awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Convolution (goal)
Goal:
f g : ℕ → ℂ
s : ℂ
n : ℕ
hn : n ≠ 0
h :
  term (f ⍟ g) s n =
    ∑' (b : (↑(↑n.divisorsAntidiagonal : Set (ℕ × ℕ)) : Type)), term f s (↑b : ℕ × ℕ).1 * term g s (↑b : ℕ × ℕ).2
h₁ : term (f ⍟ g) s n = ∑ x ∈ n.divisorsAntidiagonal, term f s x.1 * term g s x.2
⊢ term (f ⍟ g) s n =
    ∑' (b : (↑(preimage (β := ℕ) (fun (p : ℕ × ℕ) ↦ p.1 * p.2) {n}) : Type)),
      term f s (↑b : ℕ × ℕ).1 * term g s (↑b : ℕ × ℕ).2
have h₂ : Eq (α := ℂ) (∑ p ∈ n.divisorsAntidiagonal, term f s p.1 * term g s p.2)
  (∑ x ∈ n.divisorsAntidiagonal, term f s x.1 * term g s x.2)
Tactic: rw [show (fun p : ℕ × ℕ ↦ p.1 * p.2) ⁻¹' {n} = n.divisorsAntidiagonal by ext; simp [hn],
    Finset.tsum_subtype' n.divisorsAntidiagonal fun p ↦ term f s p.1 * term g s p.2,
    term_convolution f g s n]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Convolution (hypothesis)
Goal:
f g : ℕ → ℂ
s a b : ℂ
hf : LSeriesHasSum f s a
hg : LSeriesHasSum g s b
⊢ LSeriesHasSum (f ⍟ g) s (a * b)
have hsum : Summable (α := ℂ) fun (x : ℕ × ℕ) ↦ term f s x.1 * term g s x.2
Tactic: have hsum := summable_mul_of_summable_norm hf.summable.norm hg.summable.norm
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
n : ℕ
s : ℂ
⊢ HasDerivAt (fun (z : ℂ) ↦ term f z n) (-term (logMul f) s n) s
have inl : HasDerivAt (fun (z : ℂ) ↦ term f z 0) (-term (logMul f) s 0) s
Tactic: rcases eq_or_ne n 0 with rfl | hn
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
n : ℕ
s : ℂ
inl : HasDerivAt (fun (z : ℂ) ↦ term f z 0) (-term (logMul f) s 0) s
⊢ HasDerivAt (fun (z : ℂ) ↦ term f z n) (-term (logMul f) s n) s
have inr : ∀ (hn : n ≠ 0), HasDerivAt (fun (z : ℂ) ↦ term f z n) (-term (logMul f) s n) s
Tactic: rcases eq_or_ne n 0 with rfl | hn
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
n : ℕ
s : ℂ
hn : n ≠ 0
⊢ HasDerivAt (fun (z : ℂ) ↦ term f z n) (-term (logMul f) s n) s
have inr : HasDerivAt (fun (z : ℂ) ↦ f n * HPow.hPow (α := ℂ) (↑n : ℂ) (-z))
  (f n * (-log (↑n : ℂ) * HPow.hPow (α := ℂ) (↑n : ℂ) (-s))) s
Tactic: simp_rw [term_of_ne_zero hn, ← neg_div, ← neg_mul, mul_comm, mul_div_assoc, div_eq_mul_inv,
    ← cpow_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have obtain : ∀ (x : ℝ) (hxs : x < s.re) (hf : LSeriesSummable f (↑x : ℂ)),
  LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
Tactic: obtain ⟨x, hxs, hf⟩ := LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re h
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have obtain : ∀ (y : ℝ) (hxy : x < y) (hys : y < s.re),
  LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
Tactic: obtain ⟨y, hxy, hys⟩ := exists_between hxs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
Tactic: have h₀ : Summable (fun n ↦ ‖term f x n‖) := summable_norm_iff.mpr hf
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have h₁ : ∀ (n : ℕ), DifferentiableOn ℂ (F := ℂ) (fun (x : ℂ) ↦ term f x n) S
Tactic: have h₁ (n) : DifferentiableOn ℂ (term f · n) S :=
    fun z _ ↦ (hasDerivAt_term f n _).differentiableAt.differentiableWithinAt
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
h₁ : ∀ (n : ℕ), DifferentiableOn ℂ (F := ℂ) (fun (x : ℂ) ↦ term f x n) S
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have h₂ : IsOpen S
Tactic: have h₂ : IsOpen S := isOpen_lt continuous_const continuous_re
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
h₁ : ∀ (n : ℕ), DifferentiableOn ℂ (F := ℂ) (fun (x : ℂ) ↦ term f x n) S
h₂ : IsOpen S
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have h₃ : ∀ (n : ℕ) (z : ℂ) (hz : z ∈ S), ‖term f z n‖ ≤ ‖term f (↑x : ℂ) n‖
Tactic: have h₃ (n z) (hz : z ∈ S) : ‖term f z n‖ ≤ ‖term f x n‖ :=
    norm_term_le_of_re_le_re f (by simpa using (hxy.trans hz).le) n
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
h₁ : ∀ (n : ℕ), DifferentiableOn ℂ (F := ℂ) (fun (x : ℂ) ↦ term f x n) S
h₂ : IsOpen S
h₃ : ∀ (n : ℕ) (z : ℂ) (hz : z ∈ S), ‖term f z n‖ ≤ ‖term f (↑x : ℂ) n‖
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have H : HasSum (α := ℂ) (fun (i : ℕ) ↦ deriv (fun (x : ℂ) ↦ term f x i) s) (deriv (fun (w : ℂ) ↦ ∑' (i : ℕ), term f w i) s)
Tactic: have H := hasSum_deriv_of_summable_norm h₀ h₁ h₂ h₃ hys
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
h₁ : ∀ (n : ℕ), DifferentiableOn ℂ (F := ℂ) (fun (x : ℂ) ↦ term f x n) S
h₂ : IsOpen S
h₃ : ∀ (n : ℕ) (z : ℂ) (hz : z ∈ S), ‖term f z n‖ ≤ ‖term f (↑x : ℂ) n‖
H : HasSum (α := ℂ) (fun (i : ℕ) ↦ deriv (fun (x : ℂ) ↦ term f x i) s) (deriv (fun (w : ℂ) ↦ ∑' (i : ℕ), term f w i) s)
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have H₁ : HasSum (α := ℂ) (fun (i : ℕ) ↦ -term (logMul f) s i) (deriv (fun (w : ℂ) ↦ ∑' (i : ℕ), term f w i) s)
Tactic: simp_rw [(hasDerivAt_term f _ _).deriv] at H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
x : ℝ
hxs : x < s.re
hf : LSeriesSummable f (↑x : ℂ)
y : ℝ
hxy : x < y
hys : y < s.re
S : Set ℂ := {z : ℂ | y < z.re}
h₀ : Summable (α := ℝ) fun (n : ℕ) ↦ ‖term f (↑x : ℂ) n‖
h₁ : ∀ (n : ℕ), DifferentiableOn ℂ (F := ℂ) (fun (x : ℂ) ↦ term f x n) S
h₂ : IsOpen S
h₃ : ∀ (n : ℕ) (z : ℂ) (hz : z ∈ S), ‖term f z n‖ ≤ ‖term f (↑x : ℂ) n‖
H : HasSum (α := ℂ) (fun (i : ℕ) ↦ -term (logMul f) s i) (deriv (fun (w : ℂ) ↦ ∑' (i : ℕ), term f w i) s)
⊢ LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
have refine : HasDerivAt (LSeries f) (-LSeries (logMul f) s) s
Tactic: refine ⟨summable_neg_iff.mp H.summable, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
⊢ abscissaOfAbsConv (logMul f) = abscissaOfAbsConv f
have a : abscissaOfAbsConv (logMul f) ≤ abscissaOfAbsConv f
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
a : abscissaOfAbsConv (logMul f) ≤ abscissaOfAbsConv f
⊢ abscissaOfAbsConv (logMul f) = abscissaOfAbsConv f
have a₁ : abscissaOfAbsConv f ≤ abscissaOfAbsConv (logMul f)
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
⊢ abscissaOfAbsConv (logMul f) ≤ abscissaOfAbsConv f
have a : ∀ (s : ℝ) (hs : abscissaOfAbsConv f < (↑s : EReal)), LSeriesSummable (logMul f) (↑s : ℂ)
Tactic: refine abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' fun s hs ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
⊢ abscissaOfAbsConv f ≤ abscissaOfAbsConv (logMul f)
have a : ∀ (s : ℝ) (hs : abscissaOfAbsConv (logMul f) < (↑s : EReal)), LSeriesSummable f (↑s : ℂ)
Tactic: refine abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' fun s hs ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℝ
hs : abscissaOfAbsConv (logMul f) < (↑s : EReal)
⊢ LSeriesSummable f (↑s : ℂ)
have a : ∀ᶠ (i : ℕ) in Filter.atTop, ‖term f (↑s : ℂ) i‖ ≤ (fun (n : ℕ) ↦ ‖term (logMul f) (↑s : ℂ) n‖) i
Tactic: refine (LSeriesSummable_of_abscissaOfAbsConv_lt_re <| by simp [hs])
      |>.norm.of_norm_bounded_eventually_nat (g := fun n ↦ ‖term (logMul f) s n‖) ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℝ
hs : abscissaOfAbsConv (logMul f) < (↑s : EReal)
⊢ ∀ᶠ (i : ℕ) in Filter.atTop, ‖term f (↑s : ℂ) i‖ ≤ (fun (n : ℕ) ↦ ‖term (logMul f) (↑s : ℂ) n‖) i
have h : ∀ (n : ℕ) (hn : max 1 ⌈Real.exp 1⌉₊ ≤ n), ‖term f (↑s : ℂ) n‖ ≤ ‖term (logMul f) (↑s : ℂ) n‖
Tactic: filter_upwards [Filter.eventually_ge_atTop <| max 1 (Nat.ceil (Real.exp 1))] with n hn
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℝ
hs : abscissaOfAbsConv (logMul f) < (↑s : EReal)
n : ℕ
hn : max 1 ⌈Real.exp 1⌉₊ ≤ n
⊢ ‖term f (↑s : ℂ) n‖ ≤ ‖term (logMul f) (↑s : ℂ) n‖
have h : ‖f n / HPow.hPow (α := ℂ) (↑n : ℂ) (↑s : ℂ)‖ ≤ ‖Real.log (↑n : ℝ)‖ * ‖f n / HPow.hPow (α := ℂ) (↑n : ℂ) (↑s : ℂ)‖
Tactic: simp only [term_of_ne_zero (show n ≠ 0 by omega), logMul, norm_mul, mul_div_assoc,
      ← natCast_log, norm_real]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
s : ℝ
hs : abscissaOfAbsConv (logMul f) < (↑s : EReal)
n : ℕ
hn : max 1 ⌈Real.exp 1⌉₊ ≤ n
⊢ ‖f n / HPow.hPow (α := ℂ) (↑n : ℂ) (↑s : ℂ)‖ ≤ ‖Real.log (↑n : ℝ)‖ * ‖f n / HPow.hPow (α := ℂ) (↑n : ℂ) (↑s : ℂ)‖
have h : 1 ≤ Real.log (↑n : ℝ)
Tactic: refine le_mul_of_one_le_left (norm_nonneg _) (.trans ?_ <| Real.le_norm_self _)
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
m : ℕ
⊢ abscissaOfAbsConv (logMul^[m] f) = abscissaOfAbsConv f
have zero : abscissaOfAbsConv (logMul^[0] f) = abscissaOfAbsConv f
Tactic: induction m with
  | zero => simp
  | succ n ih => simp [ih, Function.iterate_succ', Function.comp_def,
      -Function.comp_apply, -Function.iterate_succ]
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
m : ℕ
zero : abscissaOfAbsConv (logMul^[0] f) = abscissaOfAbsConv f
⊢ abscissaOfAbsConv (logMul^[m] f) = abscissaOfAbsConv f
have succ : ∀ (n : ℕ) (ih : abscissaOfAbsConv (logMul^[n] f) = abscissaOfAbsConv f),
  abscissaOfAbsConv (logMul^[n + 1] f) = abscissaOfAbsConv f
Tactic: induction m with
  | zero => simp
  | succ n ih => simp [ih, Function.iterate_succ', Function.comp_def,
      -Function.comp_apply, -Function.iterate_succ]
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
m : ℕ
ih :
  ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
    iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
⊢ iteratedDeriv (m + 1) (LSeries f) s = HPow.hPow (α := ℂ) (-1) (m + 1) * LSeries (logMul^[m + 1] f) s
have ih' : Set.EqOn (iteratedDeriv m (LSeries f)) (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f))
  {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)}
Tactic: have ih' : {s | abscissaOfAbsConv f < re s}.EqOn (iteratedDeriv m (LSeries f))
        ((-1) ^ m * LSeries (logMul^[m] f)) := fun _ hs ↦ ih hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
m : ℕ
ih :
  ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
    iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
ih' :
  Set.EqOn (iteratedDeriv m (LSeries f)) (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f))
    {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)}
⊢ iteratedDeriv (m + 1) (LSeries f) s = HPow.hPow (α := ℂ) (-1) (m + 1) * LSeries (logMul^[m + 1] f) s
have this : derivWithin (iteratedDeriv m (LSeries f)) {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)} s =
  derivWithin (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f)) {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)} s
Tactic: have := derivWithin_congr ih' (ih h)
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (hypothesis)
Goal:
f : ℕ → ℂ
m : ℕ
ih :
  ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
    iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
ih' :
  Set.EqOn (iteratedDeriv m (LSeries f)) (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f))
    {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)}
this :
  derivWithin (iteratedDeriv m (LSeries f)) {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)} s =
    derivWithin (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f)) {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)}
      s
⊢ iteratedDeriv (m + 1) (LSeries f) s = HPow.hPow (α := ℂ) (-1) (m + 1) * LSeries (logMul^[m + 1] f) s
have this₁ : deriv (iteratedDeriv m (LSeries f)) s = deriv (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f)) s
Tactic: simp_rw [derivWithin_of_isOpen (isOpen_re_gt_EReal _) h] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
m : ℕ
ih :
  ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
    iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
ih' :
  Set.EqOn (iteratedDeriv m (LSeries f)) (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f))
    {s : ℂ | abscissaOfAbsConv f < (↑s.re : EReal)}
this : deriv (iteratedDeriv m (LSeries f)) s = deriv (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f)) s
⊢ iteratedDeriv (m + 1) (LSeries f) s = HPow.hPow (α := ℂ) (-1) (m + 1) * LSeries (logMul^[m + 1] f) s
have succ : deriv (HPow.hPow (α := ℂ → ℂ) (-1) m * LSeries (logMul^[m] f)) s =
  HPow.hPow (α := ℂ) (-1) (m + 1) * LSeries (logMul^[m + 1] f) s
Tactic: rw [iteratedDeriv_succ, this]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
m : ℕ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
⊢ iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s
have zero : ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
  iteratedDeriv 0 (LSeries f) s = HPow.hPow (α := ℂ) (β := ℕ) (-1) 0 * LSeries (logMul^[0] f) s
Tactic: induction m generalizing s with
  | zero => simp
  | succ m ih =>
    have ih' : {s | abscissaOfAbsConv f < re s}.EqOn (iteratedDeriv m (LSeries f))
        ((-1) ^ m * LSeries (logMul^[m] f)) := fun _ hs ↦ ih hs
    have := derivWithin_congr ih' (ih h)
    simp_rw [derivWithin_of_isOpen (isOpen_re_gt_EReal _) h] at this
    rw [iteratedDeriv_succ, this]
    simp [Pi.mul_def, pow_succ, Function.iterate_succ',
      LSeries_deriv <| absicssaOfAbsConv_logPowMul.symm ▸ h, -Function.iterate_succ]
Cleared hypotheses: m, s, h
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Deriv (goal)
Goal:
f : ℕ → ℂ
m : ℕ
s : ℂ
h : abscissaOfAbsConv f < (↑s.re : EReal)
zero :
  ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
    iteratedDeriv 0 (LSeries f) s = HPow.hPow (α := ℂ) (β := ℕ) (-1) 0 * LSeries (logMul^[0] f) s
⊢ iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s
have succ : ∀ (m : ℕ)
  (ih :
    ∀ {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
      iteratedDeriv m (LSeries f) s = HPow.hPow (α := ℂ) (-1) m * LSeries (logMul^[m] f) s)
  {s : ℂ} (h : abscissaOfAbsConv f < (↑s.re : EReal)),
  iteratedDeriv (m + 1) (LSeries f) s = HPow.hPow (α := ℂ) (-1) (m + 1) * LSeries (logMul^[m + 1] f) s
Tactic: induction m generalizing s with
  | zero => simp
  | succ m ih =>
    have ih' : {s | abscissaOfAbsConv f < re s}.EqOn (iteratedDeriv m (LSeries f))
        ((-1) ^ m * LSeries (logMul^[m] f)) := fun _ hs ↦ ih hs
    have := derivWithin_congr ih' (ih h)
    simp_rw [derivWithin_of_isOpen (isOpen_re_gt_EReal _) h] at this
    rw [iteratedDeriv_succ, this]
    simp [Pi.mul_def, pow_succ, Function.iterate_succ',
      LSeries_deriv <| absicssaOfAbsConv_logPowMul.symm ▸ h, -Function.iterate_succ]
Cleared hypotheses: m, s, h
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
⊢ (fun (n : ℕ) ↦ (1 : (a : ℕ) → ℂ) n) = δ
have h : ∀ (x : ℕ), (1 : (a : ℕ) → ℂ) x = δ x
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
⊢ ¬LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1
have refine : ∀ (h : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1),
  Summable (α := ℂ) fun (b : ℕ) ↦
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) b) : ℂ)
Tactic: refine fun h ↦ not_summable_one_div_on_primes <| summable_ofReal.mp <| .of_neg ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
h : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1
⊢ Summable (α := ℂ) fun (b : ℕ) ↦
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) b) : ℂ)
have refine : ∀ (n : ℕ),
  Set.indicator {n : ℕ | Nat.Prime n} (term (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1) n =
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) n) : ℂ)
Tactic: refine (h.indicator {n | n.Prime}).congr fun n ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
h : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1
n : ℕ
⊢ Set.indicator {n : ℕ | Nat.Prime n} (term (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1) n =
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) n) : ℂ)
have pos : ∀ (hn : Nat.Prime n),
  Set.indicator {n : ℕ | Nat.Prime n} (term (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1) n =
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) n) : ℂ)
Tactic: by_cases hn : n.Prime
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
h : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1
n : ℕ
pos :
  ∀ (hn : Nat.Prime n),
    Set.indicator {n : ℕ | Nat.Prime n} (term (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1) n =
      -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) n) : ℂ)
⊢ Set.indicator {n : ℕ | Nat.Prime n} (term (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1) n =
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) n) : ℂ)
have neg : ∀ (hn : ¬Nat.Prime n),
  Set.indicator {n : ℕ | Nat.Prime n} (term (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) 1) n =
    -(↑(Set.indicator {p : ℕ | Nat.Prime p} (fun (n : ℕ) ↦ HDiv.hDiv (α := ℝ) 1 (↑n : ℝ)) n) : ℂ)
Tactic: by_cases hn : n.Prime
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
s : ℂ
⊢ LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) s ↔ 1 < s.re
have refine_1 : ∀ (H : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) s), 1 < s.re
Tactic: refine ⟨fun H ↦ ?_, LSeriesSummable_of_bounded_of_one_lt_re (m := 1) fun n _ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
s : ℂ
refine_1 : ∀ (H : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) s), 1 < s.re
⊢ LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) s ↔ 1 < s.re
have refine_2 : ∀ (n : ℕ), n ≠ 0 → norm (E := ℂ) (↑((μ : (a : ℕ) → ℤ) n) : ℂ) ≤ 1
Tactic: refine ⟨fun H ↦ ?_, LSeriesSummable_of_bounded_of_one_lt_re (m := 1) fun n _ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
s : ℂ
H : LSeriesSummable (fun (n : ℕ) ↦ (↑((μ : (a : ℕ) → ℤ) n) : ℂ)) s
⊢ 1 < s.re
have refine_1 : ∀ (h : s.re ≤ 1), False
Tactic: by_contra! h
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
s : ℂ
n : ℕ
h : n ≠ 0
⊢ norm (E := ℂ) (↑((μ : (a : ℕ) → ℤ) n) : ℂ) ≤ 1
have refine_2 : LE.le (α := ℤ) (|(μ : (a : ℕ) → ℤ) n| : ℤ) (1 : ℤ)
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
N : ℕ
R : Type u_1
inst : CommMonoidWithZero R
χ : DirichletCharacter R N
⊢ IsMultiplicative.{u_1} (R := R) (toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N))
have refine_1 : Eq.{u_1 + 1} (α := R) ((toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N) : (a : ℕ) → R) 1 : R)
  (1 : R)
Tactic: refine IsMultiplicative.iff_ne_zero.mpr ⟨?_, fun {m} {n} hm hn _ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
N : ℕ
R : Type u_1
inst : CommMonoidWithZero R
χ : DirichletCharacter R N
refine_1 :
  Eq.{u_1 + 1} (α := R) ((toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N) : (a : ℕ) → R) 1 : R)
    (1 : R)
⊢ IsMultiplicative.{u_1} (R := R) (toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N))
have refine_2 : ∀ {m n : ℕ} (hm : m ≠ 0) (hn : n ≠ 0),
  m.Coprime n →
    (toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N) : (a : ℕ) → R) (m * n) =
      HMul.hMul (α := R) (β := R)
        ((toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N) : (a : ℕ) → R) m : R)
        ((toArithmeticFunction fun (x : ℕ) ↦ (χ : (a : ZMod N) → R) (↑x : ZMod N) : (a : ℕ) → R) n : R)
Tactic: refine IsMultiplicative.iff_ne_zero.mpr ⟨?_, fun {m} {n} hm hn _ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
R : Type u_1
inst : CommSemiring R
n : ℕ
χ : DirichletCharacter R n
f g : ℕ → R
⊢ (fun (x : ℕ) ↦ (χ : (a : ZMod n) → R) (↑x : ZMod n)) * f ⍟
      ((fun (x : ℕ) ↦ (χ : (a : ZMod n) → R) (↑x : ZMod n)) * g) =
    (fun (x : ℕ) ↦ (χ : (a : ZMod n) → R) (↑x : ZMod n)) * (f ⍟ g)
have h : ∀ (n_1 : ℕ),
  ((fun (x : ℕ) ↦ (χ : (a : ZMod n) → R) (↑x : ZMod n)) * f ⍟
        ((fun (x : ℕ) ↦ (χ : (a : ZMod n) → R) (↑x : ZMod n)) * g))
      n_1 =
    HMul.hMul (α := ℕ → R) (fun (x : ℕ) ↦ (χ : (a : ZMod n) → R) (↑x : ZMod n)) (f ⍟ g) n_1
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.Dirichlet (goal)
Goal:
R : Type u_1
inst : CommSemiring R
n_1 : ℕ
χ : DirichletCharacter R n_1
f g : ℕ → R
n : ℕ
⊢ ((fun (x : ℕ) ↦ (χ : (a : ZMod n_1) → R) (↑x : ZMod n_1)) * f ⍟
        ((fun (x : ℕ) ↦ (χ : (a : ZMod n_1) → R) (↑x : ZMod n_1)) * g))
      n =
    HMul.hMul (α := ℕ → R) (fun (x : ℕ) ↦ (χ : (a : ZMod n_1) → R) (↑x : ZMod n_1)) (f ⍟ g) n
have h : Eq.{u_1 + 1} (α := R)
  (∑ x ∈ n.divisorsAntidiagonal,
    HMul.hMul (α := R) ((χ : (a : ZMod n_1) → R) (↑x.1 : ZMod n_1) : R) (f x.1) *
      HMul.hMul (α := R) ((χ : (a : ZMod n_1) → R) (↑x.2 : ZMod n_1) : R) (g x.2))
  (∑ i ∈ n.divisorsAntidiagonal, HMul.hMul (α := R) ((χ : (a : ZMod n_1) → R) (↑n : ZMod n_1) : R) (f i.1 * g i.2))
Tactic: simp only [Pi.mul_apply, LSeries.convolution_def, Finset.mul_sum]