mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
n : ℕ
hn : 1 / (↑n + 1) < ↑c * (↑νA)⁻¹
hb₁ : 0 < (↑νA)⁻¹
⊢ ↑(1 / (↑n + 1)) < ↑c * (↑νA)⁻¹

pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
n : ℕ
hn : 1 / (↑n + 1) < ↑c * (↑νA)⁻¹
hb₁ : 0 < (↑νA)⁻¹
⊢ νA ≠ 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
n : ℕ
hn : 1 / (↑n + 1) < ↑c * (↑νA)⁻¹
hb₁ : 0 < (↑νA)⁻¹
⊢ ↑(1 / (↑n + 1)) < ↑c * (↑νA)⁻¹

α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
n : ℕ
hn : 1 / (↑n + 1) < ↑c * (↑νA)⁻¹
hb₁ : 0 < (↑νA)⁻¹
⊢ νA ≠ 0
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ (vitaliFamily volume 1).setsAt x

Tactic:
]

Goal state after:
x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ (vitaliFamily volume 1).setsAt x
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x y : ℝ
hxy : x < y
⊢ (x + y) / 2 - x ≤ 1 * ((y - x) / 2)

pp: x y : ℝ
hxy : x < y
⊢ 0 ≤ (x + y) / 2 - x

Tactic:
]

Goal state after:
x y : ℝ
hxy : x < y
⊢ (x + y) / 2 - x ≤ 1 * ((y - x) / 2)

x y : ℝ
hxy : x < y
⊢ 0 ≤ (x + y) / 2 - x
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < ↑w
w_lt : ↑w < z
n : ℕ
hn : u n < w
a : Set α
ha : ρ a < ↑(u n) * μ a
μa_pos : 0 < μ a
μa_lt_top : μ a < ∞
⊢ z * μ a ≤ z * μ a

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc x (x + ε)
⊢ Icc x y ⊆ Icc (x - ε) (x + ε)

Tactic:
]

Goal state after:
x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc x (x + ε)
⊢ Icc x y ⊆ Icc (x - ε) (x + ε)
Module: Mathlib.Order.Monotone.Extension

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
x : α
hx : x ∈ s
this : IsGreatest (Iic x ∩ s) x
⊢ f x = f x

Tactic:
]

Goal state after:
Module: Mathlib.Order.Monotone.Extension

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
x : α
hx : x ∈ s
this : IsGreatest (Iic x ∩ s) x
⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))

Tactic:
rw [if_neg this.nonempty.not_disjoint,
        ((h.mono inter_subset_right).map_isGreatest this).csSup_eq]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
n : ℕ
hn : 1 / (↑n + 1) < ↑c * (↑νA)⁻¹
hb₁ : 0 < (↑νA)⁻¹
h' : 1 / (↑n + 1) * νA < c
⊢ ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA

Tactic:
]

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
n : ℕ
hn : 1 / (↑n + 1) < ↑c * (↑νA)⁻¹
hb₁ : 0 < (↑νA)⁻¹
h' : 1 / (↑n + 1) * νA < c
⊢ ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ (vitaliFamily volume 1).setsAt y

Tactic:
]

Goal state after:
x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ (vitaliFamily volume 1).setsAt y
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x y : ℝ
hxy : x < y
⊢ y - (x + y) / 2 ≤ 1 * ((y - x) / 2)

pp: x y : ℝ
hxy : x < y
⊢ 0 ≤ y - (x + y) / 2

Tactic:
]

Goal state after:
x y : ℝ
hxy : x < y
⊢ y - (x + y) / 2 ≤ 1 * ((y - x) / 2)

x y : ℝ
hxy : x < y
⊢ 0 ≤ y - (x + y) / 2
Module: Mathlib.Order.Monotone.Extension

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : (Iic x ∩ s).Nonempty
hy : ¬Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))

Tactic:
]

Goal state after:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : (Iic x ∩ s).Nonempty
hy : ¬Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : νA = 0
⊢ μA = 0

Tactic:
]

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : νA = 0
⊢ μA = 0
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc (x - ε) x
⊢ Icc y x ⊆ Icc (x - ε) (x + ε)

Tactic:
]

Goal state after:
x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc (x - ε) x
⊢ Icc y x ⊆ Icc (x - ε) (x + ε)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬∃ s, MeasurableSet s ∧ μ s = 0 ∧ ν sᶜ = 0
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
hA₃ : ∀ (n : ℕ), μ A ≤ ↑(1 / (↑n + 1)) * ν A
hμ : μ A = 0
⊢ ∃ ε, 0 < ε ∧ ∃ E, MeasurableSet E ∧ 0 < ν E ∧ ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ μ (A ∩ E)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬∃ s, MeasurableSet s ∧ μ s = 0 ∧ ν sᶜ = 0
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
hA₃ : ∀ (n : ℕ), μ A ≤ ↑(1 / (↑n + 1)) * ν A
hμ : μ A = 0
⊢ ∃ ε, 0 < ε ∧ ∃ E, MeasurableSet E ∧ 0 < ν E ∧ ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ μ (A ∩ E)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
x a c d : ℝ
l : Filter ℝ
hl : l ≤ 𝓝[≠] x
hf : Tendsto (fun y ↦ (f y - d) / (y - x)) l (𝓝 a)
h' : Tendsto (fun y ↦ y + c * (y - x) ^ 2) l l
L : Tendsto (fun y ↦ (y + c * (y - x) ^ 2 - x) / (y - x)) l (𝓝 1)
Z :
  Tendsto
    (fun x_1 ↦
      ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦ y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))
    l (𝓝 a)
⊢ Tendsto (fun y ↦ (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a)

Tactic:
]

Goal state after:
f : ℝ → ℝ
x a c d : ℝ
l : Filter ℝ
hl : l ≤ 𝓝[≠] x
hf : Tendsto (fun y ↦ (f y - d) / (y - x)) l (𝓝 a)
h' : Tendsto (fun y ↦ y + c * (y - x) ^ 2) l l
L : Tendsto (fun y ↦ (y + c * (y - x) ^ 2 - x) / (y - x)) l (𝓝 1)
Z :
  Tendsto
    (fun x_1 ↦
      ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦ y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))
    l (𝓝 a)
⊢ Tendsto (fun y ↦ (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})
⊢ ∫⁻ (x : α) in A ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in A ∩ {x | g x < f x}, f x ∂μ ≤ ν A

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})
⊢ ∫⁻ (x : α) in A ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in A ∩ {x | g x < f x}, f x ∂μ ≤ ν A
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
y : ℝ
hxy : x < y
⊢ (↑f y - ↑f x) / (y - x) = (↑f y - ↑f x) / (y - x)

pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
y : ℝ
hxy : x < y
⊢ 0 ≤ (↑f y - ↑f x) / (y - x)

Tactic:
]

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
y : ℝ
hxy : x < y
⊢ 0 ≤ (↑f y - ↑f x) / (y - x)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Sort u_2
f : ℕ → α → ℝ≥0∞
m : ℕ
a : α
c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
d : ℝ≥0∞ := max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
hd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ∧
    max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a

Tactic:
]

Goal state after:
α : Sort u_2
f : ℕ → α → ℝ≥0∞
m : ℕ
a : α
c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
d : ℝ≥0∞ := max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
hd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ∧
    max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hxy : y < x
⊢ (↑f x - leftLim (↑f) y) / (x - y) = (↑f x - leftLim (↑f) y) / (x - y)

pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hxy : y < x
⊢ 0 ≤ (↑f x - leftLim (↑f) y) / (x - y)

Tactic:
]

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hxy : y < x
⊢ 0 ≤ (↑f x - leftLim (↑f) y) / (x - y)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case succ
α : Type u_1
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
m : ℕ
hm : (fun x ↦ ⨆ k, ⨆ (_ : k ≤ m), f k x) ∈ measurableLE μ ν
this : (fun a ↦ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a) = fun a ↦ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (a : α) in A, max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ∂μ ≤ ν A

Tactic:
]

Goal state after:
case succ
α : Type u_1
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
m : ℕ
hm : (fun x ↦ ⨆ k, ⨆ (_ : k ≤ m), f k x) ∈ measurableLE μ ν
this : (fun a ↦ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a) = fun a ↦ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (a : α) in A, max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ∂μ ≤ ν A
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hy : x - 1 < y
h'y : y < x
⊢ y + 1 * (y - x) ^ 2 < x

Tactic:
]

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hy : x - 1 < y
h'y : y < x
⊢ y + 1 * (y - x) ^ 2 < x
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case refine_4.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
this :
  Filter.Tendsto (fun n ↦ ∫⁻ (x : α), (⨆ k, ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop
    (nhds (∫⁻ (x : α), (⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k) x ∂ν))
n : ℕ
⊢ (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) ≤ ∫⁻ (x : α), (⨆ k, ⨆ (_ : k ≤ n), f k) x ∂ν

Tactic:
]

Goal state after:
case refine_4.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
this :
  Filter.Tendsto (fun n ↦ ∫⁻ (x : α), (⨆ k, ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop
    (nhds (∫⁻ (x : α), (⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k) x ∂ν))
n : ℕ
⊢ (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) ≤ ∫⁻ (x : α), (⨆ k, ⨆ (_ : k ≤ n), f k) x ∂ν
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n ↦ s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n ↦ s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
A :
  (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
this : μ sᶜ = 0
⊢ μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) =
    μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

Tactic:
]

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

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y ↦ (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
L4 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b ↦ (↑f b - ↑f x) / (b - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑f b - ↑f x) / (b - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)

Tactic:
]

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y ↦ (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
L4 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b ↦ (↑f b - ↑f x) / (b - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑f b - ↑f x) / (b - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : leftLim f x = rightLim f x
⊢ ↑hf.stieltjesFunction x = f x

Tactic:
]

Goal state after:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : leftLim f x = rightLim f x
⊢ ↑hf.stieltjesFunction x = f x
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : leftLim f x = rightLim f x
⊢ leftLim f x ≤ f x

Tactic:
]

Goal state after:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : leftLim f x = rightLim f x
⊢ leftLim f x ≤ f x