awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
| -I / 2 * log ((cos z + sin z * I) / (cos z - sin z * I))
have enter : cos (-z) + sin (-z) * I
Tactic: enter [2, 1, 2]
    rw [sub_eq_add_neg, ← neg_mul, ← sin_neg, ← cos_neg]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ -I / 2 * log ((cos z + sin z * I) / (cos z - sin z * I)) = z
have conv_lhs : -I / 2 * log ((cos z + sin z * I) / (cos (-z) + sin (-z) * I)) = z
Tactic: conv_lhs =>
    enter [2, 1, 2]
    rw [sub_eq_add_neg, ← neg_mul, ← sin_neg, ← cos_neg]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ -I / 2 * log ((cos z + sin z * I) / (cos (-z) + sin (-z) * I)) = z
have hx₁ : -π < (2 * (I * z)).im
Tactic: rw [← exp_mul_I, ← exp_mul_I, ← exp_sub, show z * I - -z * I = 2 * (I * z) by ring, log_exp,
    show -I / 2 * (2 * (I * z)) = -(I * I) * z by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
hx₁ : -π < (2 * (I * z)).im
⊢ -I / 2 * log ((cos z + sin z * I) / (cos (-z) + sin (-z) * I)) = z
have hx₂ : (2 * (I * z)).im ≤ π
Tactic: rw [← exp_mul_I, ← exp_mul_I, ← exp_sub, show z * I - -z * I = 2 * (I * z) by ring, log_exp,
    show -I / 2 * (2 * (I * z)) = -(I * I) * z by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ -π < (2 * (I * z)).im
have hx₁ : -π < 2 * z.re
Tactic: simp
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ (2 * (I * z)).im ≤ π
have hx₂ : 2 * z.re ≤ π
Tactic: simp
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
h₀ : z ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ -π < 2 * z.re
have hx₁ : -π / 2 < z.re
Tactic: rwa [← div_lt_iff₀' two_pos, neg_div]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
x : ℝ
| (↑x : ℂ).arctan
have rw : (↑(Real.tan (Real.arctan x)) : ℂ).arctan
Tactic: rw [← Real.tan_arctan x]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
x : ℝ
⊢ (↑(Real.arctan x) : ℂ) = (↑x : ℂ).arctan
have conv_rhs : (↑(Real.arctan x) : ℂ) = (↑(Real.tan (Real.arctan x)) : ℂ).arctan
Tactic: conv_rhs => rw [← Real.tan_arctan x]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
x : ℝ
⊢ (↑(Real.arctan x) : ℂ) = (↑(Real.tan (Real.arctan x)) : ℂ).arctan
have h₀ : (↑(Real.arctan x) : ℂ) ≠ (↑π : ℂ) / 2
Tactic: rw [ofReal_tan, arctan_tan]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
x : ℝ
h₀ : (↑(Real.arctan x) : ℂ) ≠ (↑π : ℂ) / 2
⊢ (↑(Real.arctan x) : ℂ) = (↑(Real.tan (Real.arctan x)) : ℂ).arctan
have h₁ : -(π / 2) < (↑(Real.arctan x) : ℂ).re
Tactic: rw [ofReal_tan, arctan_tan]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
x : ℝ
h₀ : (↑(Real.arctan x) : ℂ) ≠ (↑π : ℂ) / 2
h₁ : -(π / 2) < (↑(Real.arctan x) : ℂ).re
⊢ (↑(Real.arctan x) : ℂ) = (↑(Real.tan (Real.arctan x)) : ℂ).arctan
have h₂ : (↑(Real.arctan x) : ℂ).re ≤ π / 2
Tactic: rw [ofReal_tan, arctan_tan]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
x : ℝ
⊢ (↑(Real.arctan x) : ℂ) ≠ (↑π : ℂ) / 2
have h₀ : ¬Real.arctan x = π / 2
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
⊢ (1 + z).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
have rw : -1 < z.re ∨ 1 + z = 0
Tactic: rw [Set.mem_Ioo, ← abs_lt, abs_arg_lt_pi_div_two_iff, add_re, one_re, ← neg_lt_iff_pos_add']
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
⊢ log (1 + z * I) + -log (1 - z * I) = log ((1 + z * I) / (1 - z * I))
have z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
Tactic: have z₁ := mem_slitPlane_iff_arg.mp (mem_slitPlane_of_norm_lt_one (z := z * I) (by simpa))
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
⊢ log (1 + z * I) + -log (1 - z * I) = log ((1 + z * I) / (1 - z * I))
have z₂ : (1 + -(z * I)).arg ≠ π ∧ 1 + -(z * I) ≠ 0
Tactic: have z₂ := mem_slitPlane_iff_arg.mp (mem_slitPlane_of_norm_lt_one (z := -(z * I)) (by simpa))
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 + -(z * I)).arg ≠ π ∧ 1 + -(z * I) ≠ 0
⊢ log (1 + z * I) + -log (1 - z * I) = log ((1 + z * I) / (1 - z * I))
have z₃ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
Tactic: rw [← sub_eq_add_neg] at z₂
Cleared hypotheses: z₂
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
⊢ log (1 + z * I) + -log (1 - z * I) = log ((1 + z * I) / (1 - z * I))
have rw : (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
Tactic: rw [← log_inv _ z₂.1, ← (log_mul_eq_add_log_iff z₁.2 (inv_eq_zero.ne.mpr z₂.2)).mpr,
    div_eq_mul_inv]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
⊢ (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
have b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
Tactic: have b₁ := arg_one_add_mem_Ioo (z := z * I) (by simpa)
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
⊢ (1 - z * I)⁻¹.arg ∈ Set.Ioo (-(π / 2)) (π / 2)
have simp_rw : (1 + -(z * I)).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
Tactic: simp_rw [arg_inv, z₂.1, ite_false, Set.neg_mem_Ioo_iff, neg_neg, sub_eq_add_neg]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
⊢ (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
have b₂ : (1 - z * I)⁻¹.arg ∈ Set.Ioo (-(π / 2)) (π / 2)
Tactic: have b₂ : arg (1 - z * I)⁻¹ ∈ Set.Ioo (-(π / 2)) (π / 2) := by
    simp_rw [arg_inv, z₂.1, ite_false, Set.neg_mem_Ioo_iff, neg_neg, sub_eq_add_neg]
    exact arg_one_add_mem_Ioo (by simpa)
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
b₂ : (1 - z * I)⁻¹.arg ∈ Set.Ioo (-(π / 2)) (π / 2)
⊢ (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
have c₁ : -(π / 2) + -(π / 2) < (1 + z * I).arg + (1 - z * I)⁻¹.arg
Tactic: have c₁ := add_lt_add b₁.1 b₂.1
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
b₂ : (1 - z * I)⁻¹.arg ∈ Set.Ioo (-(π / 2)) (π / 2)
c₁ : -(π / 2) + -(π / 2) < (1 + z * I).arg + (1 - z * I)⁻¹.arg
⊢ (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
have c₂ : (1 + z * I).arg + (1 - z * I)⁻¹.arg < π / 2 + π / 2
Tactic: have c₂ := add_lt_add b₁.2 b₂.2
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
b₂ : (1 - z * I)⁻¹.arg ∈ Set.Ioo (-(π / 2)) (π / 2)
c₁ : -(π / 2) + -(π / 2) < (1 + z * I).arg + (1 - z * I)⁻¹.arg
c₂ : (1 + z * I).arg + (1 - z * I)⁻¹.arg < π / 2 + π / 2
⊢ (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
have c₃ : -π < (1 + z * I).arg + (1 - z * I)⁻¹.arg
Tactic: rw [show -(π / 2) + -(π / 2) = -π by ring] at c₁
Cleared hypotheses: c₁, c₂
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
z₁ : (1 + z * I).arg ≠ π ∧ 1 + z * I ≠ 0
z₂ : (1 - z * I).arg ≠ π ∧ 1 - z * I ≠ 0
b₁ : (1 + z * I).arg ∈ Set.Ioo (-(π / 2)) (π / 2)
b₂ : (1 - z * I)⁻¹.arg ∈ Set.Ioo (-(π / 2)) (π / 2)
c₁ : -π < (1 + z * I).arg + (1 - z * I)⁻¹.arg
c₂ : (1 + z * I).arg + (1 - z * I)⁻¹.arg < π / 2 + π / 2
⊢ (1 + z * I).arg + (1 - z * I)⁻¹.arg ∈ Set.Ioc (-π) π
have c₃ : (1 + z * I).arg + (1 - z * I)⁻¹.arg < π
Tactic: rw [show π / 2 + π / 2 = π by ring] at c₂
Cleared hypotheses: c₂
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
⊢ HasSum (fun (n : ℕ) ↦ HPow.hPow (α := ℂ) (-1) n * z ^ (2 * n + 1) / (↑(2 * n + 1) : ℂ)) z.arctan
have this : HasSum (fun (i : ℕ) ↦ -I / 2 * (HPow.hPow (α := ℂ) (-1) (i + 1) * (z * I) ^ i / (↑i : ℂ) + (z * I) ^ i / (↑i : ℂ)))
  (-I / 2 * (log (1 + z * I) + -log (1 - z * I)))
Tactic: have := ((hasSum_taylorSeries_log (z := z * I) (by simpa)).add
    (hasSum_taylorSeries_neg_log (z := z * I) (by simpa))).mul_left (-I / 2)
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum (fun (i : ℕ) ↦ -I / 2 * (HPow.hPow (α := ℂ) (-1) (i + 1) * (z * I) ^ i / (↑i : ℂ) + (z * I) ^ i / (↑i : ℂ)))
    (-I / 2 * (log (1 + z * I) + -log (1 - z * I)))
⊢ HasSum (fun (n : ℕ) ↦ HPow.hPow (α := ℂ) (-1) n * z ^ (2 * n + 1) / (↑(2 * n + 1) : ℂ)) z.arctan
have this₁ : HasSum (fun (i : ℕ) ↦ -I / 2 * ((HPow.hPow (α := ℂ) (-1) (i + 1) + 1) * (z * I) ^ i / (↑i : ℂ)))
  (-I / 2 * log ((1 + z * I) / (1 - z * I)))
Tactic: simp_rw [← add_div, ← add_one_mul, hasSum_arctan_aux hz] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum (fun (i : ℕ) ↦ -I / 2 * ((HPow.hPow (α := ℂ) (-1) (i + 1) + 1) * (z * I) ^ i / (↑i : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
⊢ HasSum (fun (n : ℕ) ↦ HPow.hPow (α := ℂ) (-1) n * z ^ (2 * n + 1) / (↑(2 * n + 1) : ℂ)) z.arctan
have this₁ : HasSum (β := ℕ × Fin 2)
  ((fun (i : ℕ) ↦ -I / 2 * ((HPow.hPow (α := ℂ) (-1) (i + 1) + 1) * (z * I) ^ i / (↑i : ℂ))) ∘
    (⇑(Equiv.symm (Nat.divModEquiv 2)) : (a : ℕ × Fin 2) → ℕ))
  (-I / 2 * log ((1 + z * I) / (1 - z * I)))
Tactic: replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (hypothesis)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (x.1 * 2 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (x.1 * 2 + (↑x.2 : ℕ)) /
          (↑(x.1 * 2 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
⊢ HasSum (fun (n : ℕ) ↦ HPow.hPow (α := ℂ) (-1) n * z ^ (2 * n + 1) / (↑(2 * n + 1) : ℂ)) z.arctan
have this₁ : HasSum
  (fun (x : ℕ × Fin 2) ↦
    -I / 2 *
      ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
        (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
  (-I / 2 * log ((1 + z * I) / (1 - z * I)))
Tactic: simp_rw [← mul_comm 2 _] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
⊢ HasSum (fun (n : ℕ) ↦ HPow.hPow (α := ℂ) (-1) n * z ^ (2 * n + 1) / (↑(2 * n + 1) : ℂ)) z.arctan
have refine : ∀ (k : ℕ),
  HasSum
    (fun (c : Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * (k, c).1 + (↑(k, c).2 : ℕ) + 1) + 1) *
            (z * I) ^ (2 * (k, c).1 + (↑(k, c).2 : ℕ)) /
          (↑(2 * (k, c).1 + (↑(k, c).2 : ℕ)) : ℂ)))
    (HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ))
Tactic: refine this.prod_fiberwise fun k => ?_
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ HasSum
    (fun (c : Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * (k, c).1 + (↑(k, c).2 : ℕ) + 1) + 1) *
            (z * I) ^ (2 * (k, c).1 + (↑(k, c).2 : ℕ)) /
          (↑(2 * (k, c).1 + (↑(k, c).2 : ℕ)) : ℂ)))
    (HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ))
have dsimp : HasSum
  (fun (c : Fin 2) ↦
    -I / 2 *
      ((HPow.hPow (α := ℂ) (-1) (2 * k + (↑c : ℕ) + 1) + 1) * (z * I) ^ (2 * k + (↑c : ℕ)) / (↑(2 * k + (↑c : ℕ)) : ℂ)))
  (HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ))
Tactic: dsimp only
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ HasSum
    (fun (c : Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * k + (↑c : ℕ) + 1) + 1) * (z * I) ^ (2 * k + (↑c : ℕ)) /
          (↑(2 * k + (↑c : ℕ)) : ℂ)))
    (HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ))
have h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ) =
  ∑ b : Fin 2,
    -I / 2 *
      ((HPow.hPow (α := ℂ) (-1) (2 * k + (↑b : ℕ) + 1) + 1) * (z * I) ^ (2 * k + (↑b : ℕ)) / (↑(2 * k + (↑b : ℕ)) : ℂ))
Tactic: convert! hasSum_fintype (_ : Fin 2 → ℂ) using 1
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ) =
    ∑ b : Fin 2,
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * k + (↑b : ℕ) + 1) + 1) * (z * I) ^ (2 * k + (↑b : ℕ)) /
          (↑(2 * k + (↑b : ℕ)) : ℂ))
have h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ) = -I * (z * I) ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ)
Tactic: rw [Fin.sum_univ_two, Fin.val_zero, Fin.val_one, Odd.neg_one_pow (n := 2 * k + 0 + 1) (by simp),
    neg_add_cancel, zero_mul, zero_div, mul_zero, zero_add,
    show 2 * k + 1 + 1 = 2 * (k + 1) by ring, Even.neg_one_pow (n := 2 * (k + 1)) (by simp),
    ← mul_div_assoc (_ / _), ← mul_assoc, show -I / 2 * (1 + 1) = -I by ring]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ) = -I * (z * I) ^ (2 * k + 1) / (↑(2 * k + 1) : ℂ)
have h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₂ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (β := ℕ) I 2 ^ k))
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
h₂ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (β := ℕ) I 2 ^ k))
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₃ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (α := ℂ) (-1) k))
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
h₂ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (β := ℕ) I 2 ^ k))
h₃ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (α := ℂ) (-1) k))
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₄ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -(I * I) * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
h₂ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (β := ℕ) I 2 ^ k))
h₃ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (α := ℂ) (-1) k))
h₄ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -(I * I) * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₅ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = - -1 * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
h₂ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (β := ℕ) I 2 ^ k))
h₃ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (α := ℂ) (-1) k))
h₄ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -(I * I) * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
h₅ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = - -1 * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₆ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = 1 * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]
-- Lean 4 proof step from Mathlib.Analysis.SpecialFunctions.Complex.Arctan (goal)
Goal:
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum
    (fun (x : ℕ × Fin 2) ↦
      -I / 2 *
        ((HPow.hPow (α := ℂ) (-1) (2 * x.1 + (↑x.2 : ℕ) + 1) + 1) * (z * I) ^ (2 * x.1 + (↑x.2 : ℕ)) /
          (↑(2 * x.1 + (↑x.2 : ℕ)) : ℂ)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
h : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * I ^ (2 * k + 1))
h₁ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * I ^ (2 * k)))
h₂ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (β := ℕ) I 2 ^ k))
h₃ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z ^ (2 * k + 1) * (I * HPow.hPow (α := ℂ) (-1) k))
h₄ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -(I * I) * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
h₅ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = - -1 * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
h₆ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = 1 * HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
⊢ HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = -I * (z * I) ^ (2 * k + 1)
have h₇ : HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1) = HPow.hPow (α := ℂ) (-1) k * z ^ (2 * k + 1)
Tactic: rw [mul_pow, pow_succ' I, pow_mul, I_sq,
    show -I * _ = -(I * I) * (-1) ^ k * z ^ (2 * k + 1) by ring, I_mul_I, neg_neg, one_mul]