ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.quasiMeasurePreserving_smul
theorem quasiMeasurePreserving_smul {r : ℝ} (hr : r ≠ 0) :
    QuasiMeasurePreserving (r • ·) μ μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {r : ℝ} (hr : r ≠ 0)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun x => r • x) μ μ
-- MeasureTheory.Measure.addHaar_preimage_smul
@[simp]
theorem addHaar_preimage_smul {r : ℝ} (hr : r ≠ 0) (s : Set E) :
    μ ((r • ·) ⁻¹' s) = ENNReal.ofReal (abs (r ^ finrank ℝ E)⁻¹) * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {r : ℝ} (hr : r ≠ 0) (s : Set E)
Goal: μ ((fun x => r • x) ⁻¹' s) = ENNReal.ofReal |(r ^ FiniteDimensional.finrank ℝ E)⁻¹| * μ s
-- MeasureTheory.Measure.addHaar_smul
@[simp]
theorem addHaar_smul (r : ℝ) (s : Set E) :
    μ (r • s) = ENNReal.ofReal (abs (r ^ finrank ℝ E)) * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (r : ℝ) (s : Set E)
Goal: μ (r • s) = ENNReal.ofReal |r ^ FiniteDimensional.finrank ℝ E| * μ s
-- MeasureTheory.Measure.addHaar_smul_of_nonneg
theorem addHaar_smul_of_nonneg {r : ℝ} (hr : 0 ≤ r) (s : Set E) :
    μ (r • s) = ENNReal.ofReal (r ^ finrank ℝ E) * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {r : ℝ} (hr : 0 ≤ r) (s : Set E)
Goal: μ (r • s) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ s
-- MeasureTheory.Measure.NullMeasurableSet.const_smul
theorem NullMeasurableSet.const_smul (hs : NullMeasurableSet s μ) (r : ℝ) :
    NullMeasurableSet (r • s) μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] {s : Set E} (hs : MeasureTheory.NullMeasurableSet s μ) (r : ℝ)
Goal: MeasureTheory.NullMeasurableSet (r • s) μ
-- MeasureTheory.Measure.addHaar_image_homothety
@[simp]
theorem addHaar_image_homothety (x : E) (r : ℝ) (s : Set E) :
    μ (AffineMap.homothety x r '' s) = ENNReal.ofReal (abs (r ^ finrank ℝ E)) * μ s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) (r : ℝ) (s : Set E)
Goal: μ (⇑(AffineMap.homothety x r) '' s) = ENNReal.ofReal |r ^ FiniteDimensional.finrank ℝ E| * μ s
-- MeasureTheory.Measure.addHaar_ball_center
theorem addHaar_ball_center {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
    (μ : Measure E) [IsAddHaarMeasure μ] (x : E) (r : ℝ) : μ (ball x r) = μ (ball (0 : E) r) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) (r : ℝ)
Goal: μ (Metric.ball x r) = μ (Metric.ball 0 r)
-- MeasureTheory.Measure.addHaar_closedBall_center
theorem addHaar_closedBall_center {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (x : E) (r : ℝ) :
    μ (closedBall x r) = μ (closedBall (0 : E) r) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) (r : ℝ)
Goal: μ (Metric.closedBall x r) = μ (Metric.closedBall 0 r)
-- MeasureTheory.Measure.addHaar_ball_mul_of_pos
theorem addHaar_ball_mul_of_pos (x : E) {r : ℝ} (hr : 0 < r) (s : ℝ) :
    μ (ball x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : 0 < r) (s : ℝ)
Goal: μ (Metric.ball x (r * s)) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.ball 0 s)
-- MeasureTheory.Measure.addHaar_ball_of_pos
theorem addHaar_ball_of_pos (x : E) {r : ℝ} (hr : 0 < r) :
    μ (ball x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : 0 < r)
Goal: μ (Metric.ball x r) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.ball 0 1)
-- MeasureTheory.Measure.addHaar_ball_mul
theorem addHaar_ball_mul [Nontrivial E] (x : E) {r : ℝ} (hr : 0 ≤ r) (s : ℝ) :
    μ (ball x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [Nontrivial E] (x : E) {r : ℝ} (hr : 0 ≤ r) (s : ℝ)
Goal: μ (Metric.ball x (r * s)) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.ball 0 s)
-- MeasureTheory.Measure.addHaar_ball
theorem addHaar_ball [Nontrivial E] (x : E) {r : ℝ} (hr : 0 ≤ r) :
    μ (ball x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [Nontrivial E] (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: μ (Metric.ball x r) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.ball 0 1)
-- MeasureTheory.Measure.addHaar_closedBall_mul_of_pos
theorem addHaar_closedBall_mul_of_pos (x : E) {r : ℝ} (hr : 0 < r) (s : ℝ) :
    μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : 0 < r) (s : ℝ)
Goal: μ (Metric.closedBall x (r * s)) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.closedBall 0 s)
-- MeasureTheory.Measure.addHaar_closedBall_mul
theorem addHaar_closedBall_mul (x : E) {r : ℝ} (hr : 0 ≤ r) {s : ℝ} (hs : 0 ≤ s) :
    μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : 0 ≤ r) {s : ℝ} (hs : 0 ≤ s)
Goal: μ (Metric.closedBall x (r * s)) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.closedBall 0 s)
-- MeasureTheory.Measure.addHaar_closedBall'
theorem addHaar_closedBall' (x : E) {r : ℝ} (hr : 0 ≤ r) :
    μ (closedBall x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: μ (Metric.closedBall x r) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.closedBall 0 1)
-- MeasureTheory.Measure.addHaar_closed_unit_ball_eq_addHaar_unit_ball
theorem addHaar_closed_unit_ball_eq_addHaar_unit_ball :
    μ (closedBall (0 : E) 1) = μ (ball 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure]
Goal: μ (Metric.closedBall 0 1) = μ (Metric.ball 0 1)
-- MeasureTheory.Measure.addHaar_closedBall
theorem addHaar_closedBall (x : E) {r : ℝ} (hr : 0 ≤ r) :
    μ (closedBall x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : 0 ≤ r)
Goal: μ (Metric.closedBall x r) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.ball 0 1)
-- MeasureTheory.Measure.addHaar_closedBall_eq_addHaar_ball
theorem addHaar_closedBall_eq_addHaar_ball [Nontrivial E] (x : E) (r : ℝ) :
    μ (closedBall x r) = μ (ball x r) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [Nontrivial E] (x : E) (r : ℝ)
Goal: μ (Metric.closedBall x r) = μ (Metric.ball x r)
-- MeasureTheory.Measure.addHaar_sphere_of_ne_zero
theorem addHaar_sphere_of_ne_zero (x : E) {r : ℝ} (hr : r ≠ 0) : μ (sphere x r) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (x : E) {r : ℝ} (hr : r ≠ 0)
Goal: μ (Metric.sphere x r) = 0
-- MeasureTheory.Measure.addHaar_sphere
theorem addHaar_sphere [Nontrivial E] (x : E) (r : ℝ) : μ (sphere x r) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [Nontrivial E] (x : E) (r : ℝ)
Goal: μ (Metric.sphere x r) = 0
-- MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul
theorem addHaar_singleton_add_smul_div_singleton_add_smul {r : ℝ} (hr : r ≠ 0) (x y : E)
    (s t : Set E) : μ ({x} + r • s) / μ ({y} + r • t) = μ s / μ t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {r : ℝ} (hr : r ≠ 0) (x y : E) (s t : Set E)
Goal: μ ({x} + r • s) / μ ({y} + r • t) = μ s / μ t
-- MeasureTheory.Measure.addHaar_parallelepiped
theorem addHaar_parallelepiped (b : Basis ι ℝ G) (v : ι → G) :
    b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v| 
Arguments: {ι : Type u_3} {G : Type u_4} [Fintype ι] [DecidableEq ι] [NormedAddCommGroup G] [NormedSpace ℝ G] [MeasurableSpace G] [BorelSpace G] (b : Basis ι ℝ G) (v : ι → G)
Goal: b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|
-- AlternatingMap.measure_def
noncomputable irreducible_def _root_.AlternatingMap.measure (ω : G [⋀^Fin n]→ₗ[ℝ] ℝ) :
    Measure G 
Arguments: {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] [MeasurableSpace G] [BorelSpace G] [FiniteDimensional ℝ G] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ G = n)] (ω : G [⋀^Fin n]→ₗ[ℝ] ℝ)
Goal: ω.measure = ‖ω ⇑(FiniteDimensional.finBasisOfFinrankEq ℝ G ⋯)‖₊ • (FiniteDimensional.finBasisOfFinrankEq ℝ G ⋯).addHaar
-- AlternatingMap.measure_parallelepiped
theorem _root_.AlternatingMap.measure_parallelepiped (ω : G [⋀^Fin n]→ₗ[ℝ] ℝ)
    (v : Fin n → G) : ω.measure (parallelepiped v) = ENNReal.ofReal |ω v| 
Arguments: {G : Type u_4} [NormedAddCommGroup G] [NormedSpace ℝ G] [MeasurableSpace G] [BorelSpace G] [FiniteDimensional ℝ G] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ G = n)] (ω : G [⋀^Fin n]→ₗ[ℝ] ℝ) (v : Fin n → G)
Goal: ω.measure (parallelepiped v) = ENNReal.ofReal |ω v|
-- MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux1
theorem tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (u : Set E) (h'u : μ u ≠ 0) (t_bound : t ⊆ closedBall 0 1) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (x : E) (h : Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t u : Set E) (h'u : μ u ≠ 0) (t_bound : t ⊆ Metric.closedBall 0 1)
Goal: Filter.Tendsto (fun r => μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0)
-- MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux2
theorem tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (u : Set E) (h'u : μ u ≠ 0) (R : ℝ) (Rpos : 0 < R) (t_bound : t ⊆ closedBall 0 R) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (x : E) (h : Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t u : Set E) (h'u : μ u ≠ 0) (R : ℝ) (Rpos : 0 < R) (t_bound : t ⊆ Metric.closedBall 0 R)
Goal: Filter.Tendsto (fun r => μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0)
-- MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (x : E) (h : Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E) (ht : MeasurableSet t) (h''t : μ t ≠ ⊤)
Goal: Filter.Tendsto (fun r => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
-- MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_aux
theorem tendsto_addHaar_inter_smul_one_of_density_one_aux (s : Set E) (hs : MeasurableSet s)
    (x : E) (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1))
    (t : Set E) (ht : MeasurableSet t) (h't : μ t ≠ 0) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (hs : MeasurableSet s) (x : E) (h : Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 1)) (t : Set E) (ht : MeasurableSet t) (h't : μ t ≠ 0) (h''t : μ t ≠ ⊤)
Goal: Filter.Tendsto (fun r => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
-- MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one
theorem tendsto_addHaar_inter_smul_one_of_density_one (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)) (t : Set E)
    (ht : MeasurableSet t) (h't : μ t ≠ 0) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (x : E) (h : Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 1)) (t : Set E) (ht : MeasurableSet t) (h't : μ t ≠ 0) (h''t : μ t ≠ ⊤)
Goal: Filter.Tendsto (fun r => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
-- MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one
theorem eventually_nonempty_inter_smul_of_density_one (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)) (t : Set E)
    (ht : MeasurableSet t) (h't : μ t ≠ 0) :
    ∀ᶠ r in 𝓝[>] (0 : ℝ), (s ∩ ({x} + r • t)).Nonempty 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (s : Set E) (x : E) (h : Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 1)) (t : Set E) (ht : MeasurableSet t) (h't : μ t ≠ 0)
Goal: ∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • t)).Nonempty
-- WithLp.prod_inner_apply
@[simp]
theorem prod_inner_apply (x y : WithLp 2 (E × F)) :
    inner (𝕜 := 𝕜) x y = inner x.fst y.fst + inner x.snd y.snd 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] (x y : WithLp 2 (E × F))
Goal: ⟪x, y⟫_𝕜 = ⟪x.1, y.1⟫_𝕜 + ⟪x.2, y.2⟫_𝕜
-- OrthonormalBasis.prod_apply
@[simp] theorem prod_apply (v : OrthonormalBasis ι₁ 𝕜 E) (w : OrthonormalBasis ι₂ 𝕜 F) :
    ∀ i : ι₁ ⊕ ι₂, v.prod w i =
      Sum.elim ((LinearMap.inl 𝕜 E F) ∘ v) ((LinearMap.inr 𝕜 E F) ∘ w) i 
Arguments: {𝕜 : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [Fintype ι₁] [Fintype ι₂] (v : OrthonormalBasis ι₁ 𝕜 E) (w : OrthonormalBasis ι₂ 𝕜 F) (i : ι₁ ⊕ ι₂)
Goal: (v.prod w) i = Sum.elim (⇑(LinearMap.inl 𝕜 E F) ∘ ⇑v) (⇑(LinearMap.inr 𝕜 E F) ∘ ⇑w) i
-- isConformalMap_iff
theorem isConformalMap_iff (f : E →L[ℝ] F) :
    IsConformalMap f ↔ ∃ c : ℝ, 0 < c ∧ ∀ u v : E, ⟪f u, f v⟫ = c * ⟪u, v⟫ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] (f : E →L[ℝ] F)
Goal: IsConformalMap f ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪f u, f v⟫_ℝ = c * ⟪u, v⟫_ℝ
-- Euclidean.ball_eq_preimage
theorem ball_eq_preimage (x : E) (r : ℝ) :
    ball x r = toEuclidean ⁻¹' Metric.ball (toEuclidean x) r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] (x : E) (r : ℝ)
Goal: Euclidean.ball x r = ⇑toEuclidean ⁻¹' Metric.ball (toEuclidean x) r
-- Euclidean.closedBall_eq_preimage
theorem closedBall_eq_preimage (x : E) (r : ℝ) :
    closedBall x r = toEuclidean ⁻¹' Metric.closedBall (toEuclidean x) r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] (x : E) (r : ℝ)
Goal: Euclidean.closedBall x r = ⇑toEuclidean ⁻¹' Metric.closedBall (toEuclidean x) r
