mathlib-initiative/mathlib-tactics huggingface dataset:
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
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
⊢ μA = 0

Tactic:
by_cases! hb : 0 < νA

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
⊢ μA = 0

case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : νA ≤ 0
⊢ μA = 0
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: case refine_1
x : ℝ
⊢ ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ∈ (vitaliFamily volume 1).setsAt x

Tactic:
filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy

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

Goal state before:
pp: x : ℝ
⊢ Tendsto (fun y ↦ Icc x y) (𝓝[>] x) ((vitaliFamily volume 1).filterAt x)

Tactic:
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
  · intro ε εpos
    filter_upwards [Icc_mem_nhdsGT <| show x < x + ε by linarith] with y hy
    rw [closedBall_eq_Icc]
    exact Icc_subset_Icc (by linarith) hy.2

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
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))
| ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ w n)

Tactic:
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))
| ↑q * μ (w n ∩ toMeasurable (ρ + μ) (u m))
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))
x : α
hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n
⊢ ∃ᶠ (a : Set α) in v.filterAt x, (q • μ) a ≤ ρ a

Tactic:
have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) :=
          tendsto_nhds_limUnder hx.2.1.1

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))
x : α
hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n
L : Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x))
⊢ ∃ᶠ (a : Set α) in v.filterAt x, (q • μ) a ≤ ρ a
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) ∩ toMeasurable (ρ + μ) (w n))

Tactic:
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) (w n) ∩ toMeasurable (ρ + μ) (u m))
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) (w n) ∩ toMeasurable (ρ + μ) (u m))

Tactic:
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 (w n ∩ toMeasurable (ρ + μ) (u m)) = ρ (toMeasurable (ρ + μ) (w n) ∩ toMeasurable (ρ + μ) (u m))
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 := ⋯
o : ℕ → Set α := ⋯
u : ℕ → Set α := ⋯
w : ℕ → Set α := ⋯
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))
B :
  ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ (ρ + μ) (toMeasurable (ρ + μ) (u m)) < ∞

Tactic:
measure_toMeasurable

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))
B :
  ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ (ρ + μ) (u m) < ∞
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
this :
  ∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] {f : α → E} {s : Set α},
    LocallyBoundedVariationOn f s →
      ∀ {a b : α}, a ∈ s → b ∈ s → variationOnFromTo f s a b = 0 → a ≤ b → edist (f a) (f b) = 0
h' : ¬a ≤ b
⊢ -0 = 0

Tactic:
neg_zero

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 : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
h : variationOnFromTo f s a b = 0
this :
  ∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] {f : α → E} {s : Set α},
    LocallyBoundedVariationOn f s →
      ∀ {a b : α}, a ∈ s → b ∈ s → variationOnFromTo f s a b = 0 → a ≤ b → edist (f a) (f b) = 0
h' : ¬a ≤ b
⊢ 0 = 0
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: case inr
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
this :
  ∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] {f : α → E} {s : Set α},
    LocallyBoundedVariationOn f s →
      ∀ {a b : α}, a ∈ s → b ∈ s → variationOnFromTo f s a b = 0 → a ≤ b → edist (f a) (f b) = 0
h' : ¬a ≤ b
⊢ edist (f a) (f b) = 0

Tactic:
rw [edist_comm]
    apply this hf hb ha _ (le_of_not_ge h')
    rw [variationOnFromTo.eq_neg_swap, h, neg_zero]

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
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
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑t ^ 2 * ρ s

Tactic:
have f_meas : Measurable f := v.limRatioMeas_measurable hρ

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
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑t ^ 2 * ρ s
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
hf : BoundedVariationOn f s
hg : BoundedVariationOn g s
B : E →L[ℝ] F →L[ℝ] G
x : α
hx : x ∈ s
y : α
hy : y ∈ s
⊢ ‖f x + (f y - f x)‖ₑ ≤ ‖f x‖ₑ + eVariationOn f s

Tactic:
enorm_add_le,

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
hf : BoundedVariationOn f s
hg : BoundedVariationOn g s
B : E →L[ℝ] F →L[ℝ] G
x : α
hx : x ∈ s
y : α
hy : y ∈ s
⊢ ‖f x‖ₑ + ‖f y - f x‖ₑ ≤ ‖f x‖ₑ + eVariationOn f s
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
hf : BoundedVariationOn f s
hg : BoundedVariationOn g s
B : E →L[ℝ] F →L[ℝ] G
x : α
hx : x ∈ s
A : ∀ y ∈ s, ‖f y‖ₑ ≤ ‖f x‖ₑ + eVariationOn f s
y : α
hy : y ∈ s
⊢ ‖g x + (g y - g x)‖ₑ ≤ ‖g x‖ₑ + eVariationOn g s

Tactic:
enorm_add_le,

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
hf : BoundedVariationOn f s
hg : BoundedVariationOn g s
B : E →L[ℝ] F →L[ℝ] G
x : α
hx : x ∈ s
A : ∀ y ∈ s, ‖f y‖ₑ ≤ ‖f x‖ₑ + eVariationOn f s
y : α
hy : y ∈ s
⊢ ‖g x‖ₑ + ‖g y - g x‖ₑ ≤ ‖g x‖ₑ + eVariationOn g s
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
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
⊢ ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s

Tactic:
have f_meas : Measurable f := v.limRatioMeas_measurable hρ

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
⊢ ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
⊢ t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝ →
    Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝)) →
      Tendsto (fun a ↦ ρ a / μ a) (v.filterAt a✝) (𝓝 (ρ.rnDeriv μ a✝))

Tactic:
Bx

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
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
Bx : t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝
⊢ Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝)) →
    Tendsto (fun a ↦ ρ a / μ a) (v.filterAt a✝) (𝓝 (ρ.rnDeriv μ a✝))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case ha
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
I : ℕ → α
hI : Monotone I
n i : ℕ
hi : i ∈ Finset.range n
⊢ 0 ≤ i

Tactic:
Finset.mem_range

Goal state after:
case ha
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
I : ℕ → α
hI : Monotone I
n i : ℕ
hi : i < n
⊢ 0 ≤ i
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case hb
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
I : ℕ → α
hI : Monotone I
n i : ℕ
hi : i ∈ Finset.range n
⊢ i + 1 ≤ n

Tactic:
Finset.mem_range

Goal state after:
case hb
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
I : ℕ → α
hI : Monotone I
n i : ℕ
hi : i < n
⊢ i + 1 ≤ n
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

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

Tactic:
rw [(by grind : u '' Iic (n + 1) = u '' Iic n ∪ {u n, u (n + 1)}), union f]
    · simp [Finset.sum_range_succ, ih]
    · simpa [IsGreatest, upperBounds] using ⟨⟨n, by simp⟩, fun a ha ↦ hu ha⟩
    · simp [IsLeast, hu n.le_succ]

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
n : ℕ
u : ℕ → β
hu : Monotone u
ut : ∀ (i : ℕ), u i ∈ t
x : ℕ
hx : x ∈ Finset.range (n, ⟨u, ⋯⟩).1
⊢ edist ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 ((n, ⟨u, ⋯⟩).1 - 1 - x + 1))) ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 ((n, ⟨u, ⋯⟩).1 - 1 - x))) =
    edist (f (↑(n, ⟨fun i ↦ φ (u (n - i)), ⋯⟩).2 (x + 1))) (f (↑(n, ⟨fun i ↦ φ (u (n - i)), ⋯⟩).2 x))

Tactic:
Finset.mem_range

Goal state after:
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
n : ℕ
u : ℕ → β
hu : Monotone u
ut : ∀ (i : ℕ), u i ∈ t
x : ℕ
hx : x < (n, ⟨u, ⋯⟩).1
⊢ edist ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 ((n, ⟨u, ⋯⟩).1 - 1 - x + 1))) ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 ((n, ⟨u, ⋯⟩).1 - 1 - x))) =
    edist (f (↑(n, ⟨fun i ↦ φ (u (n - i)), ⋯⟩).2 (x + 1))) (f (↑(n, ⟨fun i ↦ φ (u (n - i)), ⋯⟩).2 x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: 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)
⊢ cfc (fun x ↦ s • f x) a = s • cfc f a

Tactic:
by_cases ha : p a
  · rw [cfc_apply f a, cfc_apply _ a]
    simp_rw [← Pi.smul_def, ← smul_one_smul R s _]
    rw [← map_smul]
    congr
  · simp [cfc_apply_of_not_predicate a ha]

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
hf : BoundedVariationOn f s
x : α
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s] x) (𝓝 0)

Tactic:
have A : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0) := by
    simp_rw [← Iio_inter_Ici, ← inter_assoc]
    exact (hf.mono inter_subset_left).tendsto_eVariationOn_Ici_zero_of_filter (𝓝[s ∩ Iio x] x)
      (fun y hy ↦ inter_mem_nhdsWithin _ (Ici_mem_nhds hy.2))
  have B : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Ici x] x) (𝓝 0) := by
    apply tendsto_const_nhds.congr'
    filter_upwards [self_mem_nhdsWithin] with a ha using by simp [show Ico a x = ∅ by grind]
  nth_rewrite 2 [show s = (s ∩ Iio x) ∪ (s ∩ Ici x) by grind]
  rw [nhdsWithin_union, tendsto_sup]
  simp [A, B]

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
hf : BoundedVariationOn f s
x : α
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ioc x y)) (𝓝[s] x) (𝓝 0)

Tactic:
have : (fun y ↦ eVariationOn f (s ∩ Ioc x y)) =
      (fun y ↦ eVariationOn (f ∘ ofDual) (ofDual ⁻¹' s ∩ Ico (toDual y) (toDual x))) := by
    ext y
    rw [Ico_toDual, ← preimage_inter, comp_ofDual]

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 (s ∩ Ioc x y)) (𝓝[s] x) (𝓝 0)