awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
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)
⊢ ((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_add₁ : ∀ (x y : ι → ℝ),
  HPow.hPow (α := ℝ) (∑ i : ι, |(x + y) i| ^ p) (1 / p) ≤
    HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) + HPow.hPow (α := ℝ) (∑ i : ι, |y 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
-- 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
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)
nm_add₁ :
  ∀ (x y : ι → ℝ),
    HPow.hPow (α := ℝ) (∑ i : ι, |(x + y) i| ^ p) (1 / p) ≤
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) + HPow.hPow (α := ℝ) (∑ i : ι, |y 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 nm_smul₁ : ∀ (r : ℝ) (x : ι → ℝ),
  HPow.hPow (α := ℝ) (∑ i : ι, |(r • x) i| ^ p) (1 / p) ≤ |r| * 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
-- 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 : 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)
eq_zero : ∀ (x : ι → ℝ), HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) = 0 ↔ x = 0
nm_zero : HPow.hPow (α := ℝ) (∑ i : ι, HPow.hPow (α := ℝ) |0 i| p) (1 / p) = 0
nm_neg :
  ∀ (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)
nm_add :
  ∀ (x y : ι → ℝ),
    HPow.hPow (α := ℝ) (∑ i : ι, |(x + y) i| ^ p) (1 / p) ≤
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) + HPow.hPow (α := ℝ) (∑ i : ι, |y i| ^ p) (1 / p)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℝ),
    HPow.hPow (α := ℝ) (∑ i : ι, |(r • x) i| ^ p) (1 / p) ≤ |r| * 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 rw : ((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))
Tactic: rw [measure_le_eq_lt _ nm_zero (fun x ↦ nm_neg x) (fun x y ↦ nm_add x y) (eq_zero _).mp
    (fun r x => nm_smul r x), volume_sum_rpow_lt _ hp]
-- 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 : 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)
eq_zero : ∀ (x : ι → ℝ), HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) = 0 ↔ x = 0
nm_zero : HPow.hPow (α := ℝ) (∑ i : ι, HPow.hPow (α := ℝ) |0 i| p) (1 / p) = 0
nm_neg :
  ∀ (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)
nm_add :
  ∀ (x y : ι → ℝ),
    HPow.hPow (α := ℝ) (∑ i : ι, |(x + y) i| ^ p) (1 / p) ≤
      HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p) + HPow.hPow (α := ℝ) (∑ i : ι, |y i| ^ p) (1 / p)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℝ),
    HPow.hPow (α := ℝ) (∑ i : ι, |(r • x) i| ^ p) (1 / p) ≤ |r| * HPow.hPow (α := ℝ) (∑ i : ι, |x i| ^ p) (1 / p)
rw :
  ((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))
⊢ ((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 rw₁ : ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1)) =
  ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma ((↑(card ι) : ℝ) / p + 1))
Tactic: rw [measure_le_eq_lt _ nm_zero (fun x ↦ nm_neg x) (fun x y ↦ nm_add x y) (eq_zero _).mp
    (fun r x => nm_smul r x), volume_sum_rpow_lt _ hp]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(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 (goal)
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
⊢ ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
have refine : ∀ (x : ι → ℂ) (i : ι), 0 ≤ ‖x i‖ ^ p
Tactic: refine fun _ => Finset.sum_nonneg' ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
Tactic: have h₂ : ∀ x : ι → ℂ, 0 ≤ ∑ i, ‖x i‖ ^ p := by
    refine fun _ => Finset.sum_nonneg' ?_
    exact fun i => (fun _ => rpow_nonneg (norm_nonneg _) _) _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have eq_norm : ∀ (x : ι → ℂ),
  Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) =
    HPow.hPow (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) (1 / p)
Tactic: have eq_norm (x : ι → ℂ) : ‖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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) =
      HPow.hPow (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) (1 / p)
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have this₁ : Fact (1 ≤ ENNReal.ofReal p)
Tactic: have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ENNReal.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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have nm_zero : Norm.norm.{u_1} (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
Tactic: have nm_zero : ‖toLp (.ofReal p) (0 : ι → ℂ)‖ = 0 := norm_zero
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
Tactic: have eq_zero (x : ι → ℂ) : ‖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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have nm_neg : ∀ (x : ι → ℂ),
  Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (-toLp (ENNReal.ofReal p) x) =
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x)
Tactic: have nm_neg := fun x : ι → ℂ => 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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x)
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have nm_add : ∀ (x y : ι → ℂ),
  ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) +
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) y)
Tactic: have nm_add := fun x y : ι → ℂ => 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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℂ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) y)
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have nm_smul : ∀ (r : ℝ) (x : ι → ℂ),
  ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) r (toLp (ENNReal.ofReal p) x)‖ ≤
    ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) (toLp (ENNReal.ofReal p) x)
Tactic: have nm_smul := fun (r : ℝ) (x : ι → ℂ) =>
    norm_smul_le (β := PiLp (.ofReal p) (fun _ : ι => ℂ)) r (toLp (.ofReal p) x)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℂ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) y)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℂ),
    ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) r (toLp (ENNReal.ofReal p) x)‖ ≤
      ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) (toLp (ENNReal.ofReal p) x)
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have eq_zero₁ : ∀ (x : ι → ℂ), 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] 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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℂ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) y)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℂ),
    ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) r (toLp (ENNReal.ofReal p) x)‖ ≤
      ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) (toLp (ENNReal.ofReal p) x)
eq_zero₁ : ∀ (x : ι → ℂ), HPow.hPow (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) (1 / p) = 0 ↔ x = 0
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have nm_zero₁ : HPow.hPow (α := ℝ) (∑ i : ι, Norm.norm (E := ℂ) (0 i) ^ p) (1 / p) = 0
Tactic: simp_rw [← toLp_neg, ← toLp_add, ← toLp_smul, eq_norm] 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
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this_1 : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)
eq_norm :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (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 → ℂ)) (toLp (ENNReal.ofReal p) 0) = 0
eq_zero : ∀ (x : ι → ℂ), Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) = 0 ↔ x = 0
nm_neg :
  ∀ (x : ι → ℂ),
    Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (-toLp (ENNReal.ofReal p) x) =
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x)
nm_add :
  ∀ (x y : ι → ℂ),
    ‖HAdd.hAdd (α := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) (toLp (ENNReal.ofReal p) y)‖ ≤
      Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) x) +
        Norm.norm (E := WithLp (ENNReal.ofReal p) (ι → ℂ)) (toLp (ENNReal.ofReal p) y)
nm_smul :
  ∀ (r : ℝ) (x : ι → ℂ),
    ‖HSMul.hSMul (β := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) r (toLp (ENNReal.ofReal p) x)‖ ≤
      ‖r‖ * Norm.norm (E := PiLp (ENNReal.ofReal p) fun (x : ι) ↦ ℂ) (toLp (ENNReal.ofReal p) x)
eq_zero₁ : ∀ (x : ι → ℂ), HPow.hPow (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) (1 / p) = 0 ↔ x = 0
nm_zero₁ : HPow.hPow (α := ℝ) (∑ i : ι, Norm.norm (E := ℂ) (0 i) ^ p) (1 / p) = 0
⊢ ((volume : Measure (ι → ℂ)) : (a : Set (ι → ℂ)) → ℝ≥0∞) {x : ι → ℂ | LT.lt (α := ℝ) (∑ i : ι, ‖x i‖ ^ p) 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (HMul.hMul (α := ℝ) 2 (↑(card ι) : ℝ) / p + 1))
have nm_neg₁ : ∀ (x : ι → ℂ), 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] at eq_zero nm_zero nm_neg nm_add nm_smul
Cleared hypotheses: nm_zero, eq_zero, nm_neg, nm_add, nm_smul