-- Euclidean.ball_subset_closedBall
theorem ball_subset_closedBall {x : E} {r : ℝ} : ball x r ⊆ closedBall x r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ}
Goal: Euclidean.ball x r ⊆ Euclidean.closedBall x r
-- Euclidean.isOpen_ball
theorem isOpen_ball {x : E} {r : ℝ} : IsOpen (ball x r) 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ}
Goal: IsOpen (Euclidean.ball x r)
-- Euclidean.mem_ball_self
theorem mem_ball_self {x : E} {r : ℝ} (hr : 0 < r) : x ∈ ball x r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ} (hr : 0 < r)
Goal: x ∈ Euclidean.ball x r
-- Euclidean.closedBall_eq_image
theorem closedBall_eq_image (x : E) (r : ℝ) :
    closedBall x r = toEuclidean.symm '' Metric.closedBall (toEuclidean x) r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] (x : E) (r : ℝ)
Goal: Euclidean.closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r
-- Euclidean.isCompact_closedBall
nonrec theorem isCompact_closedBall {x : E} {r : ℝ} : IsCompact (closedBall x r) 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ}
Goal: IsCompact (Euclidean.closedBall x r)
-- Euclidean.isClosed_closedBall
theorem isClosed_closedBall {x : E} {r : ℝ} : IsClosed (closedBall x r) 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ}
Goal: IsClosed (Euclidean.closedBall x r)
-- Euclidean.closure_ball
nonrec theorem closure_ball (x : E) {r : ℝ} (h : r ≠ 0) : closure (ball x r) = closedBall x r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] (x : E) {r : ℝ} (h : r ≠ 0)
Goal: closure (Euclidean.ball x r) = Euclidean.closedBall x r
-- Euclidean.exists_pos_lt_subset_ball
nonrec theorem exists_pos_lt_subset_ball {R : ℝ} {s : Set E} {x : E} (hR : 0 < R) (hs : IsClosed s)
    (h : s ⊆ ball x R) : ∃ r ∈ Ioo 0 R, s ⊆ ball x r 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {R : ℝ} {s : Set E} {x : E} (hR : 0 < R) (hs : IsClosed s) (h : s ⊆ Euclidean.ball x R)
Goal: ∃ r ∈ Set.Ioo 0 R, s ⊆ Euclidean.ball x r
-- Euclidean.nhds_basis_closedBall
theorem nhds_basis_closedBall {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (closedBall x) 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E}
Goal: (𝓝 x).HasBasis (fun r => 0 < r) (Euclidean.closedBall x)
-- Euclidean.closedBall_mem_nhds
theorem closedBall_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : closedBall x r ∈ 𝓝 x 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ} (hr : 0 < r)
Goal: Euclidean.closedBall x r ∈ 𝓝 x
-- Euclidean.nhds_basis_ball
theorem nhds_basis_ball {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (ball x) 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E}
Goal: (𝓝 x).HasBasis (fun r => 0 < r) (Euclidean.ball x)
-- Euclidean.ball_mem_nhds
theorem ball_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : ball x r ∈ 𝓝 x 
Arguments: {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [T2Space E] [Module ℝ E] [ContinuousSMul ℝ E] [FiniteDimensional ℝ E] {x : E} {r : ℝ} (hr : 0 < r)
Goal: Euclidean.ball x r ∈ 𝓝 x
-- ContDiff.euclidean_dist
theorem ContDiff.euclidean_dist (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (h : ∀ x, f x ≠ g x) :
    ContDiff ℝ n fun x => Euclidean.dist (f x) (g x) 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace ℝ G] [FiniteDimensional ℝ G] {f g : F → G} {n : ℕ∞} (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (h : ∀ (x : F), f x ≠ g x)
Goal: ContDiff ℝ n fun x => Euclidean.dist (f x) (g x)
-- AnalyticAt.meromorphicAt
lemma AnalyticAt.meromorphicAt {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜 f x) :
    MeromorphicAt f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜 f x)
Goal: MeromorphicAt f x
-- MeromorphicAt.id
lemma id (x : 𝕜) : MeromorphicAt id x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: MeromorphicAt id x
-- MeromorphicAt.const
lemma const (e : E) (x : 𝕜) : MeromorphicAt (fun _ ↦ e) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (e : E) (x : 𝕜)
Goal: MeromorphicAt (fun x => e) x
-- MeromorphicAt.add
lemma add {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f + g) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x)
Goal: MeromorphicAt (f + g) x