awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
x : E
⊢ 0 ≤ (1 + ‖x‖) ^ (-r)

Tactic:
positivity
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
⊢ ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)

Tactic:
have h_pos : ∀ x : E, 0 ≤ (1 + ‖x‖) ^ (-r) := fun x ↦ by positivity
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
rw : ∫⁻ (t : ℝ) in Ioi 0, (μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} < ∞
⊢ ∫⁻ (x : E), ENNReal.ofReal ((1 + ‖x‖) ^ (-r)) ∂μ < ∞

Tactic:
rw [lintegral_eq_lintegral_meas_le μ (Eventually.of_forall h_pos) h_meas.aemeasurable]

Premises used: Measurable.aemeasurable, Filter.Eventually.of_forall, MeasureTheory.lintegral_eq_lintegral_meas_le
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
t : ℝ
ht : 0 < t
h : Eq.{u_1 + 1} (α := Set E) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
    ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)

Tactic:
congr 1
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
t : ℝ
ht : 0 < t
h :
  ∀ (x : E),
    Membership.mem (γ := Set E) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} x ↔
      Membership.mem (γ := Set E) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) x
⊢ Eq.{u_1 + 1} (α := Set E) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))

Tactic:
ext x
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
t : ℝ
ht : 0 < t
x : E
h : t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1
⊢ Membership.mem (γ := Set E) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} x ↔
    Membership.mem (γ := Set E) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) x

Tactic:
simp only [mem_setOf_eq, mem_closedBall_zero_iff]

Premises used: Set.mem_setOf_eq, mem_closedBall_zero_iff
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
t : ℝ
ht : 0 < t
x : E
⊢ t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1

Tactic:
le_rpow_one_add_norm_iff_norm_le hr (mem_Ioi.mp ht) x

Premises used: Set.mem_Ioi, le_rpow_one_add_norm_iff_norm_le, Iff.mp
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
t : ℝ
ht : 0 < t
x : E
⊢ t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1

Tactic:
exact le_rpow_one_add_norm_iff_norm_le hr (mem_Ioi.mp ht) x

Premises used: Set.mem_Ioi, le_rpow_one_add_norm_iff_norm_le, Iff.mp
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
⊢ ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)

Tactic:
have h_int : ∀ t, 0 < t → μ {a : E | t ≤ (1 + ‖a‖) ^ (-r)} =
      μ (Metric.closedBall (0 : E) (t ^ (-r⁻¹) - 1)) := fun t ht ↦ by
    congr 1
    ext x
    simp only [mem_setOf_eq, mem_closedBall_zero_iff]
    exact le_rpow_one_add_norm_iff_norm_le hr (mem_Ioi.mp ht) x
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
rw : ∫⁻ (x : ℝ) in Ioi 0, (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (x ^ (-r⁻¹) - 1)) ∂volume < ∞
⊢ ∫⁻ (t : ℝ) in Ioi 0, (μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} < ∞

Tactic:
rw [setLIntegral_congr_fun measurableSet_Ioi h_int]

Premises used: MeasureTheory.setLIntegral_congr_fun, measurableSet_Ioi
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
set :
  let f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1));
  lintegral (m := measurableSpace) (Measure.restrict volume (Ioi 0)) f < ∞
⊢ ∫⁻ (x : ℝ) in Ioi 0, (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (x ^ (-r⁻¹) - 1)) ∂volume < ∞

Tactic:
set f := fun t : ℝ ↦ μ (Metric.closedBall (0 : E) (t ^ (-r⁻¹) - 1))

Premises used: HSub.hSub, Inv.inv, HPow.hPow, Metric.closedBall, OfNat.ofNat, DFunLike.coe, Neg.neg
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
⊢ ∫⁻ (t : ℝ) in Ioi 0, f t ≤ ∫⁻ (t : ℝ) in Ioc 0 1 ∪ Ioi 1, f t

Tactic:
lintegral_mono_set Ioi_subset_Ioc_union_Ioi

Premises used: Set.Ioi_subset_Ioc_union_Ioi, MeasureTheory.lintegral_mono_set
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
⊢ ∫⁻ (t : ℝ) in Ioc 0 1 ∪ Ioi 1, f t ≤ (∫⁻ (t : ℝ) in Ioc 0 1, f t) + ∫⁻ (t : ℝ) in Ioi 1, f t

Tactic:
lintegral_union_le _ _ _

Premises used: MeasureTheory.lintegral_union_le
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
⊢ (∫⁻ (t : ℝ) in Ioc 0 1, f t) + ∫⁻ (t : ℝ) in Ioi 1, f t < ∞

Tactic:
ENNReal.add_lt_top.2 ⟨?_, ?_⟩

Premises used: Iff.mpr, And.intro, ENNReal.add_lt_top
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
calc_1 : ∫⁻ (t : ℝ) in Ioc 0 1, f t < ∞
calc_2 : ∫⁻ (t : ℝ) in Ioi 1, f t < ∞
⊢ lintegral (m := measurableSpace) (Measure.restrict volume (Ioi 0)) f < ∞

Tactic:
calc
    ∫⁻ t in Ioi 0, f t ≤ ∫⁻ t in Ioc 0 1 ∪ Ioi 1, f t := lintegral_mono_set Ioi_subset_Ioc_union_Ioi
    _ ≤ (∫⁻ t in Ioc 0 1, f t) + ∫⁻ t in Ioi 1, f t := lintegral_union_le _ _ _
    _ < ∞ := ENNReal.add_lt_top.2 ⟨?_, ?_⟩

