mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
⊢ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)

Tactic:
intro z hz

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
⊢ z ∈ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_le : μ.restrict s ≤ ν.restrict s
h_meas_s : MeasurableSet s
⊢ (μ - ν) s = 0

Tactic:
← restrict_apply_self,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_le : μ.restrict s ≤ ν.restrict s
h_meas_s : MeasurableSet s
⊢ ((μ - ν).restrict s) s = 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
heq : s.restrict (S ∩ T)ᶜ = (μ.singularPart ν).restrict (S ∩ T)ᶜ
heq' : ∀ (A : Set α), MeasurableSet A → s A = (s.restrict (S ∩ T)ᶜ) A
A : Set α
hA : MeasurableSet A
⊢ s A = (μ.singularPart ν) A

Tactic:
have hμinter : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by
    rw [← nonpos_iff_eq_zero]
    exact hT₂ ▸ measure_mono (inter_subset_right.trans inter_subset_right)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
heq : s.restrict (S ∩ T)ᶜ = (μ.singularPart ν).restrict (S ∩ T)ᶜ
heq' : ∀ (A : Set α), MeasurableSet A → s A = (s.restrict (S ∩ T)ᶜ) A
A : Set α
hA : MeasurableSet A
hμinter : (μ.singularPart ν) (A ∩ (S ∩ T)) = 0
⊢ s A = (μ.singularPart ν) A
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
s : Set α
hs : MeasurableSet s
⊢ μ.restrict s = (μ.singularPart ν).restrict s + ν.withDensity (s.indicator (μ.rnDeriv ν))

Tactic:
ext t

Goal state after:
case refine_3
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
s : Set α
hs : MeasurableSet s
t : Set α
a✝ : MeasurableSet t
⊢ (μ.restrict s) t = ((μ.singularPart ν).restrict s + ν.withDensity (s.indicator (μ.rnDeriv ν))) t
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
ρ ν : Measure α
inst✝ : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a
ε : ℝ≥0
εpos : 0 < ε
x✝ : ν s < ∞
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : ν U ≤ ν s + ↑ε
f : α → Set (Set α) := fun x ↦ {a | ρ a ≤ ν a ∧ a ⊆ U}
⊢ ρ s ≤ ν s + ↑ε

Tactic:
have h : v.FineSubfamilyOn f s := by
    apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_
    have :=
      (hs x hx).and_eventually
        ((v.eventually_filterAt_mem_setsAt x).and
          (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))
    apply Frequently.mono this
    rintro a ⟨ρa, _, aU⟩
    exact ⟨ρa, aU⟩

Goal state after:
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
ρ ν : Measure α
inst✝ : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a
ε : ℝ≥0
εpos : 0 < ε
x✝ : ν s < ∞
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : ν U ≤ ν s + ↑ε
f : α → Set (Set α) := fun x ↦ {a | ρ a ≤ ν a ∧ a ⊆ U}
h : v.FineSubfamilyOn f s
⊢ ρ s ≤ ν s + ↑ε
Module: Mathlib.MeasureTheory.Covering.OneDim

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

pp: case refine_2
x : ℝ
⊢ ∀ ε > 0, ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε

Tactic:
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy

Goal state after:
case refine_2
x : ℝ
⊢ ∀ ε > 0, ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: case refine_2
x : ℝ
⊢ ∀ ε > 0, ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε

Tactic:
intro ε εpos
    filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy
    rw [closedBall_eq_Icc]
    exact Icc_subset_Icc hy.1 (by linarith)

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
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
A :
  (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ ∑' (m : ℕ) (n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0

Tactic:
simp only [H, tsum_zero]

Goal state after:
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
this :
  Tendsto (fun y ↦ (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
y : ℝ
hy : 0 < y - x
⊢ y + -1 * (y - x) ^ 2 < y

Tactic:
by nlinarith

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) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ w n)

Tactic:
rw [← coe_nnreal_smul_apply]

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) (toMeasurable (ρ + μ) (u m) ∩ w n) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ w n)
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 : ℝ≥0
s : Set α
h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}
t : Set α := {x | Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : μ tᶜ = 0
H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t)
⊢ (p • μ) (s ∩ t) + ρ tᶜ ≤ ↑p * μ (s ∩ t)

Tactic:
simp [(hρ A)]

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)))
⊢ μ.HaveLebesgueDecomposition ν

Tactic:
have hfm (n : ℕ) : Measurable (f n) := by measurability

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)
⊢ μ.HaveLebesgueDecomposition ν
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: case inr.refine_3
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
⊢ f = (fun x ↦ (variationOnFromTo f s c x + f x) / 2) - fun x ↦ (variationOnFromTo f s c x - f x) / 2

Tactic:
ext
    simp
    ring

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

Goal state before:
pp: case h₂
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 := ⋯
⊢ edist (f (u (n + 1))) (f (u n)) ≤ edist (f (v (n + 1))) (f (v n)) + edist (f (v (n + (1 + 1)))) (f (v (n + 1)))

Tactic:
grw [add_comm (edist _ _), ← edist_triangle]
          grind

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
a b : α
h : (𝓝[>] a).NeBot
h' : ContinuousWithinAt f (Ici a) a
⊢ eVariationOn f (Ioc a b) = eVariationOn f (Icc a b)

Tactic:
rcases le_or_gt b a with hab | hab

Goal state after:
case inl
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
a b : α
h : (𝓝[>] a).NeBot
h' : ContinuousWithinAt f (Ici a) a
hab : b ≤ a
⊢ eVariationOn f (Ioc a b) = eVariationOn f (Icc a b)

case inr
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
a b : α
h : (𝓝[>] a).NeBot
h' : ContinuousWithinAt f (Ici a) a
hab : a < b
⊢ eVariationOn f (Ioc a b) = eVariationOn f (Icc a b)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → E
a b : α
h' : ContinuousWithinAt f (Ici a) a
⊢ eVariationOn f (Ioc a b) = eVariationOn f (Icc a b)

Tactic:
rcases le_or_gt b a with hab | hab

Goal state after:
case inl
α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → E
a b : α
h' : ContinuousWithinAt f (Ici a) a
hab : b ≤ a
⊢ eVariationOn f (Ioc a b) = eVariationOn f (Icc a b)

case inr
α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
f : α → E
a b : α
h' : ContinuousWithinAt f (Ici a) a
hab : a < b
⊢ eVariationOn f (Ioc a b) = eVariationOn f (Icc a b)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
⊢ P (cfc f a)

Tactic:
by_cases h : p a ∧ ContinuousOn f (spectrum R a)

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ P (cfc f a)

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ P (cfc f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g a

Tactic:
rw [cfc_apply_of_not_continuousOn a hg, cfc_apply_of_not_continuousOn]
      exact fun hf ↦ hg (hf.congr hfg.symm)

Goal state after:
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
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 → α
y_mem : ∀ y_1 ∈ s, y y_1 ∈ s ∩ Ici y_1
le_y : ∀ y_1 ∈ s, δ ≤ eVariationOn f (s ∩ Icc y_1 (y y_1))
v : ℕ → α := fun n ↦ y^[n] x₀
n : ℕ
ih : v n ∈ s
⊢ y (y^[n] x₀) ∈ s

Tactic:
exact (y_mem _ ih).1

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✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
by_cases h : p a ∧ ContinuousOn f (spectrum R a)

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
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
inst✝ : UniqueHom R A
f : R → R
a : A
hf : ContinuousOn f (star '' spectrum R a)
ha : p a
⊢ cfc (fun x ↦ f (star x)) a = cfc f (star a)

Tactic:
rw [cfc_comp' .., cfc_star_id ..]

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

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
x : S
hx : x ∈ σₙ S a
⊢ Function.extend Subtype.val
      (⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
          ({ toFun := (σₙ R a).domRestrict g, continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ })))
      0 x =
    (algebraMap R S) (g (f x))

Tactic:
lift x to σₙ S a using hx

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
x : { x // x ∈ σₙ S a }
⊢ Function.extend Subtype.val
      (⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
          ({ toFun := (σₙ R a).domRestrict g, continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ })))
      0 ↑x =
    (algebraMap R S) (g (f ↑x))