Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case succ.calc_1
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
    ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖

pp: case succ.calc_2
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖ ≤
    C₁ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖

pp: case succ.calc_3
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ C₁ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖ ≤
    C₁ + ∫ (t : ℝ) in Set.Ioc ↑m ↑n, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖

pp: case succ.calc_4
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ C₁ + ∫ (t : ℝ) in Set.Ioc ↑m ↑n, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖ ≤ C₁ + C₂

Tactic:
· exact hf _

Goal state after:
case succ.calc_2
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖ ≤
    C₁ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖

case succ.calc_3
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ C₁ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖ ≤
    C₁ + ∫ (t : ℝ) in Set.Ioc ↑m ↑n, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖

case succ.calc_4
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
m : ℕ
h_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1
hf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume
hf :
  ∀ (n : ℕ),
    ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =
      ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ - ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖
g : ℝ → ℝ
hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g
hg₂ : IntegrableAtFilter g atTop volume
C₁ : ℝ
hC₁ : ∀ (n : ℕ), ‖‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖‖ ≤ C₁
C₂ : ℝ := ∫ (t : ℝ) in Set.Ioi ↑m, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖
n : ℕ
⊢ C₁ + ∫ (t : ℝ) in Set.Ioc ↑m ↑n, ‖deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖‖ ≤ C₁ + C₂