Premises used: Iff.mpr, Set.Ioc, Set.Ioi, Set.Ioi_subset_Ioc_union_Ioi, MeasureTheory.Measure.restrict, MeasureTheory.lintegral_union_le, MeasureTheory.MeasureSpace.volume, And.intro, MeasureTheory.lintegral_mono_set, ENNReal.add_lt_top, OfNat.ofNat, MeasureTheory.lintegral
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioc 0 1) t
⊢ f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB

Tactic:
μ.addHaar_closedBall (0 : E) ?_

Premises used: MeasureTheory.Measure.addHaar_closedBall, OfNat.ofNat
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioc 0 1) t
refine : 0 ≤ t ^ (-r⁻¹) - 1
⊢ f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB

Tactic:
refine μ.addHaar_closedBall (0 : E) ?_

Premises used: MeasureTheory.Measure.addHaar_closedBall, OfNat.ofNat
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioc 0 1) t
rw : 1 ≤ t ^ (-r⁻¹)
⊢ 0 ≤ t ^ (-r⁻¹) - 1

Tactic:
rw [sub_nonneg]

Premises used: sub_nonneg
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioc 0 1) t
⊢ -r⁻¹ ≤ 0

Tactic:
simp [hr.le]

Premises used: LT.lt.le
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioc 0 1) t
⊢ 1 ≤ t ^ (-r⁻¹)

Tactic:
exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])

Premises used: Real.one_le_rpow_of_pos_of_le_one_of_nonpos, And.right, And.left
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
⊢ ∀ (t : ℝ), Membership.mem (γ := Set ℝ) (Ioc 0 1) t → f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB

Tactic:
have h_int' : ∀ t ∈ Ioc (0 : ℝ) 1,
        f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB := fun t ht ↦ by
      refine μ.addHaar_closedBall (0 : E) ?_
      rw [sub_nonneg]
      exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
h_int' : ∀ (t : ℝ), Membership.mem (γ := Set ℝ) (Ioc 0 1) t → f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB
calc_1 :
  HMul.hMul (β := ℝ≥0∞) (∫⁻ (a : ℝ) in Ioc 0 1, ENNReal.ofReal ((a ^ (-r⁻¹) - 1) ^ finrank ℝ E) ∂volume)
      ((μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1) : ℝ≥0∞) <
    ∞
⊢ ∫⁻ (t : ℝ) in Ioc 0 1, f t < ∞

Tactic:
rw [setLIntegral_congr_fun measurableSet_Ioc h_int',
      lintegral_mul_const' _ _ measure_ball_lt_top.ne]

Premises used: MeasureTheory.setLIntegral_congr_fun, MeasureTheory.measure_ball_lt_top, MeasureTheory.lintegral_mul_const', measurableSet_Ioc, LT.lt.ne
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
h_int' : ∀ (t : ℝ), Membership.mem (γ := Set ℝ) (Ioc 0 1) t → f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB
⊢ HMul.hMul (β := ℝ≥0∞) (∫⁻ (a : ℝ) in Ioc 0 1, ENNReal.ofReal ((a ^ (-r⁻¹) - 1) ^ finrank ℝ E) ∂volume)
      ((μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1) : ℝ≥0∞) <
    ∞

Tactic:
ENNReal.mul_lt_top
      (finite_integral_rpow_sub_one_pow_aux (finrank ℝ E) hnr) measure_ball_lt_top

Premises used: Real, finite_integral_rpow_sub_one_pow_aux, ENNReal.mul_lt_top, MeasureTheory.measure_ball_lt_top, Module.finrank
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
h_int' : ∀ (t : ℝ), Membership.mem (γ := Set ℝ) (Ioc 0 1) t → f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB
⊢ HMul.hMul (β := ℝ≥0∞) (∫⁻ (a : ℝ) in Ioc 0 1, ENNReal.ofReal ((a ^ (-r⁻¹) - 1) ^ finrank ℝ E) ∂volume)
      ((μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1) : ℝ≥0∞) <
    ∞

Tactic:
exact ENNReal.mul_lt_top
      (finite_integral_rpow_sub_one_pow_aux (finrank ℝ E) hnr) measure_ball_lt_top

Premises used: Real, finite_integral_rpow_sub_one_pow_aux, ENNReal.mul_lt_top, MeasureTheory.measure_ball_lt_top, Module.finrank
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
t : ℝ
ht : Membership.mem (γ := Set ℝ) (Ioi 1) t
⊢ f t = 0

Tactic:
simp only [f, closedBall_rpow_sub_one_eq_empty_aux E hr ht, measure_empty]

Premises used: MeasureTheory.measure_empty, closedBall_rpow_sub_one_eq_empty_aux
-- Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
r : ℝ
hnr : (↑(finrank ℝ E) : ℝ) < r
hr : 0 < r
h_meas : Measurable (β := ℝ) fun (ω : E) ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int :
  ∀ (t : ℝ),
    0 < t →
      Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {a : E | t ≤ (1 + ‖a‖) ^ (-r)} : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1)) : ℝ≥0∞)
f : (t : ℝ) → ℝ≥0∞ := fun (t : ℝ) ↦ (μ : (a : Set E) → ℝ≥0∞) (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) (Metric.ball 0 1)
⊢ ∀ (t : ℝ), Membership.mem (γ := Set ℝ) (Ioi 1) t → f t = 0

Tactic:
have h_int'' : ∀ t ∈ Ioi (1 : ℝ), f t = 0 := fun t ht => by
      simp only [f, closedBall_rpow_sub_one_eq_empty_aux E hr ht, measure_empty]