awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
N : ℕ
inst : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : Eq (α := ℂ) (∑ j : ZMod N, (χ : (a : ZMod N) → ℂ) j) 0
ε : ℂ := HPow.hPow (β := ℕ) I (if χ.Even then 0 else 1)
calc :
  completedLFunction χ (1 - s) =
    HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1)) ((χ : (a : ZMod N) → ℂ) (-1) : ℂ) / ε *
      ZMod.completedLFunction (N := N) ((𝓕 : (a : ZMod N → ℂ) → ZMod N → ℂ) (⇑χ : (a : ZMod N) → ℂ)) s
calc₁ :
  HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1)) ((χ : (a : ZMod N) → ℂ) (-1) : ℂ) / ε *
      ZMod.completedLFunction (N := N) ((𝓕 : (a : ZMod N → ℂ) → ZMod N → ℂ) (⇑χ : (a : ZMod N) → ℂ)) s =
    HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1)) ((χ : (a : ZMod N) → ℂ) (-1) : ℂ) / ε *
      ZMod.completedLFunction (N := N)
        (fun (j : ZMod N) ↦
          HMul.hMul (β := ℂ) (HMul.hMul (α := ℂ) ((χ⁻¹ : (a : ZMod N) → ℂ) (-1) : ℂ) (gaussSum χ stdAddChar))
            ((χ⁻¹ : (a : ZMod N) → ℂ) j : ℂ))
        s
calc₂ :
  HMul.hMul (β := ℂ) (HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1)) ((χ : (a : ZMod N) → ℂ) (-1) : ℂ) / ε *
      ZMod.completedLFunction (N := N)
        (fun (j : ZMod N) ↦
          HMul.hMul (β := ℂ) (HMul.hMul (α := ℂ) ((χ⁻¹ : (a : ZMod N) → ℂ) (-1) : ℂ) (gaussSum χ stdAddChar))
            ((χ⁻¹ : (a : ZMod N) → ℂ) j : ℂ))
        s =
    HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s *
      HMul.hMul (α := ℂ) (β := ℂ) ((χ : (a : ZMod N) → ℂ) (-1) : ℂ) ((χ⁻¹ : (a : ZMod N) → ℂ) (-1) : ℂ)
⊢ completedLFunction χ (1 - s) =
    HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1) / HPow.hPow (β := ℕ) I (if χ.Even then 0 else 1) * gaussSum χ stdAddChar *
      completedLFunction χ⁻¹ s
have calc₃ : HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s *
    HMul.hMul (α := ℂ) (β := ℂ) ((χ : (a : ZMod N) → ℂ) (-1) : ℂ) ((χ⁻¹ : (a : ZMod N) → ℂ) (-1) : ℂ) =
  HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s
Tactic: calc completedLFunction χ (1 - s)
  _ = N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (𝓕 χ) s := by
    simp only [ε]
    split_ifs with h
    · rw [pow_zero, div_one, h, mul_one, completedLFunction,
        completedLFunction_one_sub_even h.to_fun _ (.inr h_sum) (.inr <| χ.map_zero' hN)]
    · replace h : χ.Odd := χ.even_or_odd.resolve_left h
      rw [completedLFunction, completedLFunction_one_sub_odd h.to_fun,
        pow_one, h, div_I, mul_neg_one, ← neg_mul, neg_neg]
  _ = (_) * ZMod.completedLFunction (fun j ↦ χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹ j) s := by
    congr 2 with j
    rw [hχ.fourierTransform_eq_inv_mul_gaussSum, ← neg_one_mul j, map_mul, mul_right_comm]
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) := by
    rw [completedLFunction, completedLFunction_const_mul]
    ring
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s := by
    rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
N : ℕ
inst : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
⊢ completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
have inl : ∀ [inst : NeZero (R := ℕ) 1] {χ : DirichletCharacter ℂ 1} (hχ : χ.IsPrimitive),
  completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑1 : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
Tactic: classical
  -- First handle special case of Riemann zeta
  rcases eq_or_ne N 1 with rfl | hN
  · simp [completedLFunction_modOne_eq, completedRiemannZeta_one_sub, rootNumber_modOne]
  -- facts about `χ` as function
  have h_sum : ∑ j, χ j = 0 := by
    refine χ.sum_eq_zero_of_ne_one (fun h ↦ hN.symm ?_)
    rwa [IsPrimitive, h, conductor_one] at hχ
  let ε := I ^ (if χ.Even then 0 else 1)
  -- gather up powers of N
  rw [rootNumber, ← mul_comm_div, ← mul_comm_div, ← cpow_sub _ _ (NeZero.ne _), sub_sub, add_halves]
  calc completedLFunction χ (1 - s)
  _ = N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (𝓕 χ) s := by
    simp only [ε]
    split_ifs with h
    · rw [pow_zero, div_one, h, mul_one, completedLFunction,
        completedLFunction_one_sub_even h.to_fun _ (.inr h_sum) (.inr <| χ.map_zero' hN)]
    · replace h : χ.Odd := χ.even_or_odd.resolve_left h
      rw [completedLFunction, completedLFunction_one_sub_odd h.to_fun,
        pow_one, h, div_I, mul_neg_one, ← neg_mul, neg_neg]
  _ = (_) * ZMod.completedLFunction (fun j ↦ χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹ j) s := by
    congr 2 with j
    rw [hχ.fourierTransform_eq_inv_mul_gaussSum, ← neg_one_mul j, map_mul, mul_right_comm]
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) := by
    rw [completedLFunction, completedLFunction_const_mul]
    ring
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s := by
    rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one]
Cleared hypotheses: N, inst, χ, hχ
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
N : ℕ
inst : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
inl :
  ∀ [inst : NeZero (R := ℕ) 1] {χ : DirichletCharacter ℂ 1} (hχ : χ.IsPrimitive),
    completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑1 : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
⊢ completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
have inr : ∀ (hN : N ≠ 1),
  completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
Tactic: classical
  -- First handle special case of Riemann zeta
  rcases eq_or_ne N 1 with rfl | hN
  · simp [completedLFunction_modOne_eq, completedRiemannZeta_one_sub, rootNumber_modOne]
  -- facts about `χ` as function
  have h_sum : ∑ j, χ j = 0 := by
    refine χ.sum_eq_zero_of_ne_one (fun h ↦ hN.symm ?_)
    rwa [IsPrimitive, h, conductor_one] at hχ
  let ε := I ^ (if χ.Even then 0 else 1)
  -- gather up powers of N
  rw [rootNumber, ← mul_comm_div, ← mul_comm_div, ← cpow_sub _ _ (NeZero.ne _), sub_sub, add_halves]
  calc completedLFunction χ (1 - s)
  _ = N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (𝓕 χ) s := by
    simp only [ε]
    split_ifs with h
    · rw [pow_zero, div_one, h, mul_one, completedLFunction,
        completedLFunction_one_sub_even h.to_fun _ (.inr h_sum) (.inr <| χ.map_zero' hN)]
    · replace h : χ.Odd := χ.even_or_odd.resolve_left h
      rw [completedLFunction, completedLFunction_one_sub_odd h.to_fun,
        pow_one, h, div_I, mul_neg_one, ← neg_mul, neg_neg]
  _ = (_) * ZMod.completedLFunction (fun j ↦ χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹ j) s := by
    congr 2 with j
    rw [hχ.fourierTransform_eq_inv_mul_gaussSum, ← neg_one_mul j, map_mul, mul_right_comm]
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) := by
    rw [completedLFunction, completedLFunction_const_mul]
    ring
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s := by
    rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
N : ℕ
inst : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
inl :
  ∀ [inst : NeZero (R := ℕ) 1] {χ : DirichletCharacter ℂ 1} (hχ : χ.IsPrimitive),
    completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑1 : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
inr inr₁ :
  ∀ (hN : N ≠ 1),
    completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
⊢ completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
have inr₂ : ∀ (hN : N ≠ 1) (h_sum : Eq (α := ℂ) (∑ j : ZMod N, (χ : (a : ZMod N) → ℂ) j) 0),
  completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
Tactic: classical
  -- First handle special case of Riemann zeta
  rcases eq_or_ne N 1 with rfl | hN
  · simp [completedLFunction_modOne_eq, completedRiemannZeta_one_sub, rootNumber_modOne]
  -- facts about `χ` as function
  have h_sum : ∑ j, χ j = 0 := by
    refine χ.sum_eq_zero_of_ne_one (fun h ↦ hN.symm ?_)
    rwa [IsPrimitive, h, conductor_one] at hχ
  let ε := I ^ (if χ.Even then 0 else 1)
  -- gather up powers of N
  rw [rootNumber, ← mul_comm_div, ← mul_comm_div, ← cpow_sub _ _ (NeZero.ne _), sub_sub, add_halves]
  calc completedLFunction χ (1 - s)
  _ = N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (𝓕 χ) s := by
    simp only [ε]
    split_ifs with h
    · rw [pow_zero, div_one, h, mul_one, completedLFunction,
        completedLFunction_one_sub_even h.to_fun _ (.inr h_sum) (.inr <| χ.map_zero' hN)]
    · replace h : χ.Odd := χ.even_or_odd.resolve_left h
      rw [completedLFunction, completedLFunction_one_sub_odd h.to_fun,
        pow_one, h, div_I, mul_neg_one, ← neg_mul, neg_neg]
  _ = (_) * ZMod.completedLFunction (fun j ↦ χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹ j) s := by
    congr 2 with j
    rw [hχ.fourierTransform_eq_inv_mul_gaussSum, ← neg_one_mul j, map_mul, mul_right_comm]
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) := by
    rw [completedLFunction, completedLFunction_const_mul]
    ring
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s := by
    rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
N : ℕ
inst : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
inl :
  ∀ [inst : NeZero (R := ℕ) 1] {χ : DirichletCharacter ℂ 1} (hχ : χ.IsPrimitive),
    completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑1 : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
inr inr₁ :
  ∀ (hN : N ≠ 1),
    completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
inr₂ inr₃ :
  ∀ (hN : N ≠ 1) (h_sum : Eq (α := ℂ) (∑ j : ZMod N, (χ : (a : ZMod N) → ℂ) j) 0),
    completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
⊢ completedLFunction χ (1 - s) = HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
have inr₄ : ∀ (hN : N ≠ 1) (h_sum : Eq (α := ℂ) (∑ j : ZMod N, (χ : (a : ZMod N) → ℂ) j) 0),
  completedLFunction χ (1 - s) =
    HPow.hPow (α := ℂ) (↑N : ℂ) (s - 1) / HPow.hPow (β := ℕ) I (if χ.Even then 0 else 1) * gaussSum χ stdAddChar *
      completedLFunction χ⁻¹ s
Tactic: classical
  -- First handle special case of Riemann zeta
  rcases eq_or_ne N 1 with rfl | hN
  · simp [completedLFunction_modOne_eq, completedRiemannZeta_one_sub, rootNumber_modOne]
  -- facts about `χ` as function
  have h_sum : ∑ j, χ j = 0 := by
    refine χ.sum_eq_zero_of_ne_one (fun h ↦ hN.symm ?_)
    rwa [IsPrimitive, h, conductor_one] at hχ
  let ε := I ^ (if χ.Even then 0 else 1)
  -- gather up powers of N
  rw [rootNumber, ← mul_comm_div, ← mul_comm_div, ← cpow_sub _ _ (NeZero.ne _), sub_sub, add_halves]
  calc completedLFunction χ (1 - s)
  _ = N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (𝓕 χ) s := by
    simp only [ε]
    split_ifs with h
    · rw [pow_zero, div_one, h, mul_one, completedLFunction,
        completedLFunction_one_sub_even h.to_fun _ (.inr h_sum) (.inr <| χ.map_zero' hN)]
    · replace h : χ.Odd := χ.even_or_odd.resolve_left h
      rw [completedLFunction, completedLFunction_one_sub_odd h.to_fun,
        pow_one, h, div_I, mul_neg_one, ← neg_mul, neg_neg]
  _ = (_) * ZMod.completedLFunction (fun j ↦ χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹ j) s := by
    congr 2 with j
    rw [hχ.fourierTransform_eq_inv_mul_gaussSum, ← neg_one_mul j, map_mul, mul_right_comm]
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) := by
    rw [completedLFunction, completedLFunction_const_mul]
    ring
  _ = N ^ (s - 1) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s := by
    rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
⊢ LFunctionTrivChar₁ n 1 ≠ 0
have simp : Ne (α := ℂ) (∏ p ∈ n.primeFactors, HSub.hSub (α := ℂ) 1 (↑p : ℂ)⁻¹) 0
Tactic: simp only [Function.update_self]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
⊢ Ne (α := ℂ) (∏ p ∈ n.primeFactors, HSub.hSub (α := ℂ) 1 (↑p : ℂ)⁻¹) 0
have refine : ∀ (p : ℕ) (hp : p ∈ n.primeFactors), HSub.hSub (α := ℂ) 1 (↑p : ℂ)⁻¹ ≠ 0
Tactic: refine Finset.prod_ne_zero_iff.mpr fun p hp ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
⊢ Differentiable ℂ (LFunctionTrivChar₁ n)
have rw : DifferentiableOn ℂ (LFunctionTrivChar₁ n) (Set.univ \ {1}) ∧ ContinuousAt (LFunctionTrivChar₁ n) 1
Tactic: rw [← differentiableOn_univ,
    ← differentiableOn_compl_singleton_and_continuousAt_iff (c := 1) Filter.univ_mem]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
x : ℂ
hs : Membership.mem (γ := Set ℂ) (Set.univ \ {1}) x
⊢ DifferentiableAt ℂ (F := ℂ) (fun (s : ℂ) ↦ (s - 1) * LFunctionTrivChar n s) x
have funProp : x ≠ 1 ∨ Ne (α := DirichletCharacter ℂ n) 1 1
Tactic: fun_prop (disch := simp_all)
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
⊢ DifferentiableOn ℂ (LFunctionTrivChar₁ n) (Set.univ \ {1}) ∧ ContinuousAt (LFunctionTrivChar₁ n) 1
have refine : ContinuousWithinAt (LFunctionTrivChar₁ n) {1}ᶜ 1
Tactic: refine ⟨DifferentiableOn.congr (f := fun s ↦ (s - 1) * LFunctionTrivChar n s)
    (fun _ hs ↦ DifferentiableAt.differentiableWithinAt <| by fun_prop (disch := simp_all))
    fun _ hs ↦ Function.update_of_ne (Set.mem_diff_singleton.mp hs).2 ..,
    continuousWithinAt_compl_self.mp ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
⊢ deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
have refine : ∃ s_1 ∈ 𝓝 s, Set.EqOn (LFunctionTrivChar₁ n) (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s_1
Tactic: refine eventuallyEq_iff_exists_mem.mpr ?_ |>.deriv_eq
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (hypothesis)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
Tactic: have H : deriv (LFunctionTrivChar₁ n) s =
      deriv (fun w ↦ (w - 1) * LFunctionTrivChar n w) s := by
    refine eventuallyEq_iff_exists_mem.mpr ?_ |>.deriv_eq
    exact ⟨_, isOpen_ne.mem_nhds hs, fun _ hw ↦ Function.update_of_ne (Set.mem_setOf.mp hw) ..⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have rw : deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s =
  (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
Tactic: rw [H, deriv_fun_mul (by fun_prop) (differentiableAt_LFunction _ s (.inl hs)), deriv_sub_const,
    deriv_id'', one_mul, add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
rw :
  deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have rw₁ : deriv (fun (w : ℂ) ↦ w - 1) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
  (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
Tactic: rw [H, deriv_fun_mul (by fun_prop) (differentiableAt_LFunction _ s (.inl hs)), deriv_sub_const,
    deriv_id'', one_mul, add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
rw :
  deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₁ :
  deriv (fun (w : ℂ) ↦ w - 1) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have rw₂ : deriv (fun (w : ℂ) ↦ w) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
  (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
Tactic: rw [H, deriv_fun_mul (by fun_prop) (differentiableAt_LFunction _ s (.inl hs)), deriv_sub_const,
    deriv_id'', one_mul, add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
rw :
  deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₁ :
  deriv (fun (w : ℂ) ↦ w - 1) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₂ :
  deriv (fun (w : ℂ) ↦ w) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have rw₃ : (fun (x : ℂ) ↦ (1 : ℂ)) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
  (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
Tactic: rw [H, deriv_fun_mul (by fun_prop) (differentiableAt_LFunction _ s (.inl hs)), deriv_sub_const,
    deriv_id'', one_mul, add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
rw :
  deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₁ :
  deriv (fun (w : ℂ) ↦ w - 1) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₂ :
  deriv (fun (w : ℂ) ↦ w) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₃ :
  (fun (x : ℂ) ↦ (1 : ℂ)) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have rw₄ : LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
  (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
Tactic: rw [H, deriv_fun_mul (by fun_prop) (differentiableAt_LFunction _ s (.inl hs)), deriv_sub_const,
    deriv_id'', one_mul, add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
s : ℂ
hs : s ≠ 1
H : deriv (LFunctionTrivChar₁ n) s = deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s
rw :
  deriv (fun (w : ℂ) ↦ (w - 1) * LFunctionTrivChar n w) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₁ :
  deriv (fun (w : ℂ) ↦ w - 1) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₂ :
  deriv (fun (w : ℂ) ↦ w) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₃ :
  (fun (x : ℂ) ↦ (1 : ℂ)) s * LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
rw₄ :
  LFunction (N := n) 1 s + (s - 1) * deriv (LFunction (N := n) 1) s =
    (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
⊢ deriv (LFunctionTrivChar₁ n) s = (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
have rw₅ : (s - 1) * deriv (LFunction (N := n) 1) s + LFunction (N := n) 1 s =
  (s - 1) * deriv (LFunctionTrivChar n) s + LFunctionTrivChar n s
Tactic: rw [H, deriv_fun_mul (by fun_prop) (differentiableAt_LFunction _ s (.inl hs)), deriv_sub_const,
    deriv_id'', one_mul, add_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
⊢ ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunctionTrivChar₁ n) s / LFunctionTrivChar₁ n s)
    {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
have simp_rw : ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -(deriv (LFunctionTrivChar₁ n) s / LFunctionTrivChar₁ n s))
  {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
Tactic: simp_rw [neg_div]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (hypothesis)
Goal:
n : ℕ
inst : NeZero n
⊢ ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -(deriv (LFunctionTrivChar₁ n) s / LFunctionTrivChar₁ n s))
    {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
have h : Differentiable ℂ (LFunctionTrivChar₁ n)
Tactic: have h := differentiable_LFunctionTrivChar₁ n
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
⊢ ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -(deriv (LFunctionTrivChar₁ n) s / LFunctionTrivChar₁ n s))
    {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0}
have refine : ∀ (w : ℂ) (hw : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0} w), LFunctionTrivChar₁ n w ≠ 0
Tactic: refine ((h.contDiff.continuous_deriv le_rfl).continuousOn.div
    h.continuous.continuousOn fun w hw ↦ ?_).neg
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
w : ℂ
hw : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0} w
⊢ LFunctionTrivChar₁ n w ≠ 0
have inl : ∀ (hw : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0} 1), LFunctionTrivChar₁ n 1 ≠ 0
Tactic: rcases eq_or_ne w 1 with rfl | hw'
Cleared hypotheses: w, hw
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
w : ℂ
hw : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0} w
inl : ∀ (hw : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0} 1), LFunctionTrivChar₁ n 1 ≠ 0
⊢ LFunctionTrivChar₁ n w ≠ 0
have inr : ∀ (hw' : w ≠ 1), LFunctionTrivChar₁ n w ≠ 0
Tactic: rcases eq_or_ne w 1 with rfl | hw'
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.DirichletContinuation (goal)
Goal:
n : ℕ
inst : NeZero n
h : Differentiable ℂ (LFunctionTrivChar₁ n)
w : ℂ
hw : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar n s ≠ 0} w
hw' : w ≠ 1
⊢ LFunctionTrivChar₁ n w ≠ 0
have inr : w - 1 ≠ 0 ∧ LFunctionTrivChar n w ≠ 0
Tactic: rw [LFunctionTrivChar₁, Function.update_of_ne hw', mul_ne_zero_iff]