mathlib-initiative/mathlib-tactics huggingface dataset:
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
⊢ ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ w n)

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)

Tactic:
by

Goal state after:
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
⊢ ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ w n) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ w n)

Tactic:
by

Goal state after:
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
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
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
this✝ :
  Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
y : ℝ
hy : y - x < 0
this : 0 < (y - x) ^ 2
⊢ y + -1 * (y - x) ^ 2 < y

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)

Tactic:
by

Goal state after:
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
⊢ ρ (toMeasurable (ρ + μ) (u m) ∩ w n) = ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

Tactic:
by

Goal state after:
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
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
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (a : α) in A, (ξ + E.indicator fun x ↦ ↑ε) a ∂ν = ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν + ∫⁻ (a : α) in A \ E, ξ a ∂ν

Tactic:
by

Goal state after:
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B :
  ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
H : (ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ∞
⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ∞

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
⊢ ∀ᵐ (x : ℝ), HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
⊢ ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x

Tactic:
by

Goal state after:
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
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
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
⊢ (ξ + E.indicator fun x ↦ ↑ε) ∈ measurableLE ν μ

Tactic:
by

Goal state after:
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B :
  ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) <
    ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

Tactic:
by

Goal state after:
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
⊢ ∃ a b,
    MeasurableSet a ∧ MeasurableSet b ∧ {x | v.limRatio ρ x < ↑p} ⊆ a ∧ {x | ↑q < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0

Tactic:
by

Goal state after:
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
⊢ AEMeasurable (v.limRatio ρ) μ

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
s : Set ℝ
hf : MonotoneOn f s
⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

Tactic:
by

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

Goal state before:
pp: f : ℝ → ℝ
s : Set ℝ
hf : MonotoneOn f s
hs : MeasurableSet s
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict s, DifferentiableWithinAt ℝ f s x

Tactic:
by

Goal state after:
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
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t)
⊢ ρ s = ρ (s ∩ t ∪ s ∩ tᶜ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a : α
⊢ variationOnFromTo f s a a = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
h : a ≤ b
⊢ 0 ≤ variationOnFromTo f s a b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
⊢ variationOnFromTo f s a b = -variationOnFromTo f s b a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
h : b ≤ a
⊢ variationOnFromTo f s a b ≤ 0

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t)
⊢ ρ (s ∩ t) + ρ (s ∩ tᶜ) ≤ (p • μ) (s ∩ t) + ρ tᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
a b : α
⊢ |variationOnFromTo f s a b| ≤ (eVariationOn f s).toReal

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t)
⊢ (p • μ) (s ∩ t) + ρ tᶜ ≤ ↑p * μ (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
h : b ≤ a
⊢ variationOnFromTo f s a b = -(eVariationOn f (s ∩ Icc b a)).toReal

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t)
⊢ ↑p * μ (s ∩ t) ≤ ↑p * μ s

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t)
⊢ ρ s ≤ ↑p * μ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
s : Set α
hs : s.Nonempty
⊢ Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f f' : α → E
s : Set α
h : ∀ ⦃x : α⦄, x ∈ s → edist (f x) (f' x) = 0
⊢ eVariationOn f s = eVariationOn f' s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f f' : α → E
s : Set α
h : EqOn f f' s
x : α
xs : x ∈ s
⊢ edist (f x) (f' x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
⊢ ρ s ≤ ↑p * μ s

Tactic:
by

Goal state after:
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
⊢ ρ s ≤ ↑p * μ s

Tactic:
by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t) by calc
    ρ s = ρ (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
    _ ≤ ρ (s ∩ t) + ρ (s ∩ tᶜ) := measure_union_le _ _
    _ ≤ (p • μ) (s ∩ t) + ρ tᶜ := by gcongr; apply inter_subset_right
    _ ≤ p * μ (s ∩ t) := by simp [(hρ A)]
    _ ≤ p * μ s := by gcongr; apply inter_subset_left
  refine v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => ?_
  have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 hx.2).2 _ (h hx.1)
  apply I.frequently.mono fun a ha => ?_
  rw [coe_nnreal_smul_apply]
  refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
  simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ variationOnFromTo f s a b - variationOnFromTo f s a c = variationOnFromTo f s c b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ variationOnFromTo f s a b - variationOnFromTo f s c b = variationOnFromTo f s a c

Tactic:
by

Goal state after:
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
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < v.limRatioMeas hρ x}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t)
⊢ (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
hmn : m ≤ n
π : ℕ → ↑(Icc m n) := projIcc m n hmn
v : ℕ → α := fun i ↦ u ↑(π i)
i : ℕ
hi : i ∈ Finset.Ico m n
⊢ edist (f (u (i + 1))) (f (u i)) = edist (f (v (i + 1))) (f (v i))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
⊢ ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
hu : MonotoneOn u (Iic n)
us : ∀ i ≤ n, u i ∈ s
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

Tactic:
by

Goal state after:
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
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < v.limRatioMeas hρ x}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t)
⊢ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) ≤ ρ (s ∩ t) + (q • μ) tᶜ

Tactic:
by

Goal state after:
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
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < v.limRatioMeas hρ x}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t)
⊢ ρ (s ∩ t) + (q • μ) tᶜ = ρ (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ StrictMonoOn u (Iic 0)

Tactic:
by

Goal state after: