awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
linarith : p ≤ 0 → -1 + (1 - p) + (p - 0) = 0
⊢ 0 < p

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
⊢ 0 < p

Tactic:
have h₁ : 0 < p := by linarith
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
Goal:
ι : Type u_1
inst : Fintype ι
inst_1 : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
⊢ (ENNReal.ofReal p).toReal = p

Tactic:
have : (ENNReal.ofReal p).toReal = p := toReal_ofReal (le_of_lt h₁)
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
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:
simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]

Premises used: ENNReal.ofReal, WithLp.toLp, PiLp.norm_eq_sum
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (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]
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ Fact (1 ≤ ENNReal.ofReal p)

Tactic:
have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ofReal_one ▸ (ofReal_le_ofReal hp))
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ 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
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (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 _)
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (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)
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ ∀ (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)
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ ∀ (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)
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ ∀ (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

Premises used: WithLp.toLp_smul, WithLp.toLp_neg, WithLp.toLp_add, Real.norm_eq_abs
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ 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

Premises used: WithLp.toLp_smul, WithLp.toLp_neg, WithLp.toLp_add, Real.norm_eq_abs
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (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

Premises used: WithLp.toLp_smul, WithLp.toLp_neg, WithLp.toLp_add, Real.norm_eq_abs
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ ∀ (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

Premises used: WithLp.toLp_smul, WithLp.toLp_neg, WithLp.toLp_add, Real.norm_eq_abs
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ ∀ (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

Premises used: WithLp.toLp_smul, WithLp.toLp_neg, WithLp.toLp_add, Real.norm_eq_abs
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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))
rw₁ rw₂ :
  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))
⊢ ((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]

Premises used: MeasureTheory.volume_sum_rpow_lt, MeasureTheory.measure_le_eq_lt, Iff.mp
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)

Tactic:
fun _ => Finset.sum_nonneg' ?_

Premises used: Finset.sum_nonneg'
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
Goal:
ι : Type u_1
inst : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
refine : ∀ (x : ι → ℂ) (i : ι), 0 ≤ ‖x i‖ ^ p
⊢ ∀ (x : ι → ℂ), LE.le (α := ℝ) 0 (∑ i : ι, ‖x i‖ ^ p)

Tactic:
refine fun _ => Finset.sum_nonneg' ?_

Premises used: Finset.sum_nonneg'
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (i : ι), 0 ≤ ‖x i‖ ^ p

Tactic:
fun i => (fun _ => rpow_nonneg (norm_nonneg _) _) _

Premises used: Real.rpow_nonneg, norm_nonneg
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (i : ι), 0 ≤ ‖x i‖ ^ p

Tactic:
exact fun i => (fun _ => rpow_nonneg (norm_nonneg _) _) _

Premises used: Real.rpow_nonneg, norm_nonneg
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)

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 _) _) _
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
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:
simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]

Premises used: ENNReal.ofReal, WithLp.toLp, PiLp.norm_eq_sum
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ ∀ (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]
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ Fact (1 ≤ ENNReal.ofReal p)

Tactic:
have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ENNReal.ofReal_one ▸ (ofReal_le_ofReal hp))
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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)
⊢ 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
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (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 _)
-- Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
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
⊢ ∀ (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)