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
s : Set α
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ ↑⟨0, ⟨u, ⋯⟩⟩.snd ⟨0, ⟨u, ⋯⟩⟩.fst = u 0 ∧
    ∑ x ∈ Finset.range 0, edist (f (u (x + 1))) (f (u x)) =
      ∑ i ∈ Finset.range ⟨0, ⟨u, ⋯⟩⟩.fst, edist (f (↑⟨0, ⟨u, ⋯⟩⟩.snd (i + 1))) (f (↑⟨0, ⟨u, ⋯⟩⟩.snd 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
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < v.limRatioMeas hρ x}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t)
⊢ ρ (s ∩ t) ≤ ρ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < v.limRatioMeas hρ x}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t)
⊢ ↑q * μ s ≤ ρ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
h : variationOnFromTo f s a b = 0
⊢ edist (f a) (f b) = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
⊢ variationOnFromTo f s a b = variationOnFromTo f s a c ↔ variationOnFromTo f s b c = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
n m : ℕ
v : ℕ → α
v_mono : StrictMonoOn v (Iic m)
v_mem : ∀ i ∈ Iic m, v i ∈ s
hv : ↑⟨m, ⟨v, ⋯⟩⟩.snd ⟨m, ⟨v, ⋯⟩⟩.fst = u n
h'v :
  ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
    ∑ i ∈ Finset.range ⟨m, ⟨v, ⋯⟩⟩.fst, edist (f (↑⟨m, ⟨v, ⋯⟩⟩.snd (i + 1))) (f (↑⟨m, ⟨v, ⋯⟩⟩.snd i))
hn : u n < u (n + 1)
⊢ (fun i ↦ if i ≤ m then v i else u (n + 1)) (m + 1) = u (n + 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_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
n m : ℕ
v : ℕ → α
v_mono : StrictMonoOn v (Iic m)
v_mem : ∀ i ∈ Iic m, v i ∈ s
hv : ↑⟨m, ⟨v, ⋯⟩⟩.snd ⟨m, ⟨v, ⋯⟩⟩.fst = u n
h'v :
  ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
    ∑ i ∈ Finset.range ⟨m, ⟨v, ⋯⟩⟩.fst, edist (f (↑⟨m, ⟨v, ⋯⟩⟩.snd (i + 1))) (f (↑⟨m, ⟨v, ⋯⟩⟩.snd i))
hn : u n < u (n + 1)
⊢ StrictMonoOn (fun i ↦ if i ≤ m then v i else u (n + 1)) (Iic (m + 1)) ∧
    ∀ i ≤ m + 1, (fun i ↦ if i ≤ m then v i else u (n + 1)) i ∈ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
ab : a ≤ b
⊢ variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Icc a b → ∀ ⦃y : α⦄, y ∈ s ∩ Icc a b → edist (f x) (f y) = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < v.limRatioMeas hρ x}
⊢ ↑q * μ s ≤ ρ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
ba : b ≤ a
⊢ variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Icc b a → ∀ ⦃y : α⦄, y ∈ s ∩ Icc b a → edist (f x) (f y) = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
n m : ℕ
v : ℕ → α
v_mono : StrictMonoOn v (Iic m)
v_mem : ∀ i ∈ Iic m, v i ∈ s
hv : ↑⟨m, ⟨v, ⋯⟩⟩.snd ⟨m, ⟨v, ⋯⟩⟩.fst = u n
h'v :
  ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
    ∑ i ∈ Finset.range ⟨m, ⟨v, ⋯⟩⟩.fst, edist (f (↑⟨m, ⟨v, ⋯⟩⟩.snd (i + 1))) (f (↑⟨m, ⟨v, ⋯⟩⟩.snd i))
hn : u n < u (n + 1)
⊢ ∀ x ∈ Finset.range m,
    edist (f (v (x + 1))) (f (v x)) =
      edist (f (if x < m then v (x + 1) else u (n + 1))) (f (if x ≤ m then v x else u (n + 1)))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ uIcc a b → ∀ ⦃y : α⦄, y ∈ s ∩ uIcc a b → edist (f x) (f y) = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
⊢ MonotoneOn (variationOnFromTo f s a) s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
b : α
bs : b ∈ s
⊢ AntitoneOn (fun a ↦ variationOnFromTo f s a b) s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
⊢ ∃ p,
    ↑p.snd p.fst = u n ∧
      ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
        ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
as : a ∈ s
bs : b ∈ s
cs : c ∈ s
bc : b ≤ c
⊢ |f c - f b| = dist (f b) (f c)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
as : a ∈ s
bs : b ∈ s
cs : c ∈ s
bc : b ≤ c
⊢ dist (f b) (f c) ≤ variationOnFromTo f s b c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + ν.withDensity p.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a b c : α
as : a ∈ s
bs : b ∈ s
cs : c ∈ s
bc : b ≤ c
⊢ variationOnFromTo f s b c = variationOnFromTo f s a c - variationOnFromTo f s a b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
b : α
bs : b ∈ s
c : α
cs : c ∈ s
bc : b ≤ c
this : f b - f c ≤ variationOnFromTo f s a c - variationOnFromTo f s a b
⊢ (variationOnFromTo f s a + f) b ≤ (variationOnFromTo f s a + f) c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
b : α
bs : b ∈ s
c : α
cs : c ∈ s
bc : b ≤ c
⊢ f b - f c ≤ |f c - f b|

Tactic:
by

Goal state after:
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)))
n : ℕ
⊢ Measurable (f n)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
⊢ MonotoneOn (variationOnFromTo f s a + f) s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
⊢ MonotoneOn (↑⟨n, ⟨u, ⋯⟩⟩.snd) (Iic ⟨n, ⟨u, ⋯⟩⟩.fst)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
⊢ ∀ i ≤ ⟨n, ⟨u, ⋯⟩⟩.fst, ↑⟨n, ⟨u, ⋯⟩⟩.snd i ∈ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
⊢ MonotoneOn (variationOnFromTo f s a - 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
⊢ eVariationOn f s = ⨆ p, ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))

Tactic:
by

Goal state after:
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)
⊢ sum ξ ⟂ₘ ν

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 t : Set α
hst : t ⊆ s
⊢ eVariationOn f t ≤ 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
⊢ ⨆ p ∈ {p | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Icc p.1 p.2) ≤ eVariationOn f s

Tactic:
simp [iSup_le_iff, mono f inter_subset_left]

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
⊢ ⨆ p ∈ {p | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Icc p.1 p.2) ≤ eVariationOn f s

Tactic:
by

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

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
⊢ variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y)

Tactic:
by

Goal state after:
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 ξ + ν.withDensity (∑' (n : ℕ), f n) = sum fun n ↦ ξ n + ν.withDensity (f n)

Tactic:
by

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 ∩ Iio b] b) (𝓝 l)
H :
  Tendsto (fun x ↦ variationOnFromTo f s a b - variationOnFromTo f s x b) (𝓝[s ∩ Iio b] b)
    (𝓝 (variationOnFromTo f s a b - dist (f b) l))
⊢ Tendsto (variationOnFromTo f s a) (𝓝[s ∩ Iio b] b) (𝓝 (variationOnFromTo f s a b - dist (f b) l))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
hu : Monotone u
hus : ∀ (i : ℕ), u i ∈ s
⊢ ∀ i ≤ n, u i ∈ s ∩ Icc (u 0) (u n)

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
hu : Monotone u
hus : ∀ (i : ℕ), u i ∈ s
⊢ eVariationOn f (s ∩ Icc (u 0) (u n)) ≤ ⨆ p ∈ {p | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Icc p.1 p.2)

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 s = ⨆ p ∈ {p | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Icc p.1 p.2)

Tactic:
by

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 ∩ Iio b] b) (𝓝 l)
H : Tendsto (fun x ↦ (eVariationOn f (s ∩ Icc x b)).toReal) (𝓝[s ∩ Iio b] b) (𝓝 (dist (f b) l))
x : α
hx : x ∈ s ∩ Iio b
⊢ (eVariationOn f (s ∩ Icc x b)).toReal = variationOnFromTo f s x b

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 g : α → E
s : Set α
h : EqOn f g s
⊢ eVariationOn f s = eVariationOn g s

Tactic:
by

Goal state after:
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
hadd : sum ξ + ν.withDensity (∑' (n : ℕ), f n) = μ
⊢ Measurable (sum ξ, ∑' (n : ℕ), f n).2

Tactic:
by

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 ∩ Iio b] b) (𝓝 l)
H : Tendsto (fun x ↦ (eVariationOn f (s ∩ Icc x b)).toReal) (𝓝[s ∩ Iio b] b) (𝓝 (dist (f b) l))
⊢ Tendsto (fun x ↦ variationOnFromTo f s x b) (𝓝[s ∩ Iio b] b) (𝓝 (dist (f b) l))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SFinite μ
inst✝ : SigmaFinite ν
⊢ μ.HaveLebesgueDecomposition ν

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
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
by

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
this :
  (r • ν).singularPart μ + μ.withDensity ((r • ν).rnDeriv μ) = (r • ν).singularPart μ + r • μ.withDensity (ν.rnDeriv μ)
⊢ μ.withDensity ((↑r • ν).rnDeriv μ) = ↑r • μ.withDensity (ν.rnDeriv μ)

Tactic:
by

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 ∩ Iio b] b) (𝓝 l)
⊢ edist (f b) l ≠ ∞

Tactic:
by

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 ∩ Iio b] b) (𝓝 l)
⊢ Tendsto (variationOnFromTo f s a) (𝓝[s ∩ Iio b] b) (𝓝 (variationOnFromTo f s a b - dist (f b) l))

Tactic:
by

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
⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ

Tactic:
by

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 ≠ ∞
⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ

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 s = 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0

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 : (f '' s).Subsingleton
⊢ eVariationOn f s = 0

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
hs : s.Subsingleton
⊢ BoundedVariationOn 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
ι : Type u_3
F : ι → α → E
p : Filter ι
f : α → E
s : Set α
Ffs : ∀ x ∈ s, Tendsto (fun i ↦ F i x) p (𝓝 (f x))
v : ℝ≥0∞
hv : v < eVariationOn f s
n : ℕ
u : ℕ → α
um : Monotone u
us : ∀ (i : ℕ), u i ∈ s
hlt : v < ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i))
⊢ Tendsto (fun j ↦ ∑ i ∈ Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))))

Tactic:
by

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)
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 b + variationOnFromTo f s b x = variationOnFromTo f s a x

Tactic:
variationOnFromTo.add hf ha hb hx.1

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)
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
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 :
  ν.singularPart (r • μ) + (r • μ).withDensity (ν.rnDeriv (r • μ)) =
    ν.singularPart (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:
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
ι : Type u_3
F : ι → α → E
p : Filter ι
f : α → E
s : Set α
Ffs : ∀ x ∈ s, Tendsto (fun i ↦ F i x) p (𝓝 (f x))
v : ℝ≥0∞
hv : v < eVariationOn f s
⊢ ∀ᶠ (n : ι) in p, v < eVariationOn (F n) 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
s : Set α
f : α →ᵤ[singleton '' s] E
⊢ SemicontinuousAt (fun x1 x2 ↦ (fun f ↦ eVariationOn f s) x1 > x2) f

Tactic:
by

Goal state after: