awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ≠ 1
s : ℂ
hs : s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
refine_2 : Membership.mem (γ := Set ℂ) {s : ℂ | LFunction χ s ≠ 0} 1
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | LFunction χ s ≠ 0} s
have refine_2₁ : ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0), Membership.mem (γ := Set ℂ) {s : ℂ | LFunction χ s ≠ 0} s
Tactic: rcases hs with rfl | hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
⊢ ContinuousOn (LFunctionResidueClassAux a) {s : ℂ | 1 ≤ s.re}
have refine : ∀ (s : ℂ) (hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 ≤ s.re} s),
  Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s
Tactic: refine (continuousOn_LFunctionResidueClassAux' a).mono fun s hs ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
s : ℂ
hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 ≤ s.re} s
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s
have inl : ∀ (hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 ≤ s.re} 1),
  Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} 1
Tactic: rcases eq_or_ne s 1 with rfl | hs₁
Cleared hypotheses: s, hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
s : ℂ
hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 ≤ s.re} s
inl :
  ∀ (hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 ≤ s.re} 1),
    Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} 1
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s
have inr : ∀ (hs₁ : s ≠ 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s
Tactic: rcases eq_or_ne s 1 with rfl | hs₁
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
s : ℂ
hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 ≤ s.re} s
hs₁ : s ≠ 1
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s
have inr : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
Tactic: simp only [ne_eq, Set.mem_setOf_eq, hs₁, false_or]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : Membership.mem (γ := Set ℂ) {s : ℂ | 1 < s.re} s
⊢ LFunctionResidueClassAux a s =
    (fun (s : ℂ) ↦ L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s - (↑q.totient : ℂ)⁻¹ / (s - 1)) s
have hs₁ : 1 < s.re
Tactic: replace hs := Set.mem_setOf.mp hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
⊢ LFunctionResidueClassAux a s =
    (fun (s : ℂ) ↦ L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s - (↑q.totient : ℂ)⁻¹ / (s - 1)) s
have simp : (↑q.totient : ℂ)⁻¹ *
    (-deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s -
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s) =
  HMul.hMul (α := ℂ) (-(↑q.totient : ℂ)⁻¹)
      (∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s)) -
    (↑q.totient : ℂ)⁻¹ / (s - 1)
Tactic: simp only [LSeries_residueClass_eq ha hs, LFunctionResidueClassAux]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
⊢ (↑q.totient : ℂ)⁻¹ *
      (-deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s -
        ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s) =
    HMul.hMul (α := ℂ) (-(↑q.totient : ℂ)⁻¹)
        (∑ χ : DirichletCharacter ℂ q,
          HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s)) -
      (↑q.totient : ℂ)⁻¹ / (s - 1)
have rw : -(↑q.totient : ℂ)⁻¹ *
    (deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s) =
  -(↑q.totient : ℂ)⁻¹ *
    (∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1))
Tactic: rw [neg_div, ← neg_add', mul_neg, ← neg_mul, div_eq_mul_one_div (q.totient : ℂ)⁻¹,
    sub_eq_add_neg, ← neg_mul, ← mul_add]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
⊢ -(↑q.totient : ℂ)⁻¹ *
      (deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
        ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s) =
    -(↑q.totient : ℂ)⁻¹ *
      (∑ χ : DirichletCharacter ℂ q,
          HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
        1 / (s - 1))
have congrm : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
  ∑ χ : DirichletCharacter ℂ q,
      HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
    1 / (s - 1)
Tactic: congrm (_ * ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have ha' : IsUnit a⁻¹
Tactic: have ha' : IsUnit a⁻¹ := isUnit_of_dvd_one ⟨a, (ZMod.inv_mul_of_unit a ha).symm⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have rw : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
    ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
Tactic: rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
      ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
have simp : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ x ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction x) s / LFunction x s) =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
    ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
Tactic: simp only [mul_div_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ x ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction x) s / LFunction x s) =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
      ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
have congrm : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
Tactic: congrm (?_ + _)
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
have hs₁ : s ≠ 1
Tactic: have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
have rw : deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + LFunction (N := q) 1 s / ((s - 1) * LFunction (N := q) 1 s) =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
Tactic: rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
    Function.update_of_ne hs₁, LFunctionTrivChar, add_div,
    mul_div_mul_left _ _ (sub_ne_zero_of_ne hs₁)]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
| deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + LFunction (N := q) 1 s / ((s - 1) * LFunction (N := q) 1 s)
have enter : LFunction (N := q) 1 s
Tactic: enter [2, 1]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
| LFunction (N := q) 1 s
have rw : LFunction (N := q) 1 s * 1
Tactic: rw [← mul_one (LFunction ..)]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
| deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + LFunction (N := q) 1 s / ((s - 1) * LFunction (N := q) 1 s)
have enter : LFunction (N := q) 1 s * 1
Tactic: enter [2, 1]; rw [← mul_one (LFunction ..)]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
⊢ deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
      LFunction (N := q) 1 s / ((s - 1) * LFunction (N := q) 1 s) =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
have conv_lhs : deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
    LFunction (N := q) 1 s * 1 / ((s - 1) * LFunction (N := q) 1 s) =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
Tactic: conv_lhs => enter [2, 1]; rw [← mul_one (LFunction ..)]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
⊢ deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
      LFunction (N := q) 1 s * 1 / ((s - 1) * LFunction (N := q) 1 s) =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
have rw : deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
    1 * LFunction (N := q) 1 s / ((s - 1) * LFunction (N := q) 1 s) =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
Tactic: rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
hs₁ : s ≠ 1
rw :
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
      1 * LFunction (N := q) 1 s / ((s - 1) * LFunction (N := q) 1 s) =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
⊢ deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
      LFunction (N := q) 1 s * 1 / ((s - 1) * LFunction (N := q) 1 s) =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
have rw₁ : deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1)
Tactic: rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have classical : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
  HMul.hMul (α := ℂ) ((1 : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
      ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
    1 / (s - 1)
Tactic: classical -- for `Fintype.sum_eq_add_sum_compl`
  rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
  simp only [mul_div_assoc]
  congrm (?_ + _)
  have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
  rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
    Function.update_of_ne hs₁, LFunctionTrivChar, add_div,
    mul_div_mul_left _ _ (sub_ne_zero_of_ne hs₁)]
  conv_lhs => enter [2, 1]; rw [← mul_one (LFunction ..)]
  rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
classical :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    HMul.hMul (α := ℂ) ((1 : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have classical₁ : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
  1 * (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
      ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
    1 / (s - 1)
Tactic: classical -- for `Fintype.sum_eq_add_sum_compl`
  rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
  simp only [mul_div_assoc]
  congrm (?_ + _)
  have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
  rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
    Function.update_of_ne hs₁, LFunctionTrivChar, add_div,
    mul_div_mul_left _ _ (sub_ne_zero_of_ne hs₁)]
  conv_lhs => enter [2, 1]; rw [← mul_one (LFunction ..)]
  rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
classical :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    HMul.hMul (α := ℂ) ((1 : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
classical₁ :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    1 * (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have classical₂ : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
      ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
    1 / (s - 1)
Tactic: classical -- for `Fintype.sum_eq_add_sum_compl`
  rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
  simp only [mul_div_assoc]
  congrm (?_ + _)
  have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
  rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
    Function.update_of_ne hs₁, LFunctionTrivChar, add_div,
    mul_div_mul_left _ _ (sub_ne_zero_of_ne hs₁)]
  conv_lhs => enter [2, 1]; rw [← mul_one (LFunction ..)]
  rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
classical :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    HMul.hMul (α := ℂ) ((1 : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
classical₁ :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    1 * (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
classical₂ :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have classical₃ : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
    ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
Tactic: classical -- for `Fintype.sum_eq_add_sum_compl`
  rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
  simp only [mul_div_assoc]
  congrm (?_ + _)
  have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
  rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
    Function.update_of_ne hs₁, LFunctionTrivChar, add_div,
    mul_div_mul_left _ _ (sub_ne_zero_of_ne hs₁)]
  conv_lhs => enter [2, 1]; rw [← mul_one (LFunction ..)]
  rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
s : ℂ
hs : 1 < s.re
ha' : IsUnit a⁻¹
classical :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    HMul.hMul (α := ℂ) ((1 : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
classical₁ :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    1 * (deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s) +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
classical₂ :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s +
        ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s) +
      1 / (s - 1)
classical₃ classical₄ classical₅ :
  deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
      ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
      ∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s =
    ∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s / LFunction χ s) +
      1 / (s - 1)
have classical₆ : deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s +
    ∑ x ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction x) s / LFunction x s) =
  deriv (LFunction (N := q) 1) s / LFunction (N := q) 1 s + 1 / (s - 1) +
    ∑ i ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction i) s / LFunction i s)
Tactic: classical -- for `Fintype.sum_eq_add_sum_compl`
  rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
  simp only [mul_div_assoc]
  congrm (?_ + _)
  have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
  rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
    Function.update_of_ne hs₁, LFunctionTrivChar, add_div,
    mul_div_mul_left _ _ (sub_ne_zero_of_ne hs₁)]
  conv_lhs => enter [2, 1]; rw [← mul_one (LFunction ..)]
  rw [mul_comm _ 1, mul_div_mul_right _ _ <| LFunction_ne_zero_of_one_le_re 1 (.inr hs₁) hs.le]