awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (goal)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), LE.le (α := ℝ) 0 (∑ i : ι, |x i| ^ p)
h₂ : ∀ (x : ι → ℝ), 0 ≤ HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
hr : 0 < r
h :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    (Set.preimage (β := ι → ℝ) (fun (x : ι → ℝ) ↦ r⁻¹ • x) {x : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |x i| ^ p) 1})
h₃ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |(r⁻¹ • a) i| ^ p) 1}
h₄ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹ • a x| ^ p) 1}
h₅ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹ * a x| ^ p) 1}
h₆ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, (|r⁻¹| * |a x|) ^ p) 1}
h₇ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹| ^ p * |a x| ^ p) 1}
h₈ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r|⁻¹ ^ p * |a x| ^ p) 1}
h₉ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, (|r| ^ p)⁻¹ * |a x| ^ p) 1}
h₁₀ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | (|r| ^ p)⁻¹ * ∑ i : ι, |a i| ^ p < 1}
h₁₁ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | (r ^ p)⁻¹ * ∑ i : ι, |a i| ^ p < 1}
⊢ {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r} =
    HSMul.hSMul (β := Set (ι → ℝ)) r {x : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |x i| ^ p) 1}
have h₁₂ : Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
  {a : ι → ℝ | ∑ i : ι, |a i| ^ p < r ^ p * 1}
Tactic: simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hr), Set.preimage_setOf_eq, Pi.smul_apply,
      smul_eq_mul, abs_mul, mul_rpow (abs_nonneg _) (abs_nonneg _), abs_inv,
      inv_rpow (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr),
      inv_mul_lt_iff₀ (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff
      (rpow_nonneg (h₁ _) _) (le_of_lt hr) (by linarith : 0 < p), ← rpow_mul
      (h₁ _), div_mul_cancel₀ _ (ne_of_gt (by linarith) : p ≠ 0), Real.rpow_one]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (goal)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), LE.le (α := ℝ) 0 (∑ i : ι, |x i| ^ p)
h₂ : ∀ (x : ι → ℝ), 0 ≤ HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
hr : 0 < r
h :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    (Set.preimage (β := ι → ℝ) (fun (x : ι → ℝ) ↦ r⁻¹ • x) {x : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |x i| ^ p) 1})
h₃ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |(r⁻¹ • a) i| ^ p) 1}
h₄ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹ • a x| ^ p) 1}
h₅ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹ * a x| ^ p) 1}
h₆ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, (|r⁻¹| * |a x|) ^ p) 1}
h₇ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹| ^ p * |a x| ^ p) 1}
h₈ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r|⁻¹ ^ p * |a x| ^ p) 1}
h₉ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, (|r| ^ p)⁻¹ * |a x| ^ p) 1}
h₁₀ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | (|r| ^ p)⁻¹ * ∑ i : ι, |a i| ^ p < 1}
h₁₁ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | (r ^ p)⁻¹ * ∑ i : ι, |a i| ^ p < 1}
h₁₂ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | ∑ i : ι, |a i| ^ p < r ^ p * 1}
⊢ {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r} =
    HSMul.hSMul (β := Set (ι → ℝ)) r {x : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |x i| ^ p) 1}
have h₁₃ : Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
  {a : ι → ℝ | ∑ i : ι, |a i| ^ p < r ^ p}
Tactic: simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hr), Set.preimage_setOf_eq, Pi.smul_apply,
      smul_eq_mul, abs_mul, mul_rpow (abs_nonneg _) (abs_nonneg _), abs_inv,
      inv_rpow (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr),
      inv_mul_lt_iff₀ (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff
      (rpow_nonneg (h₁ _) _) (le_of_lt hr) (by linarith : 0 < p), ← rpow_mul
      (h₁ _), div_mul_cancel₀ _ (ne_of_gt (by linarith) : p ≠ 0), Real.rpow_one]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (goal)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), LE.le (α := ℝ) 0 (∑ i : ι, |x i| ^ p)
h₂ : ∀ (x : ι → ℝ), 0 ≤ HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
hr : 0 < r
h :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    (Set.preimage (β := ι → ℝ) (fun (x : ι → ℝ) ↦ r⁻¹ • x) {x : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |x i| ^ p) 1})
h₃ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |(r⁻¹ • a) i| ^ p) 1}
h₄ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹ • a x| ^ p) 1}
h₅ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹ * a x| ^ p) 1}
h₆ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, (|r⁻¹| * |a x|) ^ p) 1}
h₇ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r⁻¹| ^ p * |a x| ^ p) 1}
h₈ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, |r|⁻¹ ^ p * |a x| ^ p) 1}
h₉ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | LT.lt (α := ℝ) (∑ x : ι, (|r| ^ p)⁻¹ * |a x| ^ p) 1}
h₁₀ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | (|r| ^ p)⁻¹ * ∑ i : ι, |a i| ^ p < 1}
h₁₁ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | (r ^ p)⁻¹ * ∑ i : ι, |a i| ^ p < 1}
h₁₂ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | ∑ i : ι, |a i| ^ p < r ^ p * 1}
h₁₃ :
  Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r}
    {a : ι → ℝ | ∑ i : ι, |a i| ^ p < r ^ p}
⊢ {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) < r} =
    HSMul.hSMul (β := Set (ι → ℝ)) r {x : ι → ℝ | LT.lt (α := ℝ) (∑ i : ι, |x i| ^ p) 1}
have h₁₄ : Eq.{u_1 + 1} (α := Set (ι → ℝ)) {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ^ p < r ^ p}
  {a : ι → ℝ | ∑ i : ι, |a i| ^ p < r ^ p}
Tactic: simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hr), Set.preimage_setOf_eq, Pi.smul_apply,
      smul_eq_mul, abs_mul, mul_rpow (abs_nonneg _) (abs_nonneg _), abs_inv,
      inv_rpow (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr),
      inv_mul_lt_iff₀ (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff
      (rpow_nonneg (h₁ _) _) (le_of_lt hr) (by linarith : 0 < p), ← rpow_mul
      (h₁ _), div_mul_cancel₀ _ (ne_of_gt (by linarith) : p ≠ 0), Real.rpow_one]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (goal)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
⊢ 0 < p
have linarith : p ≤ 0 → -1 + (1 - p) + (p - 0) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have h₁ : 0 < p
Tactic: have h₁ : 0 < p := by linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have this : (ENNReal.ofReal p).toReal = p
Tactic: have : (ENNReal.ofReal p).toReal = p := toReal_ofReal (le_of_lt h₁)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have eq_norm : ∀ (x : ι → ℝ),
  Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
    HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
Tactic: have eq_norm (x : ι → ℝ) : ‖toLp (.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p) := by
    simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have this₁ : Fact (1 ≤ ENNReal.ofReal p)
Tactic: have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ofReal_one ▸ (ofReal_le_ofReal hp))
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
Tactic: have nm_zero : ‖toLp (.ofReal p) (0 : ι → ℝ)‖ = 0 := norm_zero
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
Tactic: have eq_zero (x : ι → ℝ) : ‖toLp (.ofReal p) x‖ = 0 ↔ x = 0 :=
    norm_eq_zero.trans (toLp_eq_zero _)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have nm_neg : ∀ (x : ι → ℝ),
  Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (-toLp (ENNReal.ofReal p) x) =
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x)
Tactic: have nm_neg := fun x : ι → ℝ => norm_neg (toLp (.ofReal p) x)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x)
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have nm_add : ∀ (x y : ι → ℝ),
  ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) +
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) y)
Tactic: have nm_add := fun x y : ι → ℝ => norm_add_le (toLp (.ofReal p) x) (toLp (.ofReal p) y)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℝ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) y)
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have nm_smul : ∀ (r : ℝ) (x : ι → ℝ),
  ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) r (toLp (ENNReal.ofReal p) x)‖ ≤
    ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) (toLp (ENNReal.ofReal p) x)
Tactic: have nm_smul := fun (r : ℝ) (x : ι → ℝ) =>
    norm_smul_le (β := PiLp (.ofReal p) (fun _ : ι => ℝ)) r (toLp (.ofReal p) x)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℝ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) y)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℝ),
    ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) r (toLp (ENNReal.ofReal p) x)‖ ≤
      ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) (toLp (ENNReal.ofReal p) x)
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have eq_zero₁ : ∀ (x : ι → ℝ), HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) = 0 ↔ x = 0
Tactic: simp_rw [← toLp_neg, ← toLp_add, ← toLp_smul, eq_norm,
    norm_eq_abs] at eq_zero nm_zero nm_neg nm_add nm_smul
Cleared hypotheses: nm_zero, eq_zero, nm_neg, nm_add, nm_smul
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℝ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) y)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℝ),
    ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) r (toLp (ENNReal.ofReal p) x)‖ ≤
      ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) (toLp (ENNReal.ofReal p) x)
eq_zero₁ : ∀ (x : ι → ℝ), HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) = 0 ↔ x = 0
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have nm_zero₁ : HPow.hPow (α := ℝ) (∑ i : ι, HPow.hPow (α := ℝ) |0 i| p) (1 / p) = 0
Tactic: simp_rw [← toLp_neg, ← toLp_add, ← toLp_smul, eq_norm,
    norm_eq_abs] at eq_zero nm_zero nm_neg nm_add nm_smul
Cleared hypotheses: nm_zero, eq_zero, nm_neg, nm_add, nm_smul
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
eq_norm :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℝ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℝ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℝ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℝ)) (toLp (ENNReal.ofReal p) y)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℝ),
    ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) r (toLp (ENNReal.ofReal p) x)‖ ≤
      ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℝ) (toLp (ENNReal.ofReal p) x)
eq_zero₁ : ∀ (x : ι → ℝ), HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) = 0 ↔ x = 0
nm_zero₁ : HPow.hPow (α := ℝ) (∑ i : ι, HPow.hPow (α := ℝ) |0 i| p) (1 / p) = 0
⊢ ((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ℝ≥0∞)
      {x : ι → ℝ | HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
have nm_neg₁ : ∀ (x : ι → ℝ), HPow.hPow (α := ℝ) (∑ i : ι, |(-x) i| ^ p) (1 / p) = HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
Tactic: simp_rw [← toLp_neg, ← toLp_add, ← toLp_smul, eq_norm,
    norm_eq_abs] at eq_zero nm_zero nm_neg nm_add nm_smul
Cleared hypotheses: nm_zero, eq_zero, nm_neg, nm_add, nm_smul