mathlib-initiative/mathlib-tactics huggingface dataset:
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
u : ℕ → α
hu : Monotone u
n : ℕ
⊢ eVariationOn f (u '' Iic n) = ∑ i ∈ Finset.range n, edist (f (u i)) (f (u (i + 1)))

Tactic:
by

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
f : α → ℝ≥0∞
hf : Measurable f
h'f : ∫⁻ (y : α), f y ∂μ ≠ ∞
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x))

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_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
n : ℕ
⊢ min n (s.card - 1) < s.card

Tactic:
by

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
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∫⁻ (y : α), f y ∂μ ≠ ∞
⊢ ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ∞

Tactic:
by

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
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∫⁻ (y : α), f y ∂μ ≠ ∞
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x))

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_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
u : ℕ → α := fun n ↦ (s.orderEmbOfFin ⋯) ⟨min n (s.card - 1), ⋯⟩
i : Fin s.card
⊢ ↑i ≤ s.card - 1 ∧ (s.orderEmbOfFin ⋯) ⟨min (↑i) (s.card - 1), ⋯⟩ = (s.orderEmbOfFin ⋯) 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_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
u : ℕ → α := fun n ↦ (s.orderEmbOfFin ⋯) ⟨min n (s.card - 1), ⋯⟩
i : ℕ
hi : i ≤ s.card - 1
⊢ i < s.card

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_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
u : ℕ → α := fun n ↦ (s.orderEmbOfFin ⋯) ⟨min n (s.card - 1), ⋯⟩
⊢ ↑s = u '' Iic (s.card - 1)

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_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this✝ : 0 < s.card
u : ℕ → α := fun n ↦ (s.orderEmbOfFin ⋯) ⟨min n (s.card - 1), ⋯⟩
this : ↑s = u '' Iic (s.card - 1)
x✝² x✝¹ : ℕ
x✝ : x✝² ≤ x✝¹
⊢ ⟨min x✝² (s.card - 1), ⋯⟩ ≤ ⟨min x✝¹ (s.card - 1), ⋯⟩

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_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
⊢ BoundedVariationOn f ↑s

Tactic:
by

Goal state after:
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:
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
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
n : ℕ
c : E
hi✝ : c ∈ t
⊢ Integrable ((A.set n).indicator fun x ↦ c) 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
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:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
s : Set α
t : Set β
φ : β → α
hφ : AntitoneOn φ t
φst : MapsTo φ t s
⊢ eVariationOn (f ∘ φ) t ≤ eVariationOn f s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
⊢ eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)

Tactic:
apply le_antisymm (comp_le_of_monotoneOn f φ hφ (mapsTo_image φ t))
  cases isEmpty_or_nonempty β
  · simp [Set.eq_empty_of_isEmpty]
  let ψ := φ.invFunOn t
  have ψφs : EqOn (φ ∘ ψ) id (φ '' t) := (surjOn_image φ t).rightInvOn_invFunOn
  have ψts : MapsTo ψ (φ '' t) t := (surjOn_image φ t).mapsTo_invFunOn
  have hψ : MonotoneOn ψ (φ '' t) := Function.monotoneOn_of_rightInvOn_of_mapsTo hφ ψφs ψts
  change eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t
  rw [← eq_of_eqOn (ψφs.comp_left : EqOn (f ∘ φ ∘ ψ) (f ∘ id) (φ '' t))]
  exact comp_le_of_monotoneOn _ ψ hψ ψts

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
⊢ eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)

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
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:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
⊢ eVariationOn (f ∘ φ) (t ∩ Icc x y) = eVariationOn f (φ '' t ∩ Icc (φ x) (φ y))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : AntitoneOn φ t
⊢ eVariationOn (f ∘ φ) t = eVariationOn f (φ '' 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
⊢ eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' 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 : α → E
s : Set α
hf : BoundedVariationOn f s
⊢ BoundedVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' 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
hf : LocallyBoundedVariationOn f s
⊢ LocallyBoundedVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s)

Tactic:
by

Goal state after:
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
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)
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iic a
⊢ n ∈ Iic n

Tactic:
by

Goal state after:
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
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)
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iic a
this : u n ≤ a
hn : u n < a
⊢ ∀ i ≤ n, u i ∈ s ∩ Iio a

Tactic:
by

Goal state after:
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
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)
u : ℕ → α
n : ℕ
u_mono : StrictMonoOn u (Iic (n + 1))
u_mem : ∀ i ∈ Iic (n + 1), u i ∈ s ∩ Iic a
this✝ : u (n + 1) ≤ a
hn : u (n + 1) = a
this :
  Tendsto (fun y ↦ eVariationOn f (s ∩ Iio a) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (eVariationOn f (s ∩ Iio a) + edist (f a) l))
⊢ u n < a

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
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
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ)

Tactic:
by

Goal state after:
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
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)
u : ℕ → α
n : ℕ
u_mono : StrictMonoOn u (Iic (n + 1))
u_mem : ∀ i ∈ Iic (n + 1), u i ∈ s ∩ Iic a
this✝¹ : u (n + 1) ≤ a
hn : u (n + 1) = a
this✝ :
  Tendsto (fun y ↦ eVariationOn f (s ∩ Iio a) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (eVariationOn f (s ∩ Iio a) + edist (f a) l))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
v : ℕ → α := fun i ↦ if i ≤ n then u i else if i = n + 1 then y else a
⊢ ∑ i ∈ Finset.range (n + 1), edist (f (u (i + 1))) (f (u i)) ≤
    ∑ i ∈ Finset.range (n + 2), edist (f (v (i + 1))) (f (v i))

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
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:
by

Goal state after:
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
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)
u : ℕ → α
n : ℕ
u_mono : StrictMonoOn u (Iic (n + 1))
u_mem : ∀ i ∈ Iic (n + 1), u i ∈ s ∩ Iic a
this✝¹ : u (n + 1) ≤ a
hn : u (n + 1) = a
this✝ :
  Tendsto (fun y ↦ eVariationOn f (s ∩ Iio a) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (eVariationOn f (s ∩ Iio a) + edist (f a) l))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
v : ℕ → α := fun i ↦ if i ≤ n then u i else if i = n + 1 then y else a
A :
  ∑ i ∈ Finset.range (n + 1), edist (f (u (i + 1))) (f (u i)) ≤
    ∑ i ∈ Finset.range (n + 2), edist (f (v (i + 1))) (f (v i))
⊢ ∑ i ∈ Finset.range (n + 2), edist (f (v (i + 1))) (f (v i)) ≤ eVariationOn f (s ∩ Iio a) + edist (f a) (f y)

Tactic:
by

Goal state after:
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
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)
⊢ (s ∩ Iio a).Nonempty

Tactic:
by

Goal state after:
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
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
⊢ StrictMonoOn (fun i ↦ b) (Iic 0) ∧ ∀ i ∈ Iic 0, (fun i ↦ b) i ∈ s ∩ Iio a

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
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 < ∞
⊢ ∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ ≤ ∫⁻ (y : α) in a, ‖f y - c‖ₑ + ‖f x - c‖ₑ ∂μ

Tactic:
by

Goal state after:
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
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))
⊢ u n < a

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
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 < ∞
⊢ ∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ + ∫⁻ (x_1 : α) in a, ‖f x - c‖ₑ ∂μ ≤ ε / 2 * μ a + ε / 2 * μ a

Tactic:
by

Goal state after:
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
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✝¹ : 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))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
h'y : y ∈ s ∩ Iio a
v : ℕ → α := fun i ↦ if i ≤ n then u i else if i = n + 1 then y else a
⊢ edist (f a) (f y) ≤ edist (f (v (n + (1 + 1)))) (f (v (n + 1)))

Tactic:
by

Goal state after:
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
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✝¹ : 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))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
h'y : y ∈ s ∩ Iio a
v : ℕ → α := fun i ↦ if i ≤ n then u i else if i = n + 1 then y else a
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y) ≤
    ∑ i ∈ Finset.range (n + 2), edist (f (v (i + 1))) (f (v i))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
a : A
ha : p a
inst✝ : UniqueHom R A
φ : C(↑(spectrum R a), R) →⋆ₐ[R] A
hφ₁ : Continuous[_, inst✝⁴] ⇑φ
hφ₂ : φ (restrict (spectrum R a) (ContinuousMap.id R)) = a
⊢ (cfcHom ha) (restrict (spectrum R a) (ContinuousMap.id R)) = φ (restrict (spectrum R a) (ContinuousMap.id R))

Tactic:
by

Goal state after:
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
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✝¹ : 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))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
h'y : y ∈ s ∩ Iio a
v : ℕ → α := fun i ↦ if i ≤ n then u i else if i = n + 1 then y else a
A :
  ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y) ≤
    ∑ i ∈ Finset.range (n + 2), edist (f (v (i + 1))) (f (v i))
⊢ MonotoneOn v (Iic (n + 2))

Tactic:
by

Goal state after: