awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
⊢ Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
have refine : Function.comp (β := { n : ℕ // IsPrimePow n }) (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ IsPrimePow n))
    (⇑prodNatEquiv : (a : Nat.Primes × ℕ) → { n : ℕ // IsPrimePow n }) =
  F'
Tactic: refine (Equiv.eq_comp_symm prodNatEquiv (F₀ ∘ Subtype.val) F').mpr ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
⊢ Function.comp (β := { n : ℕ // IsPrimePow n }) (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ IsPrimePow n))
      (⇑prodNatEquiv : (a : Nat.Primes × ℕ) → { n : ℕ // IsPrimePow n }) =
    F'
have h : ∀ (n : Nat.Primes × ℕ),
  Function.comp (β := { n : ℕ // IsPrimePow n }) (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ IsPrimePow n))
      (⇑prodNatEquiv : (a : Nat.Primes × ℕ) → { n : ℕ // IsPrimePow n }) n =
    F' n
Tactic: ext1 n
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
⊢ Summable (β := (↑{n : ℕ | IsPrimePow n} : Type))
    (F₀ ∘ Subtype.val (p := fun (x : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} x))
have hFF' : Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
  (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
  (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
Tactic: have hFF' :
      F₀ ∘ Subtype.val (p := fun n ↦ n ∈ {n | IsPrimePow n}) = F' ∘ ⇑prodNatEquiv.symm := by
    refine (Equiv.eq_comp_symm prodNatEquiv (F₀ ∘ Subtype.val) F').mpr ?_
    ext1 n
    simp only [Function.comp_apply, F']
    congr
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
⊢ Summable (β := (↑{n : ℕ | IsPrimePow n} : Type))
    (F₀ ∘ Subtype.val (p := fun (x : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} x))
have refine_1 : Summable (β := (↑{n : ℕ | IsPrimePow n} : Type))
  (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
Tactic: rw [hFF']
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
⊢ Summable (β := (↑{n : ℕ | IsPrimePow n} : Type))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
have refine_1 : Summable F'
Tactic: refine (Nat.Primes.prodNatEquiv.symm.summable_iff (f := F')).mpr ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
⊢ Summable F'
have hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
Tactic: have hF'₀ (p : Nat.Primes) : F' (p, 0) = 0 := by simp only [zero_add, pow_one, hF₀, F']
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
⊢ F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
have h : ∀ (x : Nat.Primes × ℕ), F'' x = (F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1) x
Tactic: ext1
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
⊢ Summable F'
have hF'₁ : F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
Tactic: have hF'₁ : F'' = F' ∘ (Prod.map _root_.id (· + 1)) := by
    ext1
    simp only [Function.comp_apply, Prod.map_fst, id_eq, Prod.map_snd, F'', F']
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
⊢ Summable F'
have refine_1 : Function.Injective (α := Nat.Primes × ℕ) (β := Nat.Primes × ℕ) (Prod.map id fun (x : ℕ) ↦ x + 1)
Tactic: refine (Function.Injective.summable_iff ?_ fun u hu ↦ ?_).mp <| hF'₁ ▸ summable_F''
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
refine_1 : Function.Injective (α := Nat.Primes × ℕ) (β := Nat.Primes × ℕ) (Prod.map id fun (x : ℕ) ↦ x + 1)
⊢ Summable F'
have refine_1₁ : ∀ (u : Nat.Primes × ℕ) (hu : u ∉ Set.range (ι := Nat.Primes × ℕ) (Prod.map id fun (x : ℕ) ↦ x + 1)), F' u = 0
Tactic: refine (Function.Injective.summable_iff ?_ fun u hu ↦ ?_).mp <| hF'₁ ▸ summable_F''
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
u : Nat.Primes × ℕ
hu : u ∉ Set.range (ι := Nat.Primes × ℕ) (Prod.map id fun (x : ℕ) ↦ x + 1)
⊢ F' u = 0
have hu₁ : u.2 = 0
Tactic: simp only [Set.range_prodMap, Set.range_id, Set.mem_prod, Set.mem_univ, Set.mem_range,
      Nat.exists_add_one_eq, true_and, not_lt, nonpos_iff_eq_zero] at hu
Cleared hypotheses: hu
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
u : Nat.Primes × ℕ
hu : u.2 = 0
⊢ F' u = 0
have refine_1 : F' u = F' (u.1, 0)
Tactic: rw [← hF'₀ u.1, ← hu]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ)
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / (↑n : ℝ) ≤ F₀ n
hF₀ : ∀ (p : Nat.Primes), F₀ (↑p : ℕ) = 0
hFF' :
  Eq (α := { n : ℕ // Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n } → ℝ)
    (F₀ ∘ Subtype.val (p := fun (n : ℕ) ↦ Membership.mem (γ := Set ℕ) {n : ℕ | IsPrimePow n} n))
    (F' ∘ (⇑prodNatEquiv.symm : (a : { n : ℕ // IsPrimePow n }) → Nat.Primes × ℕ))
hF'₀ : ∀ (p : Nat.Primes), F' (p, 0) = 0
hF'₁ : F'' = F' ∘ Prod.map id fun (x : ℕ) ↦ x + 1
u : Nat.Primes × ℕ
hu : u.2 = 0
refine_1 : F' u = F' (u.1, 0)
⊢ F' u = 0
have refine_1₁ : F' u = F' (u.1, u.2)
Tactic: rw [← hF'₀ u.1, ← hu]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
n : ℕ
⊢ (↑(residueClass a n) : ℂ) =
    HMul.hMul (α := ℂ) (↑q.totient : ℂ)⁻¹
      (∑ χ : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) (β := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) ((χ : (a : ZMod q) → ℂ) (↑n : ZMod q) : ℂ) *
          (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
have rw : (↑q.totient : ℂ) * (↑(residueClass a n) : ℂ) =
  ∑ χ : DirichletCharacter ℂ q,
    HMul.hMul (α := ℂ) (β := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) ((χ : (a : ZMod q) → ℂ) (↑n : ZMod q) : ℂ) *
      (↑((Λ : (a : ℕ) → ℝ) n) : ℂ)
Tactic: rw [eq_inv_mul_iff_mul_eq₀ <| mod_cast (Nat.totient_pos.mpr q.pos_of_neZero).ne']
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
ha : IsUnit a
⊢ (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) =
    HSMul.hSMul (α := ℂ) (β := (n : ℕ) → ℂ) (↑q.totient : ℂ)⁻¹
      (∑ χ : DirichletCharacter ℂ q,
        HSMul.hSMul (α := ℂ) (β := (n : ℕ) → ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) fun (n : ℕ) ↦
          HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) (↑n : ZMod q) : ℂ) (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
have h : ∀ (n : ℕ),
  (↑(residueClass a n) : ℂ) =
    HSMul.hSMul (α := ℂ) (β := (n : ℕ) → ℂ) (↑q.totient : ℂ)⁻¹
      (∑ χ : DirichletCharacter ℂ q,
        HSMul.hSMul (α := ℂ) (β := (n : ℕ) → ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) fun (n : ℕ) ↦
          HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) (↑n : ZMod q) : ℂ) (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
      n
Tactic: ext1 n
-- 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
⊢ L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) 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))
have simp : L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s =
  HMul.hMul (α := ℂ) (↑q.totient : ℂ)⁻¹
    (∑ x : DirichletCharacter ℂ q,
      HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
        (L
          (HMul.hMul (α := ℕ → ℂ) (fun (x_1 : ℕ) ↦ (x : (a : ZMod q) → ℂ) (↑x_1 : ZMod q)) fun (n : ℕ) ↦
            (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
          s))
Tactic: simp only [deriv_LFunction_eq_deriv_LSeries _ hs, LFunction_eq_LSeries _ hs, neg_mul, ← mul_neg,
    ← Finset.sum_neg_distrib, ← neg_div, ← LSeries_twist_vonMangoldt_eq _ 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
⊢ L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s =
    HMul.hMul (α := ℂ) (↑q.totient : ℂ)⁻¹
      (∑ x : DirichletCharacter ℂ q,
        HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
          (L
            (HMul.hMul (α := ℕ → ℂ) (fun (x_1 : ℕ) ↦ (x : (a : ZMod q) → ℂ) (↑x_1 : ZMod q)) fun (n : ℕ) ↦
              (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
            s))
have rw : (↑q.totient : ℂ) * L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s =
  ∑ x : DirichletCharacter ℂ q,
    HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
      (L
        (HMul.hMul (α := ℕ → ℂ) (fun (x_1 : ℕ) ↦ (x : (a : ZMod q) → ℂ) (↑x_1 : ZMod q)) fun (n : ℕ) ↦
          (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
        s)
Tactic: rw [eq_inv_mul_iff_mul_eq₀ <| mod_cast (Nat.totient_pos.mpr q.pos_of_neZero).ne']
-- 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) * L (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s =
    ∑ x : DirichletCharacter ℂ q,
      HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
        (L
          (HMul.hMul (α := ℕ → ℂ) (fun (x_1 : ℕ) ↦ (x : (a : ZMod q) → ℂ) (↑x_1 : ZMod q)) fun (n : ℕ) ↦
            (↑((Λ : (a : ℕ) → ℝ) n) : ℂ))
          s)
have simp_rw : L (HSMul.hSMul (α := ℂ) (β := ℕ → ℂ) (↑q.totient : ℂ) fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s =
  L
    (∑ i : DirichletCharacter ℂ q,
      HSMul.hSMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
        (HMul.hMul (α := ℕ → ℂ) (fun (n : ℕ) ↦ (i : (a : ZMod q) → ℂ) (↑n : ZMod q)) fun (n : ℕ) ↦
          (↑((Λ : (a : ℕ) → ℝ) n) : ℂ)))
    s
Tactic: simp_rw [← LSeries_smul,
    ← LSeries_sum <| fun χ _ ↦ (LSeriesSummable_twist_vonMangoldt χ hs).smul _]
-- 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
⊢ L (HSMul.hSMul (α := ℂ) (β := ℕ → ℂ) (↑q.totient : ℂ) fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) s =
    L
      (∑ i : DirichletCharacter ℂ q,
        HSMul.hSMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
          (HMul.hMul (α := ℕ → ℂ) (fun (n : ℕ) ↦ (i : (a : ZMod q) → ℂ) (↑n : ZMod q)) fun (n : ℕ) ↦
            (↑((Λ : (a : ℕ) → ℝ) n) : ℂ)))
      s
have refine : ∀ {n : ℕ},
  n ≠ 0 →
    HSMul.hSMul (α := ℂ) (β := ℕ → ℂ) (↑q.totient : ℂ) (fun (n : ℕ) ↦ (↑(residueClass a n) : ℂ)) n =
      (∑ i : DirichletCharacter ℂ q,
          HSMul.hSMul (α := ℂ) ((i : (a : ZMod q) → ℂ) a⁻¹ : ℂ)
            (HMul.hMul (α := ℕ → ℂ) (fun (n : ℕ) ↦ (i : (a : ZMod q) → ℂ) (↑n : ZMod q)) fun (n : ℕ) ↦
              (↑((Λ : (a : ℕ) → ℝ) n) : ℂ)))
        n
Tactic: refine LSeries_congr (fun {n} _ ↦ ?_) s
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
⊢ ContinuousOn (Y := ℂ)
    (fun (s : ℂ) ↦
      (↑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))
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have refine_1 : ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s)
  {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Tactic: refine continuousOn_const.mul <| ContinuousOn.add ?_ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
refine_1 :
  ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s)
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
⊢ ContinuousOn (Y := ℂ)
    (fun (s : ℂ) ↦
      (↑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))
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have refine_2 : ContinuousOn (Y := ℂ)
  (fun (s : ℂ) ↦
    -∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s)
  {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Tactic: refine continuousOn_const.mul <| ContinuousOn.add ?_ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
⊢ ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s)
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have refine_1 : ∀ (s : ℂ) (hs : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s),
  Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
Tactic: refine (continuousOn_neg_logDeriv_LFunctionTrivChar₁ q).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 : s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
have refine_1 : ∀ (hs : s = 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
Tactic: tauto
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
s : ℂ
hs : s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
refine_1 : ∀ (hs : s = 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
have refine_1₁ : ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0),
  Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
Tactic: tauto
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
s : ℂ
hs : s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
refine_1 : ∀ (hs : s = 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₁ :
  ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0),
    Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₂ : ∀ (hs : s = 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₃ :
  ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0),
    Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
have refine_1₄ : ∀ (hs : s = 1), ¬s = 1 → LFunctionTrivChar q s ≠ 0
Tactic: tauto
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
s : ℂ
hs : s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0
refine_1 : ∀ (hs : s = 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₁ :
  ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0),
    Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₂ : ∀ (hs : s = 1), Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₃ :
  ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0),
    Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
refine_1₄ : ∀ (hs : s = 1), ¬s = 1 → LFunctionTrivChar q s ≠ 0
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ LFunctionTrivChar q s ≠ 0} s
have refine_1₅ : ∀ (hs : ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0), ¬s = 1 → LFunctionTrivChar q s ≠ 0
Tactic: tauto
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
⊢ ContinuousOn (Y := ℂ)
    (fun (s : ℂ) ↦
      -∑ χ ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((χ : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (deriv (LFunction χ) s) / LFunction χ s)
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have refine_2 : ContinuousOn (Y := ℂ)
  (fun (s : ℂ) ↦
    ∑ x ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (-deriv (LFunction x) s / LFunction x s))
  {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Tactic: simp only [← Finset.sum_neg_distrib, mul_div_assoc, ← mul_neg, ← neg_div]
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (goal)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
⊢ ContinuousOn (Y := ℂ)
    (fun (s : ℂ) ↦
      ∑ x ∈ {1}ᶜ, HMul.hMul (α := ℂ) ((x : (a : ZMod q) → ℂ) a⁻¹ : ℂ) (-deriv (LFunction x) s / LFunction x s))
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have refine_2 : ∀ (χ : DirichletCharacter ℂ q) (hχ : Membership.mem (γ := Finset (DirichletCharacter ℂ q)) {1}ᶜ χ),
  ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunction χ) s / LFunction χ s)
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
Tactic: refine continuousOn_finsetSum _ fun χ hχ ↦ continuousOn_const.mul ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LSeries.PrimesInAP (hypothesis)
Goal:
q : ℕ
a : ZMod q
inst : NeZero q
χ : DirichletCharacter ℂ q
hχ : Membership.mem (γ := Finset (DirichletCharacter ℂ q)) {1}ᶜ χ
⊢ ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunction χ) s / LFunction χ s)
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have hχ₁ : χ ≠ 1
Tactic: replace hχ : χ ≠ 1 := by simpa only [ne_eq, Finset.mem_compl, Finset.mem_singleton] using hχ
Cleared hypotheses: hχ
-- 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
⊢ ContinuousOn (Y := ℂ) (fun (s : ℂ) ↦ -deriv (LFunction χ) s / LFunction χ s)
    {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
have refine_2 : ∀ (s : ℂ) (hs : Membership.mem (γ := Set ℂ) {s : ℂ | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} s),
  Membership.mem (γ := Set ℂ) {s : ℂ | LFunction χ s ≠ 0} s
Tactic: refine (continuousOn_neg_logDeriv_LFunction_of_nontriv hχ).mono fun s hs ↦ ?_
-- 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
⊢ Membership.mem (γ := Set ℂ) {s : ℂ | LFunction χ s ≠ 0} s
have refine_2 : Membership.mem (γ := Set ℂ) {s : ℂ | LFunction χ s ≠ 0} 1
Tactic: rcases hs with rfl | hs
Cleared hypotheses: s, hs