurable fun a => ‖f a‖
-- AEMeasurable.norm
@[measurability, fun_prop]
theorem AEMeasurable.norm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
    AEMeasurable (fun a => norm (f a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun a => ‖f a‖) μ
-- measurable_nnnorm
@[measurability]
theorem measurable_nnnorm : Measurable (nnnorm : α → ℝ≥0) 
Arguments: {α : Type u_1} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α]
Goal: Measurable nnnorm
-- Measurable.nnnorm
@[measurability, fun_prop]
theorem Measurable.nnnorm {f : β → α} (hf : Measurable f) : Measurable fun a => ‖f a‖₊ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f)
Goal: Measurable fun a => ‖f a‖₊
-- AEMeasurable.nnnorm
@[measurability, fun_prop]
theorem AEMeasurable.nnnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
    AEMeasurable (fun a => ‖f a‖₊) μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun a => ‖f a‖₊) μ
-- measurable_ennnorm
@[measurability]
theorem measurable_ennnorm : Measurable fun x : α => (‖x‖₊ : ℝ≥0∞) 
Arguments: {α : Type u_1} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α]
Goal: Measurable fun x => ↑‖x‖₊
-- Measurable.ennnorm
@[measurability, fun_prop]
theorem Measurable.ennnorm {f : β → α} (hf : Measurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f)
Goal: Measurable fun a => ↑‖f a‖₊
-- AEMeasurable.ennnorm
@[measurability, fun_prop]
theorem AEMeasurable.ennnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
    AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun a => ↑‖f a‖₊) μ
-- MeasureTheory.Measure.toSphere_apply_aux
theorem toSphere_apply_aux (s : Set (sphere (0 : E) 1)) (r : Ioi (0 : ℝ)) :
    μ ((↑) '' (homeomorphUnitSphereProd E ⁻¹' s ×ˢ Iio r)) = μ (Ioo (0 : ℝ) r • ((↑) '' s)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] (μ : MeasureTheory.Measure E) (s : Set ↑(Metric.sphere 0 1)) (r : ↑(Set.Ioi 0))
Goal: μ (Subtype.val '' (⇑(homeomorphUnitSphereProd E) ⁻¹' s ×ˢ Set.Iio r)) = μ (Set.Ioo 0 ↑r • Subtype.val '' s)
-- MeasureTheory.Measure.toSphere_apply'
theorem toSphere_apply' {s : Set (sphere (0 : E) 1)} (hs : MeasurableSet s) :
    μ.toSphere s = dim E * μ (Ioo (0 : ℝ) 1 • ((↑) '' s)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) {s : Set ↑(Metric.sphere 0 1)} (hs : MeasurableSet s)
Goal: μ.toSphere s = ↑(dim E) * μ (Set.Ioo 0 1 • Subtype.val '' s)
-- MeasureTheory.Measure.toSphere_apply_univ'
theorem toSphere_apply_univ' : μ.toSphere univ = dim E * μ (ball 0 1 \ {0}) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E)
Goal: μ.toSphere Set.univ = ↑(dim E) * μ (Metric.ball 0 1 \ {0})
-- MeasureTheory.Measure.toSphere_apply_univ
@[simp]
theorem toSphere_apply_univ : μ.toSphere univ = dim E * μ (ball 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure]
Goal: μ.toSphere Set.univ = ↑(dim E) * μ (Metric.ball 0 1)
-- MeasureTheory.Measure.volumeIoiPow_apply_Iio
lemma volumeIoiPow_apply_Iio (n : ℕ) (x : Ioi (0 : ℝ)) :
    volumeIoiPow n (Iio x) = ENNReal.ofReal (x.1 ^ (n + 1) / (n + 1)) 
Arguments: (n : ℕ) (x : ↑(Set.Ioi 0))
Goal: (MeasureTheory.Measure.volumeIoiPow n) (Set.Iio x) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))
-- MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
theorem measurePreserving_homeomorphUnitSphereProd :
    MeasurePreserving (homeomorphUnitSphereProd E) (μ.comap (↑))
      (μ.toSphere.prod (volumeIoiPow (dim E - 1))) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure]
Goal: MeasureTheory.MeasurePreserving (⇑(homeomorphUnitSphereProd E)) (MeasureTheory.Measure.comap Subtype.val μ)
  (μ.toSphere.prod (MeasureTheory.Measure.volumeIoiPow (dim E - 1)))
-- MeasureTheory.integral_fun_norm_addHaar
lemma integral_fun_norm_addHaar (f : ℝ → F) :
    ∫ x, f (‖x‖) ∂μ = dim E • (μ (ball 0 1)).toReal • ∫ y in Ioi (0 : ℝ), y ^ (dim E - 1) • f y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [Nontrivial E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : ℝ → F)
Goal: ∫ (x : E), f ‖x‖ ∂μ = dim E • (μ (Metric.ball 0 1)).toReal • ∫ (y : ℝ) in Set.Ioi 0, y ^ (dim E - 1) • f y
-- periodic_circleMap
theorem periodic_circleMap (c : ℂ) (R : ℝ) : Periodic (circleMap c R) (2 * π) 
Arguments: (c : ℂ) (R : ℝ)
Goal: Function.Periodic (circleMap c R) (2 * π)
-- Set.Countable.preimage_circleMap
theorem Set.Countable.preimage_circleMap {s : Set ℂ} (hs : s.Countable) (c : ℂ) {R : ℝ}
    (hR : R ≠ 0) : (circleMap c R ⁻¹' s).Countable 
Arguments: {s : Set ℂ} (hs : s.Countable) (c : ℂ) {R : ℝ} (hR : R ≠ 0)
Goal: (circleMap c R ⁻¹' s).Countable
-- circleMap_sub_center
@[simp]
theorem circleMap_sub_center (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ - c = circleMap 0 R θ 
Arguments: (c : ℂ) (R θ : ℝ)
Goal: circleMap c R θ - c = circleMap 0 R θ
-- circleMap_zero
theorem circleMap_zero (R θ : ℝ) : circleMap 0 R θ = R * exp (θ * I) 
Arguments: (R θ : ℝ)
Goal: circleMap 0 R θ = ↑R * cexp (↑θ * Complex.I)
-- abs_circleMap_zero
@[simp]
theorem abs_circleMap_zero (R : ℝ) (θ : ℝ) : abs (circleMap 0 R θ) = |R| 
Arguments: (R θ : ℝ)
Goal: Complex.abs (circleMap 0 R θ) = |R|
-- circleMap_mem_sphere'
theorem circleMap_mem_sphere' (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∈ sphere c |R| 
Arguments: (c : ℂ) (R θ : ℝ)
Goal: circleMap c R θ ∈ Metric.sphere c |R|
-- circleMap_mem_sphere
theorem circleMap_mem_sphere (c : ℂ) {R : ℝ} (hR : 0 ≤ R) (θ : ℝ) :
    circleMap c R θ ∈ sphere c R 
Arguments: (c : ℂ) {R : ℝ} (hR : 0 ≤ R) (θ : ℝ)
Goal: circleMap c R θ ∈ Metric.sphere c R
-- circleMap_mem_closedBall
theorem circleMap_mem_closedBall (c : ℂ) {R : ℝ} (hR : 0 ≤ R) (θ : ℝ) :
    circleMap c R θ ∈ closedBall c R 
Arguments: (c : ℂ) {R : ℝ} (hR : 0 ≤ R) (θ : ℝ)
Goal: circleMap c R θ ∈ Metric.closedBall c R
-- circleMap_not_mem_ball
theorem circleMap_not_mem_ball (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∉ ball c R 
Arguments: (c : ℂ) (R θ : ℝ)
Goal: circleMap c R θ ∉ Metric.ball c R
-- circleMap_ne_mem_ball
theorem circleMap_ne_mem_ball {c : ℂ} {R : ℝ} {w : ℂ} (hw : w ∈ ball c R) (θ : ℝ) :
    circleMap c R θ ≠ w 
Arguments: {c : ℂ} {R : ℝ} {w : ℂ} (hw : w ∈ Metric.ball c R) (θ : ℝ)
Goal: circleMap c R θ ≠ w
-- range_circleMap
@[simp]
theorem range_circleMap (c : ℂ) (R : ℝ) : range (circleMap c R) = sphere c |R| 
Arguments: (c : ℂ) (R : ℝ)
Goal: Set.range (circleMap c R) = Metric.sphere c |R|
-- image_circleMap_Ioc
@[simp]
theorem image_circleMap_Ioc (c : ℂ) (R : ℝ) : circleMap c R '' Ioc 0 (2 * π) = sphere c |R| 
Arguments: (c : ℂ) (R : ℝ)
Goal: circleMap c R '' Set.Ioc 0 (2 * π) = Metric.sphere c |R|
-- circleMap_eq_center_iff
@[simp]
theorem circleMap_eq_center_iff {c : ℂ} {R : ℝ} {θ : ℝ} : circleMap c R θ = c ↔ R = 0 
Arguments: {c : ℂ} {R θ : ℝ}
Goal: circleMap c R θ = c ↔ R = 0
-- circleMap_zero_radius
@[simp]
theorem circleMap_zero_radius (c : ℂ) : circleMap c 0 = const ℝ c 
Arguments: (c : ℂ)
Goal: circleMap c 0 = Function.const ℝ c
-- circleMap_ne_center
theorem circleMap_ne_center {c : ℂ} {R : ℝ} (hR : R ≠ 0) {θ : ℝ} : circleMap c R θ ≠ c 
Arguments: {c : ℂ} {R : ℝ} (hR : R ≠ 0) {θ : ℝ}
Goal: circleMap c R θ ≠ c
-- hasDerivAt_circleMap
theorem hasDerivAt_circleMap (c : ℂ) (R : ℝ) (θ : ℝ) :
    HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ 
Arguments: (c : ℂ) (R θ : ℝ)
Goal: HasDerivAt (circleMap c R) (circleMap 0 R θ * Complex.I) θ
-- differentiable_circleMap
theorem differentiable_circleMap (c : ℂ) (R : ℝ) : Differentiable ℝ (circleMap c R) 
Arguments: (c : ℂ) (R : ℝ)
Goal: Differentiable ℝ (circleMap c R)
-- continuous_circleMap
@[continuity, fun_prop]
theorem continuous_circleMap (c : ℂ) (R : ℝ) : Continuous (circleMap c R) 
Arguments: (c : ℂ) (R : ℝ)
Goal: Continuous (circleMap c R)
-- measurable_circleMap
@[fun_prop, measurability]
theorem measurable_circleMap (c : ℂ) (R : ℝ) : Measurable (circleMap c R) 
Arguments: (c : ℂ) (R : ℝ)
Goal: Measurable (circleMap c R)
-- deriv_circleMap
@[simp]
theorem deriv_circleMap (c : ℂ) (R : ℝ) (θ : ℝ) : deriv (circleMap c R) θ = circleMap 0 R θ * I 
Arguments: (c : ℂ) (R θ : ℝ)
Goal: deriv (circleMap c R) θ = circleMap 0 R θ * Complex.I
-- deriv_circleMap_eq_zero_iff
theorem deriv_circleMap_eq_zero_iff {c : ℂ} {R : ℝ} {θ : ℝ} :
    deriv (circleMap c R) θ = 0 ↔ R = 0 
Arguments: {c : ℂ} {R θ : ℝ}
Goal: deriv (circleMap c R) θ = 0 ↔ R = 0
-- deriv_circleMap_ne_zero
theorem deriv_circleMap_ne_zero {c : ℂ} {R : ℝ} {θ : ℝ} (hR : R ≠ 0) :
    deriv (circleMap c R) θ ≠ 0 
Arguments: {c : ℂ} {R θ : ℝ} (hR : R ≠ 0)
Goal: deriv (circleMap c R) θ ≠ 0
-- lipschitzWith_circleMap
theorem lipschitzWith_circleMap (c : ℂ) (R : ℝ) : LipschitzWith (Real.nnabs R) (circleMap c R) 
Arguments: (c : ℂ) (R : ℝ)
Goal: LipschitzWith (Real.nnabs R) (circleMap c R)
-- continuous_circleMap_inv
theorem continuous_circleMap_inv {R : ℝ} {z w : ℂ} (hw : w ∈ ball z R) :
    Continuous fun θ => (circleMap z R θ - w)⁻¹ 
Arguments: {R : ℝ} {z w : ℂ} (hw : w ∈ Metric.ball z R)
Goal: Continuous fun θ => (circleMap z R θ - w)⁻¹
-- circleIntegrable_const
@[simp]
theorem circleIntegrable_const (a : E) (c : ℂ) (R : ℝ) : CircleIntegrable (fun _ => a) c R 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (a : E) (c : ℂ) (R : ℝ)
Goal: CircleIntegrable (fun x => a) c R
-- CircleIntegrable.add
nonrec theorem add (hf : CircleIntegrable f c R) (hg : CircleIntegrable g c R) :
    CircleIntegrable (f + g) c R 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f g : ℂ → E} {c : ℂ} {R : ℝ} (hf : CircleIntegrable f c R) (hg : CircleIntegrable g c R)
Goal: CircleIntegrable (f + g) c R
-- CircleIntegrable.neg
nonrec theorem neg (hf : CircleIntegrable f c R) : CircleIntegrable (-f) c R 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {c : ℂ} {R : ℝ} (hf : CircleIntegrable f c R)
Goal: CircleIntegrable (-f) c R
-- CircleIntegrable.out
theorem out [NormedSpace ℂ E] (hf : CircleIntegrable f c R) :
    IntervalIntegrable (fun θ : ℝ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0
      (2 * π) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {c : ℂ} {R : ℝ} [NormedSpace ℂ E] (hf : CircleIntegrable f c R)
Goal: IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) MeasureTheory.volume 0 (2 * π)
-- circleIntegrable_zero_radius
@[simp]
theorem circleIntegrable_zero_radius {f : ℂ → E} {c : ℂ} : CircleIntegrable f c 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {c : ℂ}
Goal: CircleIntegrable f c 0
-- circleIntegrable_iff
theorem circleIntegrable_iff [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} (R : ℝ) :
    CircleIntegrable f c R ↔ IntervalIntegrable (fun θ : ℝ =>
      deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} (R : ℝ)
Goal: CircleIntegrable f c R ↔
  IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) MeasureTheory.volume 0 (2 * π)
-- ContinuousOn.circleIntegrable'
theorem ContinuousOn.circleIntegrable' {f : ℂ → E} {c : ℂ} {R : ℝ}
    (hf : ContinuousOn f (sphere c |R|)) : CircleIntegrable f c R 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {c : ℂ} {R : ℝ} (hf : ContinuousOn f (Metric.sphere c |R|))
Goal: CircleIntegrable f c R
-- ContinuousOn.circleIntegrable
theorem ContinuousOn.circleIntegrable {f : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R)
    (hf : ContinuousOn f (sphere c R)) : CircleIntegrable f c R 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R) (hf : ContinuousOn f (Metric.sphere c R))
Goal: CircleIntegrable f c R
-- circleIntegrable_sub_zpow_iff
@[simp]
theorem circleIntegrable_sub_zpow_iff {c w : ℂ} {R : ℝ} {n : ℤ} :
    CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ sphere c |R| 
Arguments: {c w : ℂ} {R : ℝ} {n : ℤ}
Goal: CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ Metric.sphere c |R|
-- circleIntegrable_sub_inv_iff
@[simp]
theorem circleIntegrable_sub_inv_iff {c w : ℂ} {R : ℝ} :
    CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ w ∉ sphere c |R| 
Arguments: {c w : ℂ} {R : ℝ}
Goal: CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ w ∉ Metric.sphere c |R|
-- circleIntegral_def_Icc
theorem circleIntegral_def_Icc (f : ℂ → E) (c : ℂ) (R : ℝ) :
    (∮ z in C(c, R), f z) = ∫ θ in Icc 0 (2 * π),
    deriv (circleMap c R) θ • f (circleMap c R θ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (c : ℂ) (R : ℝ)
Goal: (∮ (z : ℂ) in C(c, R), f z) = ∫ (θ : ℝ) in Set.Icc 0 (2 * π), deriv (circleMap c R) θ • f (circleMap c R θ)
-- circleIntegral.integral_radius_zero
@[simp]
theorem integral_radius_zero (f : ℂ → E) (c : ℂ) : (∮ z in C(c, 0), f z) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (c : ℂ)
Goal: (∮ (z : ℂ) in C(c, 0), f z) = 0
-- circleIntegral.integral_congr
theorem integral_congr {f g : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R) (h : EqOn f g (sphere c R)) :
    (∮ z in C(c, R), f z) = ∮ z in C(c, R), g z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R) (h : Set.EqOn f g (Metric.sphere c R))
Goal: (∮ (z : ℂ) in C(c, R), f z) = ∮ (z : ℂ) in C(c, R), g z
-- circleIntegral.integral_sub_inv_smul_sub_smul
theorem integral_sub_inv_smul_sub_smul (f : ℂ → E) (c w : ℂ) (R : ℝ) :
    (∮ z in C(c, R), (z - w)⁻¹ • (z - w) • f z) = ∮ z in C(c, R), f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (c w : ℂ) (R : ℝ)
Goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • (z - w) • f z) = ∮ (z : ℂ) in C(c, R), f z
-- circleIntegral.integral_undef
theorem integral_undef {f : ℂ → E} {c : ℂ} {R : ℝ} (hf : ¬CircleIntegrable f c R) :
    (∮ z in C(c, R), f z) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R : ℝ} (hf : ¬CircleIntegrable f c R)
Goal: (∮ (z : ℂ) in C(c, R), f z) = 0
-- circleIntegral.integral_sub
theorem integral_sub {f g : ℂ → E} {c : ℂ} {R : ℝ} (hf : CircleIntegrable f c R)
    (hg : CircleIntegrable g c R) :
    (∮ z in C(c, R), f z - g z) = (∮ z in C(c, R), f z) - ∮ z in C(c, R), g z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℂ → E} {c : ℂ} {R : ℝ} (hf : CircleIntegrable f c R) (hg : CircleIntegrable g c R)
Goal: (∮ (z : ℂ) in C(c, R), f z - g z) = (∮ (z : ℂ) in C(c, R), f z) - ∮ (z : ℂ) in C(c, R), g z
-- circleIntegral.norm_integral_le_of_norm_le_const'
theorem norm_integral_le_of_norm_le_const' {f : ℂ → E} {c : ℂ} {R C : ℝ}
    (hf : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ C) : ‖∮ z in C(c, R), f z‖ ≤ 2 * π * |R| * C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R C : ℝ} (hf : ∀ z ∈ Metric.sphere c |R|, ‖f z‖ ≤ C)
Goal: ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ 2 * π * |R| * C
-- circleIntegral.norm_integral_le_of_norm_le_const
theorem norm_integral_le_of_norm_le_const {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 ≤ R)
    (hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) : ‖∮ z in C(c, R), f z‖ ≤ 2 * π * R * C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 ≤ R) (hf : ∀ z ∈ Metric.sphere c R, ‖f z‖ ≤ C)
Goal: ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ 2 * π * R * C
-- circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const
theorem norm_two_pi_i_inv_smul_integral_le_of_norm_le_const {f : ℂ → E} {c : ℂ} {R C : ℝ}
    (hR : 0 ≤ R) (hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), f z‖ ≤ R * C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 ≤ R) (hf : ∀ z ∈ Metric.sphere c R, ‖f z‖ ≤ C)
Goal: ‖(2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C
-- circleIntegral.norm_integral_lt_of_norm_le_const_of_lt
theorem norm_integral_lt_of_norm_le_const_of_lt {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 < R)
    (hc : ContinuousOn f (sphere c R)) (hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C)
    (hlt : ∃ z ∈ sphere c R, ‖f z‖ < C) : ‖∮ z in C(c, R), f z‖ < 2 * π * R * C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R C : ℝ} (hR : 0 < R) (hc : ContinuousOn f (Metric.sphere c R)) (hf : ∀ z ∈ Metric.sphere c R, ‖f z‖ ≤ C) (hlt : ∃ z ∈ Metric.sphere c R, ‖f z‖ < C)
Goal: ‖∮ (z : ℂ) in C(c, R), f z‖ < 2 * π * R * C