mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_2.x
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f [[x₀, x]] x₀ =
    ((x - x₀) ^ (n + 1) / ↑(n.succ * n !)) • iteratedDerivWithin (n + 1) f [[x₀, x]] x₀

pp: case e'_2.x
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t =
    -∫ (t : ℝ) in x₀..x,
        (deriv (fun x_1 ↦ (x - x_1) ^ (n + 1)) t / ↑(n.succ * n !)) • iteratedDerivWithin (n + 1) f [[x₀, x]] t

Tactic:
·

Goal state after:
case e'_2.x
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f [[x₀, x]] x₀ =
    ((x - x₀) ^ (n + 1) / ↑(n.succ * n !)) • iteratedDerivWithin (n + 1) f [[x₀, x]] x₀
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_2.x
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t =
    -∫ (t : ℝ) in x₀..x,
        (deriv (fun x_1 ↦ (x - x_1) ^ (n + 1)) t / ↑(n.succ * n !)) • iteratedDerivWithin (n + 1) f [[x₀, x]] t

Tactic:
·

Goal state after:
case e'_2.x
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t =
    -∫ (t : ℝ) in x₀..x,
        (deriv (fun x_1 ↦ (x - x_1) ^ (n + 1)) t / ↑(n.succ * n !)) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case a.right
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝³ : a ≤ x
x✝² : x ≤ y
x✝¹ : y < x + δ
x✝ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
⊢ (↑(Finset.range (p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).1)).PairwiseDisjoint fun i ↦
    uIoc ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).1 ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).2

Tactic:
·

Goal state after:
case a.right
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝³ : a ≤ x
x✝² : x ≤ y
x✝¹ : y < x + δ
x✝ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
⊢ (↑(Finset.range (p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).1)).PairwiseDisjoint fun i ↦
    uIoc ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).1 ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).2
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝⁴ : a ≤ x
x✝³ : x ≤ y
x✝² : y < x + δ
x✝¹ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
x✝ : ℕ
⊢ x✝ ≤ x✝ + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_3
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ∫ (t : ℝ) in x₀..x, ((x - t) ^ (n + 1) / ((↑n + 1) * ↑n !)) • iteratedDerivWithin (n + 1 + 1) f [[x₀, x]] t =
    ∫ (t : ℝ) in x₀..x,
      (fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!) t • deriv (fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t) t

Tactic:
·

Goal state after:
case e'_3
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
⊢ ∫ (t : ℝ) in x₀..x, ((x - t) ^ (n + 1) / ((↑n + 1) * ↑n !)) • iteratedDerivWithin (n + 1 + 1) f [[x₀, x]] t =
    ∫ (t : ℝ) in x₀..x,
      (fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!) t • deriv (fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t) t
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case a
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝³ : a ≤ x
x✝² : x ≤ y
x✝¹ : y < x + δ
x✝ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
⊢ ∑ i ∈ Finset.range (p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).1,
      dist ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).1 ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).2 <
    δ

Tactic:
·

Goal state after:
case a
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝³ : a ≤ x
x✝² : x ≤ y
x✝¹ : y < x + δ
x✝ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
⊢ ∑ i ∈ Finset.range (p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).1,
      dist ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).1 ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 i).2 <
    δ
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_3.e_a.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
x✝¹ : ℝ
x✝ : x✝¹ ∈ uIoo x₀ x
h1 : min x₀ x < x✝¹
h2 : x✝¹ < max x₀ x
⊢ (↑n + 1) * ↑n ! = ↑(n + 1)!

pp: case e'_3.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
x✝¹ : ℝ
x✝ : x✝¹ ∈ uIoo x₀ x
h1 : min x₀ x < x✝¹
h2 : x✝¹ < max x₀ x
⊢ derivWithin (iteratedDerivWithin (n + 1) f [[x₀, x]]) [[x₀, x]] x✝¹ =
    deriv (fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t) x✝¹

Tactic:
·

Goal state after:
case e'_3.e_a.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
x✝¹ : ℝ
x✝ : x✝¹ ∈ uIoo x₀ x
h1 : min x₀ x < x✝¹
h2 : x✝¹ < max x₀ x
⊢ (↑n + 1) * ↑n ! = ↑(n + 1)!
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_3.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
x✝¹ : ℝ
x✝ : x✝¹ ∈ uIoo x₀ x
h1 : min x₀ x < x✝¹
h2 : x✝¹ < max x₀ x
⊢ derivWithin (iteratedDerivWithin (n + 1) f [[x₀, x]]) [[x₀, x]] x✝¹ =
    deriv (fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t) x✝¹

Tactic:
·

Goal state after:
case e'_3.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
x✝¹ : ℝ
x✝ : x✝¹ ∈ uIoo x₀ x
h1 : min x₀ x < x✝¹
h2 : x✝¹ < max x₀ x
⊢ derivWithin (iteratedDerivWithin (n + 1) f [[x₀, x]]) [[x₀, x]] x✝¹ =
    deriv (fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t) x✝¹
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: case inr
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hf₀ : ∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt f 0 x
this :
  ∀ {F : Type u_2} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {f : ℝ → F} {a b : ℝ},
    AbsolutelyContinuousOnInterval f a b →
      (∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt f 0 x) → a ≤ b → ∃ C, ∀ x ∈ uIcc a b, f x = C
hab : ¬a ≤ b
⊢ ∃ C, ∀ x ∈ uIcc a b, f x = C

pp: F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace ℝ F✝
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hf₀ : ∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt f 0 x
hab : a ≤ b
⊢ ∃ C, ∀ x ∈ uIcc a b, f x = C

Tactic:
·

Goal state after:
case inr
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hf₀ : ∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt f 0 x
this :
  ∀ {F : Type u_2} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {f : ℝ → F} {a b : ℝ},
    AbsolutelyContinuousOnInterval f a b →
      (∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt f 0 x) → a ≤ b → ∃ C, ∀ x ∈ uIcc a b, f x = C
hab : ¬a ≤ b
⊢ ∃ C, ∀ x ∈ uIcc a b, f x = C
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inl
f : ℝ → ℝ
x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x₀]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x₀]]) x₀ x₀
⊢ f x₀ - taylorWithinEval f n [[x₀, x₀]] x₀ x₀ =
    ∫ (t : ℝ) in x₀..x₀, (x₀ - t) ^ n / ↑n ! * iteratedDerivWithin (n + 1) f [[x₀, x₀]] t

pp: case inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
⊢ f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, (x - t) ^ n / ↑n ! * iteratedDerivWithin (n + 1) f [[x₀, x]] t

Tactic:
·

Goal state after:
case inl
f : ℝ → ℝ
x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x₀]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x₀]]) x₀ x₀
⊢ f x₀ - taylorWithinEval f n [[x₀, x₀]] x₀ x₀ =
    ∫ (t : ℝ) in x₀..x₀, (x₀ - t) ^ n / ↑n ! * iteratedDerivWithin (n + 1) f [[x₀, x₀]] t
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hf
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ (x - t) ^ k / ↑k !) x₀ x

pp: case inr.hg
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x

Tactic:
·

Goal state after:
case inr.hf
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ (x - t) ^ k / ↑k !) x₀ x
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hg
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x

Tactic:
·

Goal state after:
case inr.hg
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hg.inl
f : ℝ → ℝ
x x₀ : ℝ
this : x₀ ≠ x
k : ℕ
hf₁ : ContDiffOn ℝ (↑k) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin k f [[x₀, x]]) x₀ x
hk : k ≤ k
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x

pp: case inr.hg.inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk✝ : k ≤ n
hk : k < n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x

Tactic:
·

Goal state after:
case inr.hg.inl
f : ℝ → ℝ
x x₀ : ℝ
this : x₀ ≠ x
k : ℕ
hf₁ : ContDiffOn ℝ (↑k) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin k f [[x₀, x]]) x₀ x
hk : k ≤ k
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: case inl
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace ℝ F✝
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a d r : ℝ
hr : r > 0
had : a ≤ d
hf : AbsolutelyContinuousOnInterval f a d
hf₀ : ∀ᵐ (x : ℝ), x ∈ Icc a d → HasDerivAt f 0 x
hab : a ≤ d
hd : a ≤ d ∧ d ≤ d
⊢ dist (f d) (f d) ≤ r

pp: case inr
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace ℝ F✝
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Icc a b → HasDerivAt f 0 x
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
⊢ dist (f d) (f b) ≤ r

Tactic:
·

Goal state after:
case inl
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace ℝ F✝
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a d r : ℝ
hr : r > 0
had : a ≤ d
hf : AbsolutelyContinuousOnInterval f a d
hf₀ : ∀ᵐ (x : ℝ), x ∈ Icc a d → HasDerivAt f 0 x
hab : a ≤ d
hd : a ≤ d ∧ d ≤ d
⊢ dist (f d) (f d) ≤ r
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝³ : a ≤ x
x✝² : x ≤ y
x✝¹ : y < x + δ
x✝ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
this : ↑p.2 p.1 - ↑p.2 0 < δ
i : ℕ
⊢ i ≤ i + 1

Tactic:
by lia

Goal state after: