mathlib-initiative/mathlib-tactics huggingface dataset:
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)
y : ℝ
hy : y < x
⊢ leftLim (↑f) y - ↑f x ≤ ↑f y - ↑f x

Tactic:
simpa using

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)
y : ℝ
hy : y < x
⊢ leftLim (↑f) y ≤ ↑f y
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
this : Tendsto (fun y ↦ y + -1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -1 * (x - x) ^ 2))
⊢ Tendsto (fun y ↦ y + -1 * (y - x) ^ 2) (𝓝 x) (𝓝 x)

Tactic:
simpa using

Goal state after:
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
this : Tendsto (fun y ↦ y + -1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -1 * (x - x) ^ 2))
⊢ Tendsto (fun y ↦ y + -(y - x) ^ 2) (𝓝 x) (𝓝 x)
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 ↦ y + -1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -1 * (x - x) ^ 2))
⊢ Tendsto (fun y ↦ y + -1 * (y - x) ^ 2) (𝓝 x) (𝓝 x)

Tactic:
simpa using

Goal state after:
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 ↦ y + -1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -1 * (x - x) ^ 2))
⊢ Tendsto (fun y ↦ y + -(y - x) ^ 2) (𝓝 x) (𝓝 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 ≪ 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 simp [(hρ A)]

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:
simpa using

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hr : r ≠ ∞
h : (r.toNNReal • ν).rnDeriv μ =ᵐ[μ] r.toNNReal • ν.rnDeriv μ
⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ

Tactic:
simpa using

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hr : r ≠ ∞
h : (r.toNNReal • ν).rnDeriv μ =ᵐ[μ] r.toNNReal • ν.rnDeriv μ
⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
f : α → E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
hf : BoundedVariationOn f univ
a x : α
hx : ContinuousWithinAt f (Ici x) x
this : variationOnFromTo f univ a = fun y ↦ variationOnFromTo f univ a x + variationOnFromTo f univ x y
⊢ ContinuousWithinAt f (univ ∩ Ici x) x

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
f : α → E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
hf : BoundedVariationOn f univ
a x : α
hx : ContinuousWithinAt f (Ici x) x
this : variationOnFromTo f univ a = fun y ↦ variationOnFromTo f univ a x + variationOnFromTo f univ x y
⊢ ContinuousWithinAt f (Ici x) x
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: case hab
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ variationOnFromTo f s c x + f x ≤ variationOnFromTo f s c y + f y

Tactic:
simpa using

Goal state after:
case hab
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ variationOnFromTo f s c x + f x ≤ variationOnFromTo f s c y + f y
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: case hab
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ variationOnFromTo f s c x - f x ≤ variationOnFromTo f s c y - f y

Tactic:
simpa using

Goal state after:
case hab
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ variationOnFromTo f s c x ≤ variationOnFromTo f s c y - f y + f 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 ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
⊢ ↑t ≠ 0

Tactic:
simpa using

Goal state after:
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
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
⊢ ¬t = 0
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case a
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
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ μ (s ∩ f ⁻¹' {∞}) ≤ 0

Tactic:
exact (measure_mono inter_subset_right).trans (v.measure_limRatioMeas_top hρ).le

Goal state after:
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: α : Type u_2
inst✝ : LinearOrder α
f g : α → ℝ
C D : ℝ≥0∞
s : Set α
hf : ∀ x ∈ s, ‖f x‖ₑ ≤ C
hg : ∀ x ∈ s, ‖g x‖ₑ ≤ D
⊢ eVariationOn (f * g) s ≤ C * eVariationOn g s + D * eVariationOn f s

Tactic:
simpa using

Goal state after:
α : Type u_2
inst✝ : LinearOrder α
f g : α → ℝ
C D : ℝ≥0∞
s : Set α
hf : ∀ x ∈ s, ‖f x‖ₑ ≤ C
hg : ∀ x ∈ s, ‖g x‖ₑ ≤ D
⊢ eVariationOn (f * g) s ≤ C * eVariationOn g s + D * eVariationOn f s
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: case h
α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : PartialOrder R
inst✝⁶ : NonUnitalSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : StarOrderedRing R
inst✝³ : TopologicalSpace R
inst✝² : ContinuousStar R
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousSqrt R
f g : C(α, R)
h : ∀ (a : α), f a ≤ g a
x : α
⊢ g x =
    (f +
        star
            ({ toFun := {x | x.1 ≤ x.2}.domRestrict ContinuousSqrt.sqrt, continuous_toFun := ⋯ }.comp
              { toFun := Set.codRestrict ⇑(f.prodMk g) {x | x.1 ≤ x.2} h, continuous_toFun := ⋯ }) *
          { toFun := {x | x.1 ≤ x.2}.domRestrict ContinuousSqrt.sqrt, continuous_toFun := ⋯ }.comp
            { toFun := Set.codRestrict ⇑(f.prodMk g) {x | x.1 ≤ x.2} h, continuous_toFun := ⋯ })
      x

Tactic:
simpa using

Goal state after:
case h
α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : PartialOrder R
inst✝⁶ : NonUnitalSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : StarOrderedRing R
inst✝³ : TopologicalSpace R
inst✝² : ContinuousStar R
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousSqrt R
f g : C(α, R)
h : ∀ (a : α), f a ≤ g a
x : α
⊢ g x = f x + ContinuousSqrt.sqrt (f x, g x) * ContinuousSqrt.sqrt (f x, g x)
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : PartialOrder R
inst✝⁶ : NonUnitalSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : StarOrderedRing R
inst✝³ : TopologicalSpace R
inst✝² : ContinuousStar R
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousSqrt R
⊢ StarOrderedRing C(α, R)

Tactic:
by
  refine StarOrderedRing.of_le_iff ?_
  intro f g
  constructor
  · rw [ContinuousMap.le_def]
    intro h
    use (mk _ ContinuousSqrt.continuousOn_sqrt.domRestrict).comp
      ⟨_, map_continuous (f.prodMk g) |>.codRestrict (s := {x | x.1 ≤ x.2}) (by exact h)⟩
    ext x
    simpa [IsSelfAdjoint.star_eq <| .of_nonneg (ContinuousSqrt.sqrt_nonneg (f x, g x) (h x))]
      using ContinuousSqrt.sqrt_mul_sqrt (f x, g x) (h x)
  · rintro ⟨p, rfl⟩
    exact fun x ↦ le_add_of_nonneg_right (star_mul_self_nonneg (p x))

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
p✝ p : C(α, R)
p_mem : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hp' : ∀ (f g : C(α, R)₀), ↑g = ↑f + p → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
f g : C(α, R)₀
s : C(α, R)
hp : ↑g = ↑f + (star s * s + p)
⊢ (star s * s + p) 0 = 0

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
p✝ p : C(α, R)
p_mem : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hp' : ∀ (f g : C(α, R)₀), ↑g = ↑f + p → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
f g : C(α, R)₀
s : C(α, R)
hp : ↑g = ↑f + (star s * s + p)
⊢ star (s 0) * s 0 + p 0 = 0
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
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})
⊢ μ (s ∩ f ⁻¹' {∞}) ≤ 0

Tactic:
exact (measure_mono inter_subset_right).trans (v.measure_limRatioMeas_top hρ).le

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

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
this : ∀ (a b : α), a ≤ b → eVariationOn f {a, b} = edist (f a) (f b)
hab : ¬a ≤ b
⊢ eVariationOn f {a, b} = edist (f a) (f b)

Tactic:
simpa using

Goal state after:
case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
this : ∀ (a b : α), a ≤ b → eVariationOn f {a, b} = edist (f a) (f b)
hab : ¬a ≤ b
⊢ eVariationOn f {b, a} = edist (f a) (f b)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
this : ∀ (a b : α), a ≤ b → eVariationOn f {a, b} = edist (f a) (f b)
hab : ¬a ≤ b
⊢ eVariationOn f {a, b} = edist (f a) (f b)

Tactic:
simpa [edist_comm, pair_comm] using this b a (le_of_not_ge hab)

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

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (μ.restrict s)
x : α
hx : Tendsto (fun a ↦ (μ.restrict s) a / μ a) (v.filterAt x) (𝓝 ((μ.restrict s).rnDeriv μ x))
h'x : (μ.restrict s).rnDeriv μ x = s.indicator 1 x
⊢ Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x))

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (μ.restrict s)
x : α
hx : Tendsto (fun a ↦ (μ.restrict s) a / μ a) (v.filterAt x) (𝓝 ((μ.restrict s).rnDeriv μ x))
h'x : (μ.restrict s).rnDeriv μ x = s.indicator 1 x
⊢ Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case succ.hs
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
u : ℕ → α
hu : Monotone u
n : ℕ
ih : eVariationOn f (u '' Iic n) = ∑ i ∈ Finset.range n, edist (f (u i)) (f (u (i + 1)))
⊢ IsGreatest (u '' Iic n) ?m.77

Tactic:
simpa using

Goal state after:
case succ.hs
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
u : ℕ → α
hu : Monotone u
n : ℕ
ih : eVariationOn f (u '' Iic n) = ∑ i ∈ Finset.range n, edist (f (u i)) (f (u (i + 1)))
⊢ (∃ x ≤ n, u x = ?m.77) ∧ ∀ a ≤ n, u a ≤ ?m.77
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_3
inst✝¹ : PseudoMetricSpace E
f : α → E
s : Set α
inst✝ : Finite ↑s
⊢ BoundedVariationOn f s

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝² : LinearOrder α
E : Type u_3
inst✝¹ : PseudoMetricSpace E
f : α → E
s : Set α
inst✝ : Finite ↑s
⊢ BoundedVariationOn f s
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
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
c : E
hc : c ∈ t
⊢ ∃ n, x ∈ A.set n

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
c : E
hc : c ∈ t
⊢ ∃ n, x ∈ A.set n
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
hf : BoundedVariationOn f s
⊢ BoundedVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s)

Tactic:
simpa using

Goal state after:
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
⊢ ¬eVariationOn f s = ∞
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
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x✝ : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x✝)
        (𝓝 ‖f x✝ - (A.set n).indicator (fun x ↦ c) x✝‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x✝, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x✝) (𝓝 ‖f x✝ - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
c : E
ct : c ∈ t
xc : ‖f x✝ - c‖ₑ < ε / 2
a : Set α
ha : (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a < ε / 2
h'a : 0 < μ a
h''a : μ a < ∞
x : α
⊢ ‖f x - f x✝‖ₑ ≤ ‖f x - c‖ₑ + ‖f x✝ - c‖ₑ

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x✝ : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x✝)
        (𝓝 ‖f x✝ - (A.set n).indicator (fun x ↦ c) x✝‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x✝, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x✝) (𝓝 ‖f x✝ - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
c : E
ct : c ∈ t
xc : ‖f x✝ - c‖ₑ < ε / 2
a : Set α
ha : (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a < ε / 2
h'a : 0 < μ a
h''a : μ a < ∞
x : α
⊢ edist (f x) (f x✝) ≤ edist (f x) c + edist (f x✝) c