mathlib-initiative/mathlib-tactics huggingface dataset:
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 ≠ 0
⊢ (↑r • μ).withDensity (ν.rnDeriv (↑r • μ)) = ↑r⁻¹ • (↑r • μ).withDensity (ν.rnDeriv μ)

Tactic:
]

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 ≠ 0
⊢ (↑r • μ).withDensity (ν.rnDeriv (↑r • μ)) = ↑r⁻¹ • (↑r • μ).withDensity (ν.rnDeriv μ)
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 ≠ 0
this : (r • μ).withDensity (ν.rnDeriv (r • μ)) = r⁻¹ • (r • μ).withDensity (ν.rnDeriv μ)
⊢ (↑r • μ).withDensity (ν.rnDeriv (↑r • μ)) = ↑r⁻¹ • (↑r • μ).withDensity (ν.rnDeriv μ)

Tactic:
]

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 ≠ 0
this : (r • μ).withDensity (ν.rnDeriv (r • μ)) = r⁻¹ • (r • μ).withDensity (ν.rnDeriv μ)
⊢ (↑r • μ).withDensity (ν.rnDeriv (↑r • μ)) = ↑r⁻¹ • (↑r • μ).withDensity (ν.rnDeriv μ)
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
s : Set α
E : Type u_3
inst✝² : PseudoMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
l : E
a b : α
ha : a ∈ s
hb : b ∈ s
hf : LocallyBoundedVariationOn f s
h'f : Tendsto f (𝓝[s ∩ Ioi b] b) (𝓝 l)
H :
  Tendsto (fun x ↦ variationOnFromTo f s a b + variationOnFromTo f s b x) (𝓝[s ∩ Ioi b] b)
    (𝓝 (variationOnFromTo f s a b + dist (f b) l))
x : α
hx : x ∈ s ∩ Ioi b
⊢ variationOnFromTo f s a x = variationOnFromTo f s a x

Tactic:
]

Goal state after:
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 ≠ 0
⊢ ν = ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)

Tactic:
]

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 ≠ 0
⊢ ν = ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
s : Set α
f : α →ᵤ[{s}] E
this : ∀ s_1 ∈ {s}, TendstoUniformlyOn (⇑(UniformOnFun.toFun {s}) ∘ id) ((UniformOnFun.toFun {s}) f) (𝓝 f) s_1
⊢ ∀ x ∈ s, Tendsto (fun i ↦ id i x) (𝓝 f) (𝓝 (f x))

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
s : Set α
f : α →ᵤ[{s}] E
this : ∀ s_1 ∈ {s}, TendstoUniformlyOn (⇑(UniformOnFun.toFun {s}) ∘ id) ((UniformOnFun.toFun {s}) f) (𝓝 f) s_1
⊢ ∀ x ∈ s, Tendsto (fun i ↦ id i x) (𝓝 f) (𝓝 (f x))
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 ≠ 0
hr_ne_top : r ≠ ∞
⊢ ¬(r = 0 ∨ r = ∞)

Tactic:
]

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 ≠ 0
hr_ne_top : r ≠ ∞
⊢ ¬(r = 0 ∨ r = ∞)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Set α
h : BoundedVariationOn f s
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Set α
h : BoundedVariationOn f s
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : BoundedVariationOn f s
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ dist (f x) (f y) ≤ (eVariationOn f s).toReal

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : BoundedVariationOn f s
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ dist (f x) (f y) ≤ (eVariationOn f s).toReal
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 ≠ 0
hr_ne_top : r ≠ ∞
h : ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ
⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ

Tactic:
]

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 ≠ 0
hr_ne_top : r ≠ ∞
h : ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ
⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ μ.withDensity ((ν₁ + ν₂).rnDeriv μ) = μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)

pp: case hf
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable ((ν₁ + ν₂).rnDeriv μ) μ

pp: case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ μ.withDensity ((ν₁ + ν₂).rnDeriv μ) = μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)

case hf
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable ((ν₁ + ν₂).rnDeriv μ) μ

case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
this : μ.withDensity ((ν₁ + ν₂).rnDeriv μ) = μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)
⊢ μ.withDensity ((ν₁ + ν₂).rnDeriv μ) = μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
this : μ.withDensity ((ν₁ + ν₂).rnDeriv μ) = μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)
⊢ μ.withDensity ((ν₁ + ν₂).rnDeriv μ) = μ.withDensity (ν₁.rnDeriv μ + ν₂.rnDeriv μ)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ ν₁ + ν₂ = ν₁ + ν₂

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
s : Set α
E : Type u_3
inst✝² : PseudoMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
l : E
a b : α
ha : a ∈ s
hb : b ∈ s
hf : LocallyBoundedVariationOn f s
h'f : Tendsto f (𝓝[s ∩ Ioi b] b) (𝓝 l)
⊢ Tendsto (fun x ↦ (eVariationOn f (s ∩ Icc b x)).toReal) (𝓝[s ∩ Ioi b] b) (𝓝 (edist (f b) l).toReal)

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
s : Set α
E : Type u_3
inst✝² : PseudoMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
l : E
a b : α
ha : a ∈ s
hb : b ∈ s
hf : LocallyBoundedVariationOn f s
h'f : Tendsto f (𝓝[s ∩ Ioi b] b) (𝓝 l)
⊢ Tendsto (fun x ↦ (eVariationOn f (s ∩ Icc b x)).toReal) (𝓝[s ∩ Ioi b] b) (𝓝 (edist (f b) l).toReal)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
⊢ 0 + (ν.restrict s - ξ.restrict s) = 0 + ν.restrict s - ξ.restrict s

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
⊢ 0 + (ν.restrict s - ξ.restrict s) = 0 + ν.restrict s - ξ.restrict s
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
⊢ μ.restrict sᶜ + (ν.restrict sᶜ - 0) = μ.restrict sᶜ + ν.restrict sᶜ - 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
⊢ μ.restrict sᶜ + (ν.restrict sᶜ - 0) = μ.restrict sᶜ + ν.restrict sᶜ - 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
h_le_s_compl : μ.restrict sᶜ + (ν - ξ).restrict sᶜ = μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ
⊢ μ + (ν - ξ) = μ + (ν - ξ)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
h_le_s_compl : μ.restrict sᶜ + (ν - ξ).restrict sᶜ = μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ
⊢ μ.restrict s + ν.restrict s - ξ.restrict s + (μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ) =
    μ.restrict s + ν.restrict s - ξ.restrict s + (μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ)

Tactic:
]

Goal state after:
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
y : α
⊢ variationOnFromTo f univ a y = variationOnFromTo f univ a y

Tactic:
]

Goal state after:
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 (fun y ↦ variationOnFromTo f univ a x + variationOnFromTo f univ x y) (Ici x) x

Tactic:
]

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 (fun y ↦ variationOnFromTo f univ a x + variationOnFromTo f univ x y) (Ici x) x
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
h : μ ⟂ₘ ξ
s : Set α := h.nullSet
hs : MeasurableSet s
h_le_s : μ.restrict s + (ν - ξ).restrict s = μ.restrict s + ν.restrict s - ξ.restrict s
h_le_s_compl : μ.restrict sᶜ + (ν - ξ).restrict sᶜ = μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ
⊢ μ + ν - ξ = μ + ν - ξ

Tactic:
]

Goal state after:
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
⊢ Tendsto (fun y ↦ (eVariationOn f (univ ∩ Icc x y)).toReal) (𝓝[≥] x) (𝓝 (ENNReal.toReal 0))

Tactic:
]

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
⊢ Tendsto (fun y ↦ (eVariationOn f (univ ∩ Icc x y)).toReal) (𝓝[≥] x) (𝓝 (ENNReal.toReal 0))
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: α : Type u_2
inst✝⁶ : LinearOrder α
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set α
f : α → E
g : α → F
C D : ℝ≥0∞
hf : ∀ x ∈ s, ‖f x‖ₑ ≤ C
hg : ∀ x ∈ s, ‖g x‖ₑ ≤ D
B : E →L[ℝ] F →L[ℝ] G
n : ℕ
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ ∑ i ∈ Finset.range n, edist ((B (f (u (i + 1)))) (g (u (i + 1)))) ((B (f (u i))) (g (u i))) ≤
    ∑ x ∈ Finset.range n,
      (edist ((B (f (u (x + 1)))) (g (u (x + 1)))) ((B (f (u x))) (g (u (x + 1)))) +
        edist ((B (f (u x))) (g (u (x + 1)))) ((B (f (u x))) (g (u x))))

Tactic:
]

Goal state after:
α : Type u_2
inst✝⁶ : LinearOrder α
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set α
f : α → E
g : α → F
C D : ℝ≥0∞
hf : ∀ x ∈ s, ‖f x‖ₑ ≤ C
hg : ∀ x ∈ s, ‖g x‖ₑ ≤ D
B : E →L[ℝ] F →L[ℝ] G
n : ℕ
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ ∑ i ∈ Finset.range n, edist ((B (f (u (i + 1)))) (g (u (i + 1)))) ((B (f (u i))) (g (u i))) ≤
    ∑ x ∈ Finset.range n,
      (edist ((B (f (u (x + 1)))) (g (u (x + 1)))) ((B (f (u x))) (g (u (x + 1)))) +
        edist ((B (f (u x))) (g (u (x + 1)))) ((B (f (u x))) (g (u 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
x : α
x✝ : x ∈ v.limRatioMeas hρ ⁻¹' {0}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : μ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o
μs : μ s ≠ ∞
A : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s
⊢ Tendsto (fun a ↦ a) (𝓝[>] 0) (𝓝 0)

Tactic:
]

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
x : α
x✝ : x ∈ v.limRatioMeas hρ ⁻¹' {0}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : μ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o
μs : μ s ≠ ∞
A : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s
⊢ Tendsto (fun a ↦ a) (𝓝[>] 0) (𝓝 0)
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
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : N = 0
⊢ ∑ i ∈ Finset.Ico 0 n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =
    ∑ i ∈ Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i))

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : N = 0
⊢ ∑ i ∈ Finset.Ico 0 n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =
    ∑ i ∈ Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i))
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: case inr.refine_4
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
⊢ (fun x ↦ (variationOnFromTo f s c x + f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x + f x) / 2) x +
      ((fun x ↦ (variationOnFromTo f s c x - f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x - f x) / 2) x) =
    -variationOnFromTo f s c x + variationOnFromTo f s c y

Tactic:
]

Goal state after:
case inr.refine_4
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
⊢ (fun x ↦ (variationOnFromTo f s c x + f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x + f x) / 2) x +
      ((fun x ↦ (variationOnFromTo f s c x - f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x - f x) / 2) x) =
    -variationOnFromTo f s c x + variationOnFromTo f s c y
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
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : N = 0
⊢ ∑ i ∈ Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i)) ≤
    ∑ j ∈ Finset.Ico 0 (n + 1), edist (f (w (j + 1))) (f (w j))

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : N = 0
⊢ ∑ i ∈ Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i)) ≤
    ∑ j ∈ Finset.Ico 0 (n + 1), edist (f (w (j + 1))) (f (w j))
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 : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ 0 ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})

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:
]

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_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})
⊢ 0 ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})

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
Module: Mathlib.Analysis.Complex.Spectrum

Goal state before:
pp: A : Type u_1
inst✝³ : NonUnitalRing A
inst✝² : Module ℂ A
inst✝¹ : IsScalarTower ℂ A A
inst✝ : SMulCommClass ℂ A A
a : A
⊢ (∀ x ∈ spectrum ℂ ↑a, x = ↑x.re) ↔ ∀ x ∈ spectrum ℂ ↑a, x = ↑x.re

Tactic:
]

Goal state after:
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:
]

Goal state after:
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
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
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : 0 < N
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) =
    ∑ i ∈ Finset.Ico 0 (N - 1), edist (f (u (i + 1))) (f (u i)) +
        ∑ i ∈ Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) +
      ∑ i ∈ Finset.Ico N n, edist (f (u (i + 1))) (f (u i))

Tactic:
rw [Finset.sum_Ico_consecutive, Finset.sum_Ico_consecutive, Finset.range_eq_Ico] <;> grind

Goal state after:
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: α : Type u_2
inst✝⁶ : LinearOrder α
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set α
f : α → E
g : α → F
C D : ℝ≥0∞
hf : ∀ x ∈ s, ‖f x‖ₑ ≤ C
hg : ∀ x ∈ s, ‖g x‖ₑ ≤ D
B : E →L[ℝ] F →L[ℝ] G
n : ℕ
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ ‖B‖ₑ * D * eVariationOn f s + ‖B‖ₑ * C * eVariationOn g s = ‖B‖ₑ * (C * eVariationOn g s + D * eVariationOn f s)

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: α : Type u_2
inst✝⁷ : LinearOrder α
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_6
f : α → 𝕜
g : α → F
inst✝⁴ : NormedRing 𝕜
inst✝³ : NormedAlgebra ℝ 𝕜
inst✝² : Module 𝕜 F
inst✝¹ : NormSMulClass 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
C D : ℝ≥0∞
s : Set α
hf : ∀ x ∈ s, ‖f x‖ₑ ≤ C
hg : ∀ x ∈ s, ‖g x‖ₑ ≤ D
⊢ C * eVariationOn g s + D * eVariationOn f s ≤ C * eVariationOn g s + D * eVariationOn f s

Tactic:
]

Goal state after: