awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
t : ℝ
ht : t ∈ Set.Ioc 1 T
refine_2 : 1 ≤ t
⊢ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1) ≤ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1

Tactic:
gcongr
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
t : ℝ
ht : t ∈ Set.Ioc 1 T
⊢ 1 ≤ t

Tactic:
ht.1.le

Premises used: LT.lt.le, And.left
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
t : ℝ
ht : t ∈ Set.Ioc 1 T
⊢ 1 ≤ t

Tactic:
exact ht.1.le

Premises used: LT.lt.le, And.left
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
⊢ Cs ≤ C

Tactic:
have h₄ : Cs ≤ C := by
    refine setIntegral_mono_on ?_ h₂ measurableSet_Ioc fun t ht ↦ ?_
    · exact h₁.mono_set <| Set.Ioc_subset_Ioi_self.trans Set.Ioi_subset_Ici_self
    · gcongr
      exact ht.1.le
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
rwa rwa₁ rwa₂ rwa₃ : 1 < s
⊢ 1 < (↑s : ℂ).re

Tactic:
rwa [ofReal_re]

Premises used: Complex.ofReal_re
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
rw :
  IsBigO (E := ℂ) (F := ℝ) atTop (fun (n : ℕ) ↦ ∑ k ∈ Icc 1 n, f k) fun (n : ℕ) ↦ HPow.hPow (α := ℝ) (β := ℝ) (↑n : ℝ) 1
⊢ ‖((↑s : ℂ) - 1) * LSeries f (↑s : ℂ) - (↑s : ℂ) * l‖ =
    ‖(((↑s : ℂ) - 1) * (↑s : ℂ) *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
            S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1)) -
        l * (↑s : ℂ) *
          (((↑s : ℂ) - 1) *
            @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
              (↑t : ℂ) ^ (-(↑s : ℂ)))‖

Tactic:
rw [h₃, mul_one, mul_comm l, LSeries_eq_mul_integral _ zero_le_one (by rwa [ofReal_re])
        (hfS _ hs), neg_add', mul_assoc]

Premises used: mul_assoc, zero_le_one, neg_add', LSeries_eq_mul_integral, mul_comm, mul_one
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑(HPow.hPow (α := ℝ) (β := ℝ) (↑n : ℝ) 1) : ℂ)) atTop
    (𝓝 l)

Tactic:
simpa using hlim
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ IsBigO (E := ℂ) (F := ℝ) atTop (fun (n : ℕ) ↦ ∑ k ∈ Icc 1 n, f k) fun (n : ℕ) ↦ HPow.hPow (α := ℝ) (β := ℝ) (↑n : ℝ) 1

Tactic:
exact isBigO_atTop_natCast_rpow_of_tendsto_div_rpow (a := l) (by simpa using hlim)

Premises used: Asymptotics.isBigO_atTop_natCast_rpow_of_tendsto_div_rpow
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
rw :
  ‖(((↑s : ℂ) - 1) * (↑s : ℂ) *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
            S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1)) -
        l * (↑s : ℂ) *
          (((↑s : ℂ) - 1) *
            @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
              (↑t : ℂ) ^ (-(↑s : ℂ)))‖ =
    ‖((↑s : ℂ) - 1) * (↑s : ℂ) *
        ((@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
            S a * (↑a : ℂ) ^ (-(↑s : ℂ) - 1)) -
          l *
            @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
              (↑a : ℂ) ^ (-(↑s : ℂ)))‖
hf :
  @Integrable ℂ _ _ _ MeasureSpace.toMeasurableSpace (fun (t : ℝ) ↦ S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1))
    (Measure.restrict volume (Set.Ioi 1))
hg :
  @Integrable ℂ _ _ _ MeasureSpace.toMeasurableSpace (fun (t : ℝ) ↦ l * (↑t : ℂ) ^ (-(↑s : ℂ)))
    (Measure.restrict volume (Set.Ioi 1))
⊢ ‖(((↑s : ℂ) - 1) * (↑s : ℂ) *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
            S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1)) -
        l * (↑s : ℂ) *
          (((↑s : ℂ) - 1) *
            @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
              (↑t : ℂ) ^ (-(↑s : ℂ)))‖ =
    ‖((↑s : ℂ) - 1) * (↑s : ℂ) *
        @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
          S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1) - l * (↑t : ℂ) ^ (-(↑s : ℂ))‖

Tactic:
rw [integral_sub, integral_const_mul]

Premises used: MeasureTheory.integral_const_mul, MeasureTheory.integral_sub
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
e_a :
  (((↑s : ℂ) - 1) * (↑s : ℂ) *
        @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
          S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1)) -
      l * (↑s : ℂ) *
        (((↑s : ℂ) - 1) *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
            (↑t : ℂ) ^ (-(↑s : ℂ))) =
    ((↑s : ℂ) - 1) * (↑s : ℂ) *
      ((@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
          S a * (↑a : ℂ) ^ (-(↑s : ℂ) - 1)) -
        l *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
            (↑a : ℂ) ^ (-(↑s : ℂ)))
⊢ ‖(((↑s : ℂ) - 1) * (↑s : ℂ) *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
            S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1)) -
        l * (↑s : ℂ) *
          (((↑s : ℂ) - 1) *
            @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
              (↑t : ℂ) ^ (-(↑s : ℂ)))‖ =
    ‖((↑s : ℂ) - 1) * (↑s : ℂ) *
        ((@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
            S a * (↑a : ℂ) ^ (-(↑s : ℂ) - 1)) -
          l *
            @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
              (↑a : ℂ) ^ (-(↑s : ℂ)))‖

Tactic:
congr
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ (((↑s : ℂ) - 1) * (↑s : ℂ) *
        @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
          S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1)) -
      l * (↑s : ℂ) *
        (((↑s : ℂ) - 1) *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
            (↑t : ℂ) ^ (-(↑s : ℂ))) =
    ((↑s : ℂ) - 1) * (↑s : ℂ) *
      ((@integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
          S a * (↑a : ℂ) ^ (-(↑s : ℂ) - 1)) -
        l *
          @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (a : ℝ) ↦
            (↑a : ℂ) ^ (-(↑s : ℂ)))

Tactic:
ring
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ @Integrable ℂ _ _ _ MeasureSpace.toMeasurableSpace (fun (t : ℝ) ↦ S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1))
    (Measure.restrict volume (Set.Ioi 1))

Tactic:
(lemma₁ hlim hs).mono_set Set.Ioi_subset_Ici_self

Premises used: Set.Ioi_subset_Ici_self, MeasureTheory.IntegrableOn, _private.0.lemma₁, MeasureTheory.IntegrableOn.mono_set
-- Module: Mathlib.NumberTheory.LSeries.SumCoeff
Goal:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun (n : ℕ) ↦ HDiv.hDiv (α := ℂ) (∑ k ∈ Icc 1 n, f k) (↑n : ℂ)) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f (↑s : ℂ)
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * (↑y : ℂ)‖ < ε * y
S : ℝ → ℂ := fun (t : ℝ) ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (ε := ℂ) (fun (t : ℝ) ↦ S t - l * (↑t : ℂ)) (Set.Ici 1) volume
h₁ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace) (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1))
    (Set.Ici 1) volume
h₂ :
  IntegrableOn (ε := ℝ) (mα := MeasureSpace.toMeasurableSpace)
    (fun (t : ℝ) ↦ ‖S t - l * (↑t : ℂ)‖ * t ^ HSub.hSub (α := ℝ) (-1) 1) (Set.Ioc 1 T) volume
h₃ :
  (((↑s : ℂ) - 1) *
      @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioi 1)) fun (t : ℝ) ↦
        (↑t : ℂ) ^ (-(↑s : ℂ))) =
    1
Cs : ℝ :=
  @integral _ _ _ _ MeasureSpace.toMeasurableSpace (Measure.restrict volume (Set.Ioc 1 T)) fun (t : ℝ) ↦
    ‖S t - l * (↑t : ℂ)‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ @Integrable ℂ _ _ _ MeasureSpace.toMeasurableSpace (fun (t : ℝ) ↦ S t * (↑t : ℂ) ^ (-(↑s : ℂ) - 1))
    (Measure.restrict volume (Set.Ioi 1))

Tactic:
exact (lemma₁ hlim hs).mono_set Set.Ioi_subset_Ici_self

Premises used: Set.Ioi_subset_Ici_self, _private.0.lemma₁, MeasureTheory.IntegrableOn.mono_set