-- Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
theorem Complex.volume_sum_rpow_lt_one {p : ℝ} (hp : 1 ≤ p) :
    volume {x : ι → ℂ | ∑ i, ‖x i‖ ^ p < 1} =
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1))

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, ‖0 x‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖(-x) x_1‖ ^ p) ^ (1 / p) = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
nm_add :
  ∀ (x y : ι → ℂ),
    (∑ x_1 : ι, ‖(x + y) x_1‖ ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) + (∑ x : ι, ‖y x‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ x_1 : ι, ‖(r • x) x_1‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
x✝ : ι → ℂ
⊢ 0 ≤ ∑ i : ι, ‖x✝ i‖ ^ p

Tactic:
exact Finset.sum_nonneg' (fun _ => rpow_nonneg (norm_nonneg _) _)

Result:
no goals

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ENNReal.ofReal_one ▸ (ofReal_le_ofReal hp))

Result:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, ‖0 x‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖(-x) x_1‖ ^ p) ^ (1 / p) = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
nm_add :
  ∀ (x y : ι → ℂ),
    (∑ x_1 : ι, ‖(x + y) x_1‖ ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) + (∑ x : ι, ‖y x‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ x_1 : ι, ‖(r • x) x_1‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
⊢ 0 < p

Tactic:
linarith

Result:
no goals

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

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 _) _) _

Result:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
⊢ ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p

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

Result:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
x✝ : ι → ℂ
⊢ ∀ (i : ι), 0 ≤ ‖x✝ i‖ ^ p

Proof state:
case h.e'_3.h.e'_1.h.e'_5
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, ‖0 x‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖(-x) x_1‖ ^ p) ^ (1 / p) = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
nm_add :
  ∀ (x y : ι → ℂ),
    (∑ x_1 : ι, ‖(x + y) x_1‖ ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) + (∑ x : ι, ‖y x‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ x_1 : ι, ‖(r • x) x_1‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
⊢ (π * Real.Gamma (2 / p + 1)) ^ card ι = ∫ (x : ι → ℂ), ∏ x_1 : ι, rexp (-‖x x_1‖ ^ p)

Tactic:
rw [integral_fintype_prod_eq_pow ι fun x : ℂ => Real.exp (- ‖x‖ ^ p),
  Complex.integral_exp_neg_rpow hp]

Result:
no goals

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, ‖0 x‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖(-x) x_1‖ ^ p) ^ (1 / p) = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
nm_add :
  ∀ (x y : ι → ℂ),
    (∑ x_1 : ι, ‖(x + y) x_1‖ ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) + (∑ x : ι, ‖y x‖ ^ p) ^ (1 / p)
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have nm_smul := fun (r : ℝ) (x : ι → ℂ) =>
  norm_smul_le (β := PiLp (.ofReal p) (fun _ : ι => ℂ)) r x

Result:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, ‖0 x‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ x_1 : ι, ‖(-x) x_1‖ ^ p) ^ (1 / p) = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
nm_add :
  ∀ (x y : ι → ℂ),
    (∑ x_1 : ι, ‖(x + y) x_1‖ ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p) + (∑ x : ι, ‖y x‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), ‖r • x‖ ≤ ‖r‖ * ‖x‖
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have nm_zero := norm_zero (E := PiLp (.ofReal p) (fun _ : ι => ℂ))

Result:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Proof state:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℂ), ‖x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℂ), ‖-x‖ = ‖x‖
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have nm_add := fun x y : ι → ℂ => norm_add_le (E := PiLp (.ofReal p) (fun _ : ι => ℂ)) x y

Result:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℂ), ‖x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℂ), ‖-x‖ = ‖x‖
nm_add : ∀ (x y : ι → ℂ), ‖x + y‖ ≤ ‖x‖ + ‖y‖
⊢ volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))