ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.measurePreserving_prod_add_swap_right
@[to_additive measurePreserving_prod_add_swap_right
" The map `(x, y) ↦ (y, x + y)` sends the measure `μ × ν` to `ν × μ`. "]
theorem measurePreserving_prod_mul_swap_right [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.2, z.1 * z.2)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.1 + z.2)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_prod_mul_swap_right
@[to_additive measurePreserving_prod_add_swap_right
" The map `(x, y) ↦ (y, x + y)` sends the measure `μ × ν` to `ν × μ`. "]
theorem measurePreserving_prod_mul_swap_right [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.2, z.1 * z.2)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [μ.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.1 * z.2)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_add_prod
@[to_additive measurePreserving_add_prod
" The map `(x, y) ↦ (x + y, y)` preserves the measure `μ × ν`. "]
theorem measurePreserving_mul_prod [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.1 * z.2, z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1 + z.2, z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_mul_prod
@[to_additive measurePreserving_add_prod
" The map `(x, y) ↦ (x + y, y)` preserves the measure `μ × ν`. "]
theorem measurePreserving_mul_prod [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.1 * z.2, z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [μ.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1 * z.2, z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_div
@[to_additive measurePreserving_prod_sub "The map `(x, y) ↦ (x, y - x)` is measure-preserving."]
theorem measurePreserving_prod_div [IsMulRightInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.2 / z.1)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [ν.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.2 / z.1)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_sub
@[to_additive measurePreserving_prod_sub "The map `(x, y) ↦ (x, y - x)` is measure-preserving."]
theorem measurePreserving_prod_div [IsMulRightInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.2 / z.1)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [ν.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.2 - z.1)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_div_swap
@[to_additive measurePreserving_prod_sub_swap
      "The map `(x, y) ↦ (y, x - y)` sends `μ × ν` to `ν × μ`."]
theorem measurePreserving_prod_div_swap [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.2, z.1 / z.2)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.1 / z.2)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_prod_sub_swap
@[to_additive measurePreserving_prod_sub_swap
      "The map `(x, y) ↦ (y, x - y)` sends `μ × ν` to `ν × μ`."]
theorem measurePreserving_prod_div_swap [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.2, z.1 / z.2)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.1 - z.2)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_sub_prod
@[to_additive measurePreserving_sub_prod
" The map `(x, y) ↦ (x - y, y)` preserves the measure `μ × ν`. "]
theorem measurePreserving_div_prod [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.1 / z.2, z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1 - z.2, z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_div_prod
@[to_additive measurePreserving_sub_prod
" The map `(x, y) ↦ (x - y, y)` preserves the measure `μ × ν`. "]
theorem measurePreserving_div_prod [IsMulRightInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.1 / z.2, z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1 / z.2, z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_add_prod_neg_right
@[to_additive measurePreserving_add_prod_neg_right
"The map `(x, y) ↦ (x + y, - x)` is measure-preserving."]
theorem measurePreserving_mul_prod_inv_right [IsMulRightInvariant μ] [IsMulRightInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1 * z.2, z.1⁻¹)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant] [ν.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1 + z.2, -z.1)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_mul_prod_inv_right
@[to_additive measurePreserving_add_prod_neg_right
"The map `(x, y) ↦ (x + y, - x)` is measure-preserving."]
theorem measurePreserving_mul_prod_inv_right [IsMulRightInvariant μ] [IsMulRightInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1 * z.2, z.1⁻¹)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant] [ν.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1 * z.2, z.1⁻¹)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.quasiMeasurePreserving_neg_of_right_invariant
@[to_additive]
theorem quasiMeasurePreserving_inv_of_right_invariant [IsMulRightInvariant μ] :
    QuasiMeasurePreserving (Inv.inv : G → G) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving Neg.neg μ μ
-- MeasureTheory.quasiMeasurePreserving_inv_of_right_invariant
@[to_additive]
theorem quasiMeasurePreserving_inv_of_right_invariant [IsMulRightInvariant μ] :
    QuasiMeasurePreserving (Inv.inv : G → G) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving Inv.inv μ μ
-- MeasureTheory.quasiMeasurePreserving_div_left
@[to_additive]
theorem quasiMeasurePreserving_div_left [IsMulLeftInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => g / h) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => g / h) μ μ
-- MeasureTheory.quasiMeasurePreserving_sub_left
@[to_additive]
theorem quasiMeasurePreserving_div_left [IsMulLeftInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => g / h) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => g - h) μ μ
-- MeasureTheory.quasiMeasurePreserving_sub_left_of_right_invariant
@[to_additive]
theorem quasiMeasurePreserving_div_left_of_right_invariant [IsMulRightInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => g / h) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => g - h) μ μ
-- MeasureTheory.quasiMeasurePreserving_div_left_of_right_invariant
@[to_additive]
theorem quasiMeasurePreserving_div_left_of_right_invariant [IsMulRightInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => g / h) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => g / h) μ μ
-- MeasureTheory.quasiMeasurePreserving_sub_of_right_invariant
@[to_additive]
theorem quasiMeasurePreserving_div_of_right_invariant [IsMulRightInvariant μ] :
    QuasiMeasurePreserving (fun p : G × G => p.1 / p.2) (μ.prod ν) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun p => p.1 - p.2) (μ.prod ν) μ
-- MeasureTheory.quasiMeasurePreserving_div_of_right_invariant
@[to_additive]
theorem quasiMeasurePreserving_div_of_right_invariant [IsMulRightInvariant μ] :
    QuasiMeasurePreserving (fun p : G × G => p.1 / p.2) (μ.prod ν) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun p => p.1 / p.2) (μ.prod ν) μ
-- MeasureTheory.quasiMeasurePreserving_sub
@[to_additive]
theorem quasiMeasurePreserving_div [IsMulLeftInvariant μ] :
    QuasiMeasurePreserving (fun p : G × G => p.1 / p.2) (μ.prod ν) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun p => p.1 - p.2) (μ.prod ν) μ
-- MeasureTheory.quasiMeasurePreserving_div
@[to_additive]
theorem quasiMeasurePreserving_div [IsMulLeftInvariant μ] :
    QuasiMeasurePreserving (fun p : G × G => p.1 / p.2) (μ.prod ν) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun p => p.1 / p.2) (μ.prod ν) μ
-- MeasureTheory.quasiMeasurePreserving_add_right
@[to_additive
"A *left*-invariant measure is quasi-preserved by *right*-addition.
This should not be confused with `(measurePreserving_add_right μ g).quasiMeasurePreserving`. "]
theorem quasiMeasurePreserving_mul_right [IsMulLeftInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => h * g) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => h + g) μ μ
-- MeasureTheory.quasiMeasurePreserving_mul_right
@[to_additive
"A *left*-invariant measure is quasi-preserved by *right*-addition.
This should not be confused with `(measurePreserving_add_right μ g).quasiMeasurePreserving`. "]
theorem quasiMeasurePreserving_mul_right [IsMulLeftInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => h * g) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => h * g) μ μ
-- MeasureTheory.quasiMeasurePreserving_mul_left
@[to_additive
"A *right*-invariant measure is quasi-preserved by *left*-addition.
This should not be confused with `(measurePreserving_add_left μ g).quasiMeasurePreserving`. "]
theorem quasiMeasurePreserving_mul_left [IsMulRightInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => g * h) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulRightInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => g * h) μ μ
-- MeasureTheory.quasiMeasurePreserving_add_left
@[to_additive
"A *right*-invariant measure is quasi-preserved by *left*-addition.
This should not be confused with `(measurePreserving_add_left μ g).quasiMeasurePreserving`. "]
theorem quasiMeasurePreserving_mul_left [IsMulRightInvariant μ] (g : G) :
    QuasiMeasurePreserving (fun h : G => g * h) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddRightInvariant] (g : G)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (fun h => g + h) μ μ
-- MeasureTheory.measure_unitBall_eq_integral_div_gamma
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) 
Arguments: {E : Type u_1} {p : ℝ} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hp : 0 < p)
Goal: μ (Metric.ball 0 1) =
  ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / p + 1))
-- MeasureTheory.measure_lt_one_eq_integral_div_gamma
theorem MeasureTheory.measure_lt_one_eq_integral_div_gamma {p : ℝ} (hp : 0 < p) :
    μ {x : E | g x < 1} =
      .ofReal ((∫ (x : E), Real.exp (- (g x) ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [FiniteDimensional ℝ E] [mE : MeasurableSpace E] [tE : TopologicalSpace E] [TopologicalAddGroup E] [BorelSpace E] [T2Space E] [ContinuousSMul ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {g : E → ℝ} (h1 : g 0 = 0) (h2 : ∀ (x : E), g (-x) = g x) (h3 : ∀ (x y : E), g (x + y) ≤ g x + g y) (h4 : ∀ {x : E}, g x = 0 → x = 0) (h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x) {p : ℝ} (hp : 0 < p)
Goal: μ {x | g x < 1} =
  ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / p + 1))
-- MeasureTheory.measure_le_eq_lt
theorem MeasureTheory.measure_le_eq_lt [Nontrivial E] (r : ℝ) :
    μ {x : E | g x ≤ r} = μ {x : E | g x < r} 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [FiniteDimensional ℝ E] [mE : MeasurableSpace E] [tE : TopologicalSpace E] [TopologicalAddGroup E] [BorelSpace E] [T2Space E] [ContinuousSMul ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {g : E → ℝ} (h1 : g 0 = 0) (h2 : ∀ (x : E), g (-x) = g x) (h3 : ∀ (x y : E), g (x + y) ≤ g x + g y) (h4 : ∀ {x : E}, g x = 0 → x = 0) (h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x) [Nontrivial E] (r : ℝ)
Goal: μ {x | g x ≤ r} = μ {x | g x < r}
-- MeasureTheory.volume_sum_rpow_lt_one
theorem MeasureTheory.volume_sum_rpow_lt_one :
    volume {x : ι → ℝ | ∑ i, |x i| ^ p < 1} =
      .ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) 
Arguments: (ι : Type u_1) [Fintype ι] {p : ℝ} (hp : 1 ≤ p)
Goal: MeasureTheory.volume {x | ∑ i : ι, |x i| ^ p < 1} =
  ENNReal.ofReal ((2 * Real.Gamma (1 / p + 1)) ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / p + 1))
-- MeasureTheory.volume_sum_rpow_lt
theorem MeasureTheory.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) < r} = (.ofReal r) ^ card ι *
      .ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) 
Arguments: (ι : Type u_1) [Fintype ι] [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ)
Goal: MeasureTheory.volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) < r} =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal ((2 * Real.Gamma (1 / p + 1)) ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / p + 1))
-- MeasureTheory.volume_sum_rpow_le
theorem MeasureTheory.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ card ι *
      .ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) 
Arguments: (ι : Type u_1) [Fintype ι] [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ)
Goal: MeasureTheory.volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal ((2 * Real.Gamma (1 / p + 1)) ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / p + 1))
-- Complex.volume_sum_rpow_lt_one
theorem Complex.volume_sum_rpow_lt_one {p : ℝ} (hp : 1 ≤ p) :
    volume {x : ι → ℂ | ∑ i, ‖x i‖ ^ p < 1} =
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) 
Arguments: (ι : Type u_1) [Fintype ι] {p : ℝ} (hp : 1 ≤ p)
Goal: MeasureTheory.volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
  ENNReal.ofReal ((Real.pi * Real.Gamma (2 / p + 1)) ^ Fintype.card ι / Real.Gamma (2 * ↑(Fintype.card ι) / p + 1))
-- Complex.volume_sum_rpow_lt
theorem Complex.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = (.ofReal r) ^ (2 * card ι) *
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) 
Arguments: (ι : Type u_1) [Fintype ι] [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ)
Goal: MeasureTheory.volume {x | (∑ i : ι, ‖x i‖ ^ p) ^ (1 / p) < r} =
  ENNReal.ofReal r ^ (2 * Fintype.card ι) *
    ENNReal.ofReal ((Real.pi * Real.Gamma (2 / p + 1)) ^ Fintype.card ι / Real.Gamma (2 * ↑(Fintype.card ι) / p + 1))
-- Complex.volume_sum_rpow_le
theorem Complex.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ (2 * card ι) *
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) 
Arguments: (ι : Type u_1) [Fintype ι] [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ)
Goal: MeasureTheory.volume {x | (∑ i : ι, ‖x i‖ ^ p) ^ (1 / p) ≤ r} =
  ENNReal.ofReal r ^ (2 * Fintype.card ι) *
    ENNReal.ofReal ((Real.pi * Real.Gamma (2 / p + 1)) ^ Fintype.card ι / Real.Gamma (2 * ↑(Fintype.card ι) / p + 1))
-- EuclideanSpace.volume_ball
theorem EuclideanSpace.volume_ball (x : EuclideanSpace ℝ ι) (r : ℝ) :
    volume (Metric.ball x r) = (.ofReal r) ^ card ι *
      .ofReal (Real.sqrt π ^ card ι / Gamma (card ι / 2 + 1)) 
Arguments: (ι : Type u_1) [Nonempty ι] [Fintype ι] (x : EuclideanSpace ℝ ι) (r : ℝ)
Goal: MeasureTheory.volume (Metric.ball x r) =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal (√Real.pi ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / 2 + 1))
-- EuclideanSpace.volume_closedBall
theorem EuclideanSpace.volume_closedBall (x : EuclideanSpace ℝ ι) (r : ℝ) :
    volume (Metric.closedBall x r) = (.ofReal r) ^ card ι *
      .ofReal (sqrt π ^ card ι / Gamma (card ι / 2 + 1)) 
Arguments: (ι : Type u_1) [Nonempty ι] [Fintype ι] (x : EuclideanSpace ℝ ι) (r : ℝ)
Goal: MeasureTheory.volume (Metric.closedBall x r) =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal (√Real.pi ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / 2 + 1))
-- InnerProductSpace.volume_ball
theorem InnerProductSpace.volume_ball (x : E) (r : ℝ) :
    volume (Metric.ball x r) = (.ofReal r) ^ finrank ℝ E *
      .ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] (x : E) (r : ℝ)
Goal: MeasureTheory.volume (Metric.ball x r) =
  ENNReal.ofReal r ^ FiniteDimensional.finrank ℝ E *
    ENNReal.ofReal (√Real.pi ^ FiniteDimensional.finrank ℝ E / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / 2 + 1))
-- InnerProductSpace.volume_closedBall
theorem InnerProductSpace.volume_closedBall (x : E) (r : ℝ) :
    volume (Metric.closedBall x r) = (.ofReal r) ^ finrank ℝ E *
      .ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] (x : E) (r : ℝ)
Goal: MeasureTheory.volume (Metric.closedBall x r) =
  ENNReal.ofReal r ^ FiniteDimensional.finrank ℝ E *
    ENNReal.ofReal (√Real.pi ^ FiniteDimensional.finrank ℝ E / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / 2 + 1))
-- Complex.volume_ball
@[simp]
theorem Complex.volume_ball (a : ℂ) (r : ℝ) :
    volume (Metric.ball a r) = .ofReal r ^ 2 * NNReal.pi 
Arguments: (a : ℂ) (r : ℝ)
Goal: MeasureTheory.volume (Metric.ball a r) = ENNReal.ofReal r ^ 2 * ↑NNReal.pi
-- Complex.volume_closedBall
@[simp]
theorem Complex.volume_closedBall (a : ℂ) (r : ℝ) :
    volume (Metric.closedBall a r) = .ofReal r ^ 2 * NNReal.pi 
Arguments: (a : ℂ) (r : ℝ)
Goal: MeasureTheory.volume (Metric.closedBall a r) = ENNReal.ofReal r ^ 2 * ↑NNReal.pi
-- FormalMultilinearSeries.radius_eq_liminf
theorem radius_eq_liminf :
    p.radius = liminf (fun n => (1 / (‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop 
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] (p : FormalMultilinearSeries 𝕜 E F)
Goal: p.radius = Filter.liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) Filter.atTop
-- MeasureTheory.FinMeasAdditive.zero
theorem zero : FinMeasAdditive μ (0 : Set α → β) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β]
Goal: MeasureTheory.FinMeasAdditive μ 0
-- MeasureTheory.FinMeasAdditive.add
theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
    FinMeasAdditive μ (T + T') 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] {T T' : Set α → β} (hT : MeasureTheory.FinMeasAdditive μ T) (hT' : MeasureTheory.FinMeasAdditive μ T')
Goal: MeasureTheory.FinMeasAdditive μ (T + T')
-- MeasureTheory.FinMeasAdditive.smul
theorem smul [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) :
    FinMeasAdditive μ fun s => c • T s 
Arguments: {α : Type u_1} {𝕜 : Type u_6} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set α → β} [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : MeasureTheory.FinMeasAdditive μ T) (c : 𝕜)
Goal: MeasureTheory.FinMeasAdditive μ fun s => c • T s