awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℂ
H :
  Eq (α := FormalMultilinearSeries ℂ ℂ ℂ)
    (FormalMultilinearSeries.compContinuousLinearMap (binomialSeries ℂ (-a) : FormalMultilinearSeries ℂ ℂ ℂ) (-1) :
      FormalMultilinearSeries ℂ ℂ ℂ)
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ Ring.choose (a + (↑n : ℂ) - 1) n : FormalMultilinearSeries ℂ ℂ ℂ)
this : HasFPowerSeriesOnBall (fun (x : ℂ) ↦ (1 + x) ^ (-a)) (binomialSeries ℂ (-a)) (-0) 1
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ a)
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ Ring.choose (a + (↑n : ℂ) - 1) n) 0 1

Tactic:
simpa [cpow_neg, Function.comp_def, ← sub_eq_add_neg, H] using
    this.compContinuousLinearMap (u := -1) (x := (0 : ℂ))

Premises used: sub_eq_add_neg, HasFPowerSeriesOnBall.compContinuousLinearMap, Complex.cpow_neg, Function.comp_def
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ HAdd.hAdd (α := ℂ) (↑a : ℂ) 1)
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ Ring.choose (HAdd.hAdd (α := ℂ) (↑a : ℂ) 1 + (↑n : ℂ) - 1) n) 0 1

Tactic:
one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero (a + 1)

Premises used: Nat.cast, HAdd.hAdd, Complex.one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, OfNat.ofNat
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
h : ∀ (z : ℂ), (1 - z) ^ (a + 1) = (1 - z) ^ HAdd.hAdd (α := ℂ) (↑a : ℂ) 1
h₁ : ∀ (n : ℕ), (↑((a + n).choose a) : ℂ) = Ring.choose (HAdd.hAdd (α := ℂ) (↑a : ℂ) 1 + (↑n : ℂ) - 1) n
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ)) 0 1

Tactic:
convert! one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero (a + 1) using 3 with z n

Premises used: Nat.cast, HAdd.hAdd, Complex.one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, OfNat.ofNat
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
h h₁ h₂ h₃ : (1 - z) ^ (a + 1) = (1 - z) ^ (a + 1)
⊢ (1 - z) ^ (a + 1) = (1 - z) ^ HAdd.hAdd (α := ℂ) (↑a : ℂ) 1

Tactic:
norm_cast
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a n : ℕ
h : Ring.choose (HAdd.hAdd (α := ℂ) (↑a : ℂ) 1 + (↑n : ℂ) - 1) n = (↑((a + n).choose a) : ℂ)
h₁ : Ring.choose (HAdd.hAdd (α := ℂ) (↑a : ℂ) (↑n : ℂ) + 1 - 1) n = (↑((a + n).choose a) : ℂ)
h₂ : Ring.choose (HAdd.hAdd (α := ℂ) (↑a : ℂ) (↑n : ℂ)) n = (↑((a + n).choose a) : ℂ)
h₃ : Eq (α := ℂ) (Ring.choose (↑(a + n) : ℂ) n) (↑((a + n).choose a) : ℂ)
h₄ : Eq (α := ℂ) (↑((a + n).choose n) : ℂ) (↑((a + n).choose a) : ℂ)
h₅ h₆ : Eq (α := ℂ) (↑((a + n).choose n) : ℂ) (↑((a + n).choose n) : ℂ)
⊢ (↑((a + n).choose a) : ℂ) = Ring.choose (HAdd.hAdd (α := ℂ) (↑a : ℂ) 1 + (↑n : ℂ) - 1) n

Tactic:
rw [eq_comm, add_right_comm, add_sub_cancel_right, ← Nat.cast_add,
      Ring.choose_natCast, Nat.choose_symm_add]

Premises used: add_right_comm, eq_comm, Nat.choose_symm_add, Nat.cast_add, Ring.choose_natCast, add_sub_cancel_right
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ)) 0 1

Tactic:
have := one_div_one_sub_pow_hasFPowerSeriesOnBall_zero a
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ)) 0 1
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1

Tactic:
rw [← map_zero (z⁻¹ • 1 : ℂ →L[ℂ] ℂ)] at this

Premises used: map_zero, Inv.inv, HSMul.hSMul, OfNat.ofNat
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
⊢ HasFPowerSeriesOnBall ((fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1)) ∘ (⇑(HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) : (a : ℂ) → ℂ))
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 (1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ)

Tactic:
have := this.compContinuousLinearMap
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
this :
  HasFPowerSeriesOnBall ((fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1)) ∘ (⇑(HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) : (a : ℂ) → ℂ))
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 (1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ)
⊢ 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ

Tactic:
simp [enorm_smul, enorm_inv, hz]

Premises used: enorm_smul, enorm_inv
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
this :
  HasFPowerSeriesOnBall ((fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1)) ∘ (⇑(HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) : (a : ℂ) → ℂ))
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 (1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ)
⊢ 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ

Tactic:
have H : 1 / ‖(z⁻¹ • 1 : ℂ →L[ℂ] ℂ)‖ₑ = ‖z‖ₑ := by simp [enorm_smul, enorm_inv, hz]
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
this :
  HasFPowerSeriesOnBall ((fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1)) ∘ (⇑(HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) : (a : ℂ) → ℂ))
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 (1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ)
H : 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ
⊢ HasFPowerSeriesOnBall (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 ‖z‖ₑ

Tactic:
simp only [one_div, ContinuousLinearMap.coe_smul', H, Function.comp_def] at this

Premises used: ContinuousLinearMap.coe_smul', Function.comp_def, one_div
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
H : 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 ‖z‖ₑ
⊢ HasFPowerSeriesOnBall (fun (x : ℂ) ↦ 1 / (z - x) ^ (a + 1))
    (HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (z ^ (a + 1))⁻¹
      (FormalMultilinearSeries.compContinuousLinearMap
          (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
          (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) :
        FormalMultilinearSeries ℂ ℂ ℂ))
    0 ‖z‖ₑ

Tactic:
(this.const_smul (c := (z ^ (a + 1))⁻¹)).congr ?_

Premises used: HasFPowerSeriesOnBall.congr, HasFPowerSeriesOnBall.const_smul
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
H : 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 ‖z‖ₑ
h :
  (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (z ^ (n + a + 1))⁻¹ * (↑((a + n).choose a) : ℂ)) =
    HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (z ^ (a + 1))⁻¹
      (FormalMultilinearSeries.compContinuousLinearMap
          (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
          (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) :
        FormalMultilinearSeries ℂ ℂ ℂ)
convert_2 :
  Set.EqOn
    (HSMul.hSMul (β := ℂ → ℂ) (z ^ (a + 1))⁻¹ fun (x : ℂ) ↦
      ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (fun (x : ℂ) ↦ 1 / (z - x) ^ (a + 1)) (Metric.eball 0 ‖z‖ₑ)
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (z - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (z ^ (n + a + 1))⁻¹ * (↑((a + n).choose a) : ℂ)) 0 ‖z‖ₑ

Tactic:
convert! (this.const_smul (c := (z ^ (a + 1))⁻¹)).congr ?_ using 2

Premises used: HasFPowerSeriesOnBall.congr, HasFPowerSeriesOnBall.const_smul
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
H : 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 ‖z‖ₑ
h :
  ∀ (n : ℕ),
    Eq (α := ℂ)
      ((FormalMultilinearSeries.ofScalars ℂ (fun (n : ℕ) ↦ (z ^ (n + a + 1))⁻¹ * (↑((a + n).choose a) : ℂ)) n :
            (a : (i : Fin n) → ℂ) → ℂ)
          fun (x : Fin n) ↦ 1 :
        ℂ)
      ((HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (z ^ (a + 1))⁻¹
              (FormalMultilinearSeries.compContinuousLinearMap
                  (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) :
                    FormalMultilinearSeries ℂ ℂ ℂ)
                  (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) :
                FormalMultilinearSeries ℂ ℂ ℂ)
              n :
            (a : (i : Fin n) → ℂ) → ℂ)
          fun (x : Fin n) ↦ 1 :
        ℂ)
⊢ (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (z ^ (n + a + 1))⁻¹ * (↑((a + n).choose a) : ℂ)) =
    HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (z ^ (a + 1))⁻¹
      (FormalMultilinearSeries.compContinuousLinearMap
          (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
          (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) :
        FormalMultilinearSeries ℂ ℂ ℂ)

Tactic:
ext n
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
H : 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 ‖z‖ₑ
n : ℕ
⊢ ((FormalMultilinearSeries.ofScalars ℂ (fun (n : ℕ) ↦ (z ^ (n + a + 1))⁻¹ * (↑((a + n).choose a) : ℂ)) n :
        (a : (i : Fin n) → ℂ) → ℂ)
      fun (x : Fin n) ↦ 1) =
    HSMul.hSMul (β := ℂ) (z ^ (a + 1))⁻¹
      ((FormalMultilinearSeries.ofScalars ℂ (fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ)) n : (a : (i : Fin n) → ℂ) → ℂ)
          (Function.comp (β := ℂ) (⇑(HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1) : (a : ℂ) → ℂ) fun (x : Fin n) ↦ 1) :
        ℂ)

Tactic:
simp [add_assoc, pow_add _ _ (a + 1), mul_assoc]

Premises used: add_assoc, mul_assoc, pow_add, HAdd.hAdd, OfNat.ofNat
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a : ℕ
z : ℂ
hz : z ≠ 0
this_1 :
  HasFPowerSeriesOnBall (𝕜 := ℂ) (E := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
    ((HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1 : (a : ℂ) → ℂ) 0 : ℂ) 1
H : 1 / ‖HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹)
    (FormalMultilinearSeries.compContinuousLinearMap
      (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (↑((a + n).choose a) : ℂ) : FormalMultilinearSeries ℂ ℂ ℂ)
      (HSMul.hSMul (β := ℂ →L[ℂ] ℂ) z⁻¹ 1))
    0 ‖z‖ₑ
w : ℂ
hw : Membership.mem (γ := Set ℂ) (Metric.eball 0 ‖z‖ₑ) w
⊢ HSMul.hSMul (β := ℂ → ℂ) (z ^ (a + 1))⁻¹
      (fun (x : ℂ) ↦ ((1 - HSMul.hSMul (β := (a : ℂ) → ℂ) z⁻¹ (⇑1 : (a : ℂ) → ℂ) x) ^ (a + 1))⁻¹) w =
    (fun (x : ℂ) ↦ 1 / (z - x) ^ (a + 1)) w

Tactic:
simp [← mul_inv_rev, ← mul_pow, sub_mul, mul_right_comm _ w, hz]

Premises used: sub_mul, mul_right_comm, mul_inv_rev, mul_pow
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (z - x))
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (z ^ (n + 1))⁻¹) 0 ‖z‖ₑ

Tactic:
simpa using one_div_sub_pow_hasFPowerSeriesOnBall_zero (a := 0) hz

Premises used: Complex.one_div_sub_pow_hasFPowerSeriesOnBall_zero, OfNat.ofNat
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (z - x) 2)
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ (z ^ (n + 2))⁻¹ * HAdd.hAdd (α := ℂ) (↑n : ℂ) 1) 0 ‖z‖ₑ

Tactic:
simpa [add_comm 1] using one_div_sub_pow_hasFPowerSeriesOnBall_zero 1 hz

Premises used: add_comm, Complex.one_div_sub_pow_hasFPowerSeriesOnBall_zero, OfNat.ofNat
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / (1 - x)) (FormalMultilinearSeries.ofScalars ℂ 1) 0 1

Tactic:
simpa using one_div_sub_hasFPowerSeriesOnBall_zero (z := 1)

Premises used: Complex.one_div_sub_hasFPowerSeriesOnBall_zero
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (1 - x) 2)
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ HAdd.hAdd (α := ℂ) (↑n : ℂ) 1) 0 1

Tactic:
simpa using one_div_sub_sq_hasFPowerSeriesOnBall_zero (z := 1)

Premises used: Complex.one_div_sub_sq_hasFPowerSeriesOnBall_zero
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a b : ℂ
⊢ HasFPowerSeriesOnBall
    ((HSMul.hSMul (β := ℂ → ℂ) (b - a) fun (x : ℂ) ↦ 1 / (1 - x)) +
      HSMul.hSMul (β := ℂ → ℂ) a fun (x : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (1 - x) 2)
    (HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (b - a)
        (FormalMultilinearSeries.ofScalars ℂ 1 : FormalMultilinearSeries ℂ ℂ ℂ) +
      HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) a
        (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ HAdd.hAdd (α := ℂ) (↑n : ℂ) 1 :
          FormalMultilinearSeries ℂ ℂ ℂ))
    0 1

Tactic:
(one_div_one_sub_hasFPowerSeriesOnBall_zero.const_smul (c := b - a)).add
      (one_div_one_sub_sq_hasFPowerSeriesOnBall_zero.const_smul (c := a))

Premises used: HasFPowerSeriesOnBall.add, HasFPowerSeriesOnBall.const_smul, Complex.one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, Complex.one_div_one_sub_hasFPowerSeriesOnBall_zero
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a b : ℂ
h :
  ∀ (x : ℂ),
    (b - a) / (1 - x) + a / HPow.hPow (β := ℕ) (1 - x) 2 =
      ((HSMul.hSMul (β := ℂ → ℂ) (b - a) fun (x : ℂ) ↦ 1 / (1 - x)) +
          HSMul.hSMul (β := ℂ → ℂ) a fun (x : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (1 - x) 2)
        x
h₁ :
  (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ a * (↑n : ℂ) + b) =
    HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (b - a)
        (FormalMultilinearSeries.ofScalars ℂ 1 : FormalMultilinearSeries ℂ ℂ ℂ) +
      HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) a
        (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ HAdd.hAdd (α := ℂ) (↑n : ℂ) 1 :
          FormalMultilinearSeries ℂ ℂ ℂ)
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (x : ℂ) ↦ (b - a) / (1 - x) + a / HPow.hPow (β := ℕ) (1 - x) 2)
    (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ a * (↑n : ℂ) + b) 0 1

Tactic:
convert!
    (one_div_one_sub_hasFPowerSeriesOnBall_zero.const_smul (c := b - a)).add
      (one_div_one_sub_sq_hasFPowerSeriesOnBall_zero.const_smul (c := a)) using 2

Premises used: HasFPowerSeriesOnBall.add, HasFPowerSeriesOnBall.const_smul, Complex.one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, Complex.one_div_one_sub_hasFPowerSeriesOnBall_zero
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a b x : ℂ
⊢ (b - a) / (1 - x) + a / HPow.hPow (β := ℕ) (1 - x) 2 =
    ((HSMul.hSMul (β := ℂ → ℂ) (b - a) fun (x : ℂ) ↦ 1 / (1 - x)) +
        HSMul.hSMul (β := ℂ → ℂ) a fun (x : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (1 - x) 2)
      x

Tactic:
simp [div_eq_mul_inv]

Premises used: div_eq_mul_inv
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a b : ℂ
h :
  ∀ (n : ℕ),
    Eq (α := ℂ)
      ((FormalMultilinearSeries.ofScalars ℂ (fun (n : ℕ) ↦ a * (↑n : ℂ) + b) n : (a : (i : Fin n) → ℂ) → ℂ)
          fun (x : Fin n) ↦ 1 :
        ℂ)
      (((HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (b - a)
                  (FormalMultilinearSeries.ofScalars ℂ 1 : FormalMultilinearSeries ℂ ℂ ℂ) +
                HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) a
                  (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ HAdd.hAdd (α := ℂ) (↑n : ℂ) 1 :
                    FormalMultilinearSeries ℂ ℂ ℂ))
              n :
            (a : (i : Fin n) → ℂ) → ℂ)
          fun (x : Fin n) ↦ 1 :
        ℂ)
⊢ (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ a * (↑n : ℂ) + b) =
    HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (b - a)
        (FormalMultilinearSeries.ofScalars ℂ 1 : FormalMultilinearSeries ℂ ℂ ℂ) +
      HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) a
        (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ HAdd.hAdd (α := ℂ) (↑n : ℂ) 1 :
          FormalMultilinearSeries ℂ ℂ ℂ)

Tactic:
ext
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a b : ℂ
n : ℕ
h : a * (↑n : ℂ) + b = b - a + a * HAdd.hAdd (α := ℂ) (↑n : ℂ) 1
⊢ Eq (α := ℂ)
    ((FormalMultilinearSeries.ofScalars ℂ (fun (n : ℕ) ↦ a * (↑n : ℂ) + b) n : (a : (i : Fin n) → ℂ) → ℂ)
        fun (x : Fin n) ↦ 1 :
      ℂ)
    (((HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) (b - a)
                (FormalMultilinearSeries.ofScalars ℂ 1 : FormalMultilinearSeries ℂ ℂ ℂ) +
              HSMul.hSMul (β := FormalMultilinearSeries ℂ ℂ ℂ) a
                (FormalMultilinearSeries.ofScalars ℂ fun (n : ℕ) ↦ HAdd.hAdd (α := ℂ) (↑n : ℂ) 1 :
                  FormalMultilinearSeries ℂ ℂ ℂ))
            n :
          (a : (i : Fin n) → ℂ) → ℂ)
        fun (x : Fin n) ↦ 1 :
      ℂ)

Tactic:
simp
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
a b : ℂ
n : ℕ
⊢ a * (↑n : ℂ) + b = b - a + a * HAdd.hAdd (α := ℂ) (↑n : ℂ) 1

Tactic:
ring
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
w x : ℂ
hw : w ≠ x
rw :
  HasFPowerSeriesOnBall
    (HSub.hSub (α := (i : ℂ) → ℂ) (fun (z : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (z - w) 2) fun (z : ℂ) ↦
      1 / HPow.hPow (β := ℕ) w 2)
    (HSub.hSub (α := FormalMultilinearSeries ℂ ℂ ℂ) (β := FormalMultilinearSeries ℂ ℂ ℂ)
      (FormalMultilinearSeries.ofScalars ℂ fun (i : ℕ) ↦
          HAdd.hAdd (α := ℂ) (↑i : ℂ) 1 * HPow.hPow (β := ℤ) (w - x) (-(↑(i + 2) : ℤ)) :
        FormalMultilinearSeries ℂ ℂ ℂ)
      (FormalMultilinearSeries.ofScalars ℂ fun (i : ℕ) ↦
          Nat.casesOn (motive := fun (x : ℕ) ↦ ℂ) i (HPow.hPow (β := ℤ) w (-2)) 0 :
        FormalMultilinearSeries ℂ ℂ ℂ))
    x ‖w - x‖ₑ
⊢ HasFPowerSeriesOnBall (𝕜 := ℂ) (F := ℂ) (fun (z : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (z - w) 2 - 1 / HPow.hPow (β := ℕ) w 2)
    (FormalMultilinearSeries.ofScalars ℂ fun (i : ℕ) ↦
      HSub.hSub (β := ℂ) (HAdd.hAdd (α := ℂ) (↑i : ℂ) 1 * HPow.hPow (β := ℤ) (w - x) (-(↑(i + 2) : ℤ)))
        (Nat.casesOn (motive := fun (x : ℕ) ↦ ℂ) i (HPow.hPow (β := ℤ) w (-2)) 0 : ℂ))
    x ‖w - x‖ₑ

Tactic:
rw [← Pi.sub_def, ← Pi.sub_def, FormalMultilinearSeries.ofScalars_sub]

Premises used: FormalMultilinearSeries.ofScalars_sub, Pi.sub_def
-- Module: Mathlib.Analysis.Analytic.Binomial
Goal:
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall
    (HSub.hSub (α := (i : ℂ) → ℂ) (fun (z : ℂ) ↦ 1 / HPow.hPow (β := ℕ) (z - w) 2) fun (z : ℂ) ↦
      1 / HPow.hPow (β := ℕ) w 2)
    (HSub.hSub (α := FormalMultilinearSeries ℂ ℂ ℂ) (β := FormalMultilinearSeries ℂ ℂ ℂ)
      (FormalMultilinearSeries.ofScalars ℂ fun (i : ℕ) ↦
          HAdd.hAdd (α := ℂ) (↑i : ℂ) 1 * HPow.hPow (β := ℤ) (w - x) (-(↑(i + 2) : ℤ)) :
        FormalMultilinearSeries ℂ ℂ ℂ)
      (FormalMultilinearSeries.ofScalars ℂ fun (i : ℕ) ↦
          Nat.casesOn (motive := fun (x : ℕ) ↦ ℂ) i (HPow.hPow (β := ℤ) w (-2)) 0 :
        FormalMultilinearSeries ℂ ℂ ℂ))
    x ‖w - x‖ₑ

Tactic:
.sub ?_ ?_

Premises used: HasFPowerSeriesOnBall.sub