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))
⊢ ρ (toMeasurable (ρ + μ) (u m) ∩ w n) = ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

Tactic:
conv_rhs => rw [inter_comm]

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
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) (w n) ∩ toMeasurable (ρ + μ) (u m))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν : Measure α
inst✝¹ : SigmaFinite ν
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : ℕ → Set α := disjointed (spanningSets ν)
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : ∀ (n : ℕ), Fact (ν (s n) < ∞)
ξ : ℕ → Measure α := fun n ↦ (μ.restrict (s n)).singularPart (ν.restrict (s n))
f : ℕ → α → ℝ≥0∞ := fun n ↦ (s n).indicator ((μ.restrict (s n)).rnDeriv (ν.restrict (s n)))
hfm : ∀ (n : ℕ), Measurable (f n)
hξ : sum ξ ⟂ₘ ν
⊢ (sum fun n ↦ ξ n + ν.withDensity (f n)) = sum fun n ↦ μ.restrict (s n)

Tactic:
[

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν : Measure α
inst✝¹ : SigmaFinite ν
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : ℕ → Set α := disjointed (spanningSets ν)
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : ∀ (n : ℕ), Fact (ν (s n) < ∞)
ξ : ℕ → Measure α := fun n ↦ (μ.restrict (s n)).singularPart (ν.restrict (s n))
f : ℕ → α → ℝ≥0∞ := fun n ↦ (s n).indicator ((μ.restrict (s n)).rnDeriv (ν.restrict (s n)))
hfm : ∀ (n : ℕ), Measurable (f n)
hξ : sum ξ ⟂ₘ ν
⊢ (sum fun n ↦ ξ n + ν.withDensity (f n)) = sum fun n ↦ μ.restrict (s n)
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
⊢ μ.withDensity ((r • ν).rnDeriv μ) = μ.withDensity (r • 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
⊢ μ.withDensity ((r • ν).rnDeriv μ) = μ.withDensity (r • 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
⊢ (r • μ).withDensity (ν.rnDeriv (r • μ)) = (r • μ).withDensity (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
⊢ (r • μ).withDensity (ν.rnDeriv (r • μ)) = (r • μ).withDensity (r⁻¹ • 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.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
⊢ f x - f y ≤ (eVariationOn f s).toReal

Tactic:
apply (le_abs_self _).trans

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
⊢ |f x - f y| ≤ (eVariationOn f s).toReal
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 (variationOnFromTo f univ a) (Ici x) x

Tactic:
this

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.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
h : LocallyBoundedVariationOn f ∅
x : α
hx : x ∈ ∅
y : α
hy : y ∈ ∅
⊢ f y - f x + (0 y - 0 x) = variationOnFromTo f ∅ x y

Tactic:
simp at hx

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

Goal state before:
pp: case e_a.e_a
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.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) = edist (f (u N)) (f (u (N - 1)))

Tactic:
have A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
            have : Finset.Ico (N - 1) N = {N - 1} := by rw [← Nat.Ico_succ_singleton, A]
            simp only [this, A, Finset.sum_singleton]

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 t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)

Tactic:
by
  by_cases hs : s = ∅
  · simp [hs]
  have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } :=
    nonempty_monotone_mem (nonempty_iff_ne_empty.2 hs)
  by_cases ht : t = ∅
  · simp [ht]
  have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ t } :=
    nonempty_monotone_mem (nonempty_iff_ne_empty.2 ht)
  refine ENNReal.iSup_add_iSup_le ?_
  /- We start from two sequences `u` and `v` along `s` and `t` respectively, and we build a new
    sequence `w` along `s ∪ t` by juxtaposing them. Its variation is larger than the sum of the
    variations. -/
  rintro ⟨n, ⟨u, hu, us⟩⟩ ⟨m, ⟨v, hv, vt⟩⟩
  let w i := if i ≤ n then u i else v (i - (n + 1))
  calc
    ((∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) +
          ∑ i ∈ Finset.range m, edist (f (v (i + 1))) (f (v i))) =
        (∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i))) +
          ∑ i ∈ Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) := by
      dsimp only [w]
      congr 1
      · grind [Finset.sum_congr]
      · grind
    _ = (∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i))) +
          ∑ i ∈ Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) := by
      congr 1
      rw [Finset.range_eq_Ico]
      convert!
          Finset.sum_Ico_add (fun i : ℕ => edist (f (w (i + 1))) (f (w i))) 0 m (n + 1) using 3 <;>
        abel
    _ ≤ ∑ i ∈ Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i)) := by
      rw [← Finset.sum_union]
      · gcongr; grind
      · exact Finset.disjoint_left.2 (by grind)
    _ ≤ eVariationOn f (s ∪ t) := sum_le (by grind [Monotone]) (by grind)

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

Goal state before:
pp: case refine_1
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
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ρ s

Tactic:
have : Tendsto (fun t : ℝ≥0 =>
        ((t : ℝ≥0∞) ^ 2 * ρ s : ℝ≥0∞)) (𝓝[>] 1) (𝓝 ((1 : ℝ≥0∞) ^ 2 * ρ s)) := by
      refine ENNReal.Tendsto.mul ?_ ?_ tendsto_const_nhds ?_
      · exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds)
      · simp
      · simp only [one_pow, Ne, or_true, ENNReal.one_ne_top, not_false_iff]
    simp only [one_pow, one_mul] at this
    refine ge_of_tendsto this ?_
    filter_upwards [self_mem_nhdsWithin] with _ ht
    exact v.withDensity_le_mul hρ hs ht

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
a b : α
hab : a ≤ b
⊢ eVariationOn f {a, b} = edist (f a) (f b)

Tactic:
· apply le_antisymm _ (edist_le f (by simp) (by simp))
    simp only [eVariationOn_eq_strictMonoOn, iSup_le_iff]
    rintro ⟨n, u, hmono, hi⟩
    rcases (by omega : n = 0 ∨ n = 1 ∨ 2 ≤ n) with rfl | rfl | hn
    · simp
    · have := hmono (by simp) (by simp) zero_lt_one
      simp [(by grind : u 0 = a), (by grind : u 1 = b), edist_comm]
    · have := hmono (by simp) (by grind) zero_lt_one
      have := hmono (by grind) (by grind) one_lt_two
      grind

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
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
⊢ ∀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)

Tactic:
[

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
⊢ ∀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)
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 ∈ t, ‖f x - c‖ₑ < ε / 2

Tactic:
[

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 ∈ t, ‖f x - c‖ₑ < ε / 2
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
this✝ : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iio a
this :
  Tendsto (fun y ↦ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) l))
⊢ ∑ i ∈ Finset.range ⟨n, ⟨u, ⋯⟩⟩.fst, edist (f (↑⟨n, ⟨u, ⋯⟩⟩.snd (i + 1))) (f (↑⟨n, ⟨u, ⋯⟩⟩.snd i)) + edist (f a) l ≤
    eVariationOn f (s ∩ Iic a)

Tactic:
apply le_of_tendsto this

Goal state after:
case refine_2
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
a : α
l : E
h : (𝓝[s ∩ Iio a] a).NeBot
ha : a ∈ s
h'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)
b : α
hb : b ∈ s ∩ Iio a
this✝ : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iio a
this :
  Tendsto (fun y ↦ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) l))
⊢ ∀ᶠ (c : α) in 𝓝[s ∩ Iio a] a,
    ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f c) ≤ eVariationOn f (s ∩ Iic a)
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
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α
hy : y ∈ s
y' : α
hy' : ε ≤ eVariationOn f (s ∩ Ici y')
y'_mem : y' ∈ s ∩ Ici y
a : α
ha : a ∈ s ∩ Ici y'
b : α
hb : b ∈ s ∩ Ici y'
hab : a < b
h : δ < eVariationOn f (s ∩ (Ici y' ∩ Icc a b))
this : Ici y' ∩ Icc a b = Icc a b
⊢ δ ≤ eVariationOn f (s ∩ Icc y b)

Tactic:
this

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
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α
hy : y ∈ s
y' : α
hy' : ε ≤ eVariationOn f (s ∩ Ici y')
y'_mem : y' ∈ s ∩ Ici y
a : α
ha : a ∈ s ∩ Ici y'
b : α
hb : b ∈ s ∩ Ici y'
hab : a < b
h : δ < eVariationOn f (s ∩ Icc a b)
this : Ici y' ∩ Icc a b = Icc a b
⊢ δ ≤ eVariationOn f (s ∩ Icc y b)
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 : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ IntegrableOn (fun y ↦ ↑‖f y - f x‖₊) a μ

Tactic:
[

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 : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ IntegrableOn (fun y ↦ ↑‖f y - f x‖₊) a μ
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ ‖⨍ (y : α) in a, f y ∂μ - ⨍ (x_1 : α) in a, f x ∂μ‖ ≤ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ ‖⨍ (y : α) in a, f y ∂μ - ⨍ (x_1 : α) in a, f x ∂μ‖ ≤ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0)

Tactic:
[

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict fun x ↦ s • f x, continuous_toFun := ⋯ } =
    s • (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }

Tactic:
[

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict fun x ↦ s • f x, continuous_toFun := ⋯ } =
    s • (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
this :
  (fun y ↦ eVariationOn f (s ∩ Ioc x y)) = fun y ↦
    eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ Ico (toDual y) (toDual x))
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ioc x y)) (𝓝[s] x) (𝓝 0)

Tactic:
this

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
this :
  (fun y ↦ eVariationOn f (s ∩ Ioc x y)) = fun y ↦
    eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ Ico (toDual y) (toDual x))
⊢ Tendsto (fun y ↦ eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ Ico (toDual y) (toDual x))) (𝓝[s] x) (𝓝 0)