mathlib-initiative/mathlib-tactics huggingface dataset:
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

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ (1 - x) ^ 2 + y ^ 2 ≤ (1 - x) ^ 2 + (s * x) ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ √((1 - x) ^ 2 + (s * x) ^ 2) = √(1 - 2 * x + (1 + s ^ 2) * x * x)

Tactic:
by

Goal state after:
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 ∈ 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:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ √(1 - 2 * x + (1 + s ^ 2) * x * x) ≤ √(1 - 2 * x + (1 + s ^ 2) * (1 / (1 + s ^ 2)) * x)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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 ∈ 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
x : ℝ
a✝¹ : x ∈ Icc a b → HasDerivAt f 0 x
a✝ : x ∈ Ioo d b
⊢ HasDerivAt f 0 x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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 ∈ 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
⊢ ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ √(1 - 2 * x + (1 + s ^ 2) * (1 / (1 + s ^ 2)) * x) = √(1 - x)

Tactic:
by

Goal state after:
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
⊢ ∑ i ∈ Finset.range p.1, dist (f (↑p.2 i)) (f (↑p.2 (i + 1))) < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ 0 < 1 + s ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
⊢ 0 < r / (b - d)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ √(1 - x) ≤ 1 - x / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2

Tactic:
by

Goal state after:
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
vf : ∑ i ∈ Finset.range p.1, dist (f (↑p.2 i)) (f (↑p.2 (i + 1))) < 1
⊢ ∀ a ∈ Finset.range p.1, edist (f (↑p.2 (a + 1))) (f (↑p.2 a)) ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(x ^ 2 + y ^ 2) ≤ √(x ^ 2 + (s * x) ^ 2)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
⊢ 0 ≤ g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(x ^ 2 + (s * x) ^ 2) = √((1 + s ^ 2) * x ^ 2)

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ↑k ≤ ↑(n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
z : ↑u
hz : z ∈ s
⊢ ↑z ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √((1 + s ^ 2) * x ^ 2) = √(1 + s ^ 2) * x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(1 + s ^ 2) * x = 2 * √(1 + s ^ 2) * (x / 2)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
z : ↑u
hz : z ∈ s
slope_bound : dist (slope f (↑z).1 (↑z).2) 0 ≤ r / (b - d)
⊢ ↑z ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
z : ↑u
hz : z ∈ s
slope_bound : dist (slope f (↑z).1 (↑z).2) 0 ≤ r / (b - d)
⊢ 0 < (↑z).2 - (↑z).1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ 2 * √(1 + s ^ 2) * (x / 2) < (2 * √(1 + s ^ 2) + 1) * (x / 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ (2 * √(1 + s ^ 2) + 1) * (x / 2) ≤ (2 * √(1 + s ^ 2) + 1) * (1 - √((1 - x) ^ 2 + y ^ 2))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s : ℝ
⊢ ∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
t : ℝ
ht : t ∈ Ioo (min x₀ x) (max x₀ x)
⊢ uIoo x₀ x ∈ 𝓝 t

Tactic:
by

Goal state after:
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
vf : ∑ i ∈ Finset.range p.1, dist (f (↑p.2 i)) (f (↑p.2 (i + 1))) < 1
⊢ (∑ i ∈ Finset.range p.1, edist (f (↑p.2 (i + 1))) (f (↑p.2 i))).toReal =
    ∑ i ∈ Finset.range p.1, dist (f (↑p.2 i)) (f (↑p.2 (i + 1)))

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
t : ℝ
ht : t ∈ Ioo (min x₀ x) (max x₀ x)
⊢ ↑k < ↑(n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
z : ↑u
hz : z ∈ s
⊢ g z ≤ r / (b - d) * ((↑z).2 - (↑z).1)

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
t : ℝ
ht : t ∈ Ioo (min x₀ x) (max x₀ x)
⊢ uIoo x₀ x ⊆ [[x₀, x]]

Tactic:
by

Goal state after:
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
vf : ∑ i ∈ Finset.range p.1, dist (f (↑p.2 i)) (f (↑p.2 (i + 1))) < 1
veq :
  (∑ i ∈ Finset.range p.1, edist (f (↑p.2 (i + 1))) (f (↑p.2 i))).toReal =
    ∑ i ∈ Finset.range p.1, dist (f (↑p.2 i)) (f (↑p.2 (i + 1)))
⊢ ∑ i ∈ Finset.range p.1, edist (f (↑p.2 (i + 1))) (f (↑p.2 i)) ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
z : ℂ
hzl : (1 - z).re < ε
hzr : |z.im| < s * (1 - z).re
H : √((1 - z).re ^ 2 + z.im ^ 2) < M * (1 - √((1 - (1 - z).re) ^ 2 + z.im ^ 2))
⊢ z.im ^ 2 = (1 - z).im ^ 2

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ↑(k + 1) ≤ ↑(n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ ∃ M ε, 0 < M ∧ 0 < ε ∧ {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ MeasurableSet (Ι x₀ x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k : Tendsto (fun x ↦ ∑ i ∈ range x, f i - ∑ i ∈ range x, f i * z ^ i) atTop (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))

Tactic:
← sum_sub_distrib,

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k : Tendsto (fun x ↦ ∑ x ∈ range x, (f x - f x * z ^ x)) atTop (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[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:
by

Goal state after:
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
⊢ eVariationOn f (Icc x y) ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
this : ∀ z ∈ s, g z ≤ r / (b - d) * ((↑z).2 - (↑z).1)
⊢ ∀ i ∉ s, 0 ≤ (↑i).2 - (↑i).1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
h : SameRay ℝ x y
⊢ ‖x + y‖ = ‖x‖ + ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
h : SameRay ℝ x y
⊢ ‖x - y‖ = |‖x‖ - ‖y‖|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun x ↦ (1 - z) * ∑ x_1 ∈ range x, (∑ i ∈ Ico (x_1 + 1) x, f i) * z ^ x_1) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
n j : ℕ
hj : j ∈ range n
⊢ (∑ i ∈ Ico (j + 1) n, f i) * z ^ j = (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
h : SameRay ℝ x y
⊢ ‖x‖ • y = ‖y‖ • x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
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
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
⊢ ∑ z ∈ s, g z ≤ r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x : F
hx : x ≠ 0
⊢ Set.InjOn Norm.norm {y | SameRay ℝ x y}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
y : F
hy : y ≠ 0
⊢ Set.InjOn Norm.norm {x | SameRay ℝ x y}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ SameRay ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : F
⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y

Tactic:
by

Goal state after: