Theorem: hasDerivAt_ofReal_cpow
File: Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean

tactic: rw [Ne.def, ← add_eq_zero_iff_eq_neg, ← Ne.def] at hr
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx : x ≠ 0
r : ℂ
hr : r ≠ -1
⊢ HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

tactic: rcases lt_or_gt_of_ne hx.symm with (hx | hx)
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
⊢ HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

tactic: apply HasDerivAt.comp_ofReal (e := fun y => (y : ℂ) ^ (r + 1) / (r + 1))
state_before: case inl
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

tactic: convert HasDerivAt.div_const (𝕜 := ℂ) ?_ (r + 1) using 1
state_before: case inl
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ HasDerivAt (fun y => y ^ (r + 1) / (r + 1)) (↑x ^ r) ↑x

tactic: exact (mul_div_cancel _ hr).symm
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ ↑x ^ r = ?inl.convert_3 / (r + 1)

tactic: convert HasDerivAt.cpow_const ?_ ?_ using 1
state_before: case inl.convert_4
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ HasDerivAt (fun y => y ^ (r + 1)) (↑x ^ r * (r + 1)) ↑x

tactic: rw [add_sub_cancel, mul_comm]
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ ↑x ^ r * (r + 1) = (r + 1) * ↑x ^ (r + 1 - 1) * ?inl.convert_4.convert_2

tactic: exact (mul_one _).symm
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ (r + 1) * ↑x ^ r = (r + 1) * ↑x ^ r * ?inl.convert_4.convert_2

case inl.convert_4.convert_2
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ ℂ

case inl.convert_4.convert_2
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ ℂ

tactic: exact hasDerivAt_id (x : ℂ)
state_before: case inl.convert_4.convert_5
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ HasDerivAt (fun y => y) 1 ↑x

tactic: simp [hx]
state_before: case inl.convert_4.convert_6
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 < x
⊢ 0 < (↑x).re ∨ (↑x).im ≠ 0

tactic: have : ∀ᶠ y : ℝ in nhds x,
    (y : ℂ) ^ (r + 1) / (r + 1) = (-y : ℂ) ^ (r + 1) * exp (π * I * (r + 1)) / (r + 1) := by
  refine' Filter.eventually_of_mem (Iio_mem_nhds hx) fun y hy => _
  rw [ofReal_cpow_of_nonpos (le_of_lt hy)]
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
⊢ HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

tactic: refine' HasDerivAt.congr_of_eventuallyEq _ this
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

tactic: rw [ofReal_cpow_of_nonpos (le_of_lt hx)]
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun x => (-↑x) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)) (↑x ^ r) x

tactic: suffices HasDerivAt (fun y : ℝ => (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)))
    ((r + 1) * (-↑x) ^ r * exp (↑π * I * r)) x by
  convert this.div_const (r + 1) using 1
  conv_rhs => rw [mul_assoc, mul_comm, mul_div_cancel _ hr]
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun x => (-↑x) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)) ((-↑x) ^ r * exp (↑π * I * r)) x

tactic: rw [mul_add ((π : ℂ) * _), mul_one, exp_add, exp_pi_mul_I, mul_comm (_ : ℂ) (-1 : ℂ),
  neg_one_mul]
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1))) ((r + 1) * (-↑x) ^ r * exp (↑π * I * r)) x

tactic: simp_rw [mul_neg, ← neg_mul, ← ofReal_neg]
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => (-↑y) ^ (r + 1) * -exp (↑π * I * r)) ((r + 1) * (-↑x) ^ r * exp (↑π * I * r)) x

tactic: suffices HasDerivAt (fun y : ℝ => ↑(-y) ^ (r + 1)) (-(r + 1) * ↑(-x) ^ r) x by
  convert this.neg.mul_const _ using 1; ring
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => -↑(-y) ^ (r + 1) * exp (↑π * I * r)) ((r + 1) * ↑(-x) ^ r * exp (↑π * I * r)) x

tactic: suffices HasDerivAt (fun y : ℝ => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x) by
  convert @HasDerivAt.scomp ℝ _ ℂ _ _ x ℝ _ _ _ _ _ _ _ _ this (hasDerivAt_neg x) using 1
  rw [real_smul, ofReal_neg 1, ofReal_one]; ring
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => ↑(-y) ^ (r + 1)) (-(r + 1) * ↑(-x) ^ r) x

tactic: suffices HasDerivAt (fun y : ℂ => y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) ↑(-x) by
  exact this.comp_ofReal
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)

tactic: conv in ↑_ ^ _ => rw [(by ring : r = r + 1 - 1)]
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) ↑(-x)

tactic: convert HasDerivAt.cpow_const ?_ ?_ using 1
state_before: case inr
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => y ^ (r + 1 - 1 + 1)) ((r + 1) * ↑(-x) ^ r) ↑(-x)

tactic: refine' Filter.eventually_of_mem (Iio_mem_nhds hx) fun y hy => _
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
⊢ ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)

tactic: rw [ofReal_cpow_of_nonpos (le_of_lt hy)]
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
y : ℝ
hy : y ∈ Set.Iio 0
⊢ ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)

tactic: convert this.div_const (r + 1) using 1
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1))) ((r + 1) * (-↑x) ^ r * exp (↑π * I * r)) x
⊢ HasDerivAt (fun x => (-↑x) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)) ((-↑x) ^ r * exp (↑π * I * r)) x

tactic: conv_rhs => rw [mul_assoc, mul_comm, mul_div_cancel _ hr]
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1))) ((r + 1) * (-↑x) ^ r * exp (↑π * I * r)) x
⊢ (-↑x) ^ r * exp (↑π * I * r) = (r + 1) * (-↑x) ^ r * exp (↑π * I * r) / (r + 1)

tactic: convert this.neg.mul_const _ using 1
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => ↑(-y) ^ (r + 1)) (-(r + 1) * ↑(-x) ^ r) x
⊢ HasDerivAt (fun y => -↑(-y) ^ (r + 1) * exp (↑π * I * r)) ((r + 1) * ↑(-x) ^ r * exp (↑π * I * r)) x

tactic: ring
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => ↑(-y) ^ (r + 1)) (-(r + 1) * ↑(-x) ^ r) x
⊢ (r + 1) * ↑(-x) ^ r * exp (↑π * I * r) = -(-(r + 1) * ↑(-x) ^ r) * exp (↑π * I * r)

tactic: convert @HasDerivAt.scomp ℝ _ ℂ _ _ x ℝ _ _ _ _ _ _ _ _ this (hasDerivAt_neg x) using 1
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)
⊢ HasDerivAt (fun y => ↑(-y) ^ (r + 1)) (-(r + 1) * ↑(-x) ^ r) x

tactic: rw [real_smul, ofReal_neg 1, ofReal_one]
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)
⊢ -(r + 1) * ↑(-x) ^ r = -1 • ((r + 1) * ↑(-x) ^ r)

tactic: ring
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)
⊢ -(r + 1) * ↑(-x) ^ r = -1 * ((r + 1) * ↑(-x) ^ r)

tactic: exact this.comp_ofReal
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
this : HasDerivAt (fun y => y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) ↑(-x)
⊢ HasDerivAt (fun y => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)

tactic: ring
state_before: f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
y : ℂ
⊢ r = r + 1 - 1

tactic: rw [add_sub_cancel, add_sub_cancel]
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ (r + 1) * ↑(-x) ^ r = (r + 1 - 1 + 1) * ↑(-x) ^ (r + 1 - 1 + 1 - 1) * ?inr.convert_2

tactic: exact (mul_one _).symm
state_before: case h.e'_7
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ (r + 1) * ↑(-x) ^ r = (r + 1) * ↑(-x) ^ r * ?inr.convert_2

case inr.convert_2
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ ℂ

case inr.convert_2
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ ℂ

tactic: exact hasDerivAt_id ((-x : ℝ) : ℂ)
state_before: case inr.convert_5
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => y) 1 ↑(-x)

tactic: simp [hx]
state_before: case inr.convert_6
f g : ℂ → ℂ
s : Set ℂ
f' g' x✝ c : ℂ
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * exp (↑π * I * (r + 1)) / (r + 1)
⊢ 0 < (↑(-x)).re ∨ (↑(-x)).im ≠ 0