-- Mathlib/NumberTheory/AbelSummation.lean
theorem _root_.sum_mul_eq_sub_sub_integral_mul (ha : 0 ≤ a) (hab : a ≤ b)
    (hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t)
    (hf_int : IntegrableOn (deriv f) (Set.Icc a b)) :
    ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f k * c k =
      f b * (∑ k ∈ Icc 0 ⌊b⌋₊, c k) - f a * (∑ k ∈ Icc 0 ⌊a⌋₊, c k) -
        ∫ t in Set.Ioc a b, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k

Proof state:
case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Tactic:
have aux6 : a ≤ ⌊b⌋₊ := Nat.floor_lt ha |>.mp hb |>.le

Result:
case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Proof state:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
⊢ 1 ≤ ⌊b⌋₊

Tactic:
omega

Result:
no goals

Proof state:
case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Tactic:
have aux3 : a ≤ ⌊a⌋₊ + 1 := (Nat.lt_floor_add_one _).le

Result:
case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Proof state:
case inr.hab
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x => deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) volume a ↑⌊b⌋₊

Tactic:
rw [intervalIntegrable_iff_integrableOn_Icc_of_le aux6]

Result:
case inr.hab
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntegrableOn (fun x => deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) (Set.Icc a ↑⌊b⌋₊) volume

Proof state:
case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ f ↑⌊b⌋₊ * ∑ x ∈ Icc 0 ⌊b⌋₊, c x - f (↑⌊a⌋₊ + 1) * ∑ x ∈ Icc 0 ⌊a⌋₊, c x -
      ((∫ (x : ℝ) in a..↑⌊b⌋₊, deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) - (f (↑⌊a⌋₊ + 1) - f a) * ∑ k ∈ Icc 0 ⌊a⌋₊, c k) =
    f b * ∑ x ∈ Icc 0 ⌊b⌋₊, c x - f a * ∑ x ∈ Icc 0 ⌊a⌋₊, c x -
      ((∫ (x : ℝ) in a..↑⌊b⌋₊, deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) + (f b - f ↑⌊b⌋₊) * ∑ k ∈ Icc 0 ⌊b⌋₊, c k)

Tactic:
ring

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Tactic:
have aux2 : b ≤ ⌊b⌋₊ + 1 := (Nat.lt_floor_add_one _).le

Result:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Proof state:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Tactic:
have aux1 : ⌊a⌋₊ ≤ a := Nat.floor_le ha

Result:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
⊢ ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, f ↑k * c k =
    f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f a * ∑ k ∈ Icc 0 ⌊a⌋₊, c k - ∫ (x : ℝ) in a..b, deriv f x * ∑ k ∈ Icc 0 ⌊x⌋₊, c k

Proof state:
case inr.hab
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntegrableOn (fun x => deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) (Set.Icc a ↑⌊b⌋₊) volume

Tactic:
exact (integrableOn_mul_sum_Icc c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux5)

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
⊢ ↑⌊a⌋₊ + 1 ≤ b

Tactic:
rwa [← Nat.cast_add_one,  ← Nat.le_floor_iff (ha.trans hab)]

Result:
no goals