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
r : ℝ≥0
⊢ (r • μ).singularPart ν = r • μ.singularPart ν

Tactic:
by_cases hr : r = 0
  · rw [hr, zero_smul, zero_smul, singularPart_zero]
  by_cases hl : HaveLebesgueDecomposition μ ν
  · refine (eq_singularPart ((measurable_rnDeriv μ ν).const_smul (r : ℝ≥0∞))
          (MutuallySingular.smul r (mutuallySingular_singularPart _ _)) ?_).symm
    rw [withDensity_smul _ (measurable_rnDeriv _ _), ← smul_add,
      ← haveLebesgueDecomposition_add μ ν, ENNReal.smul_def]
  · rw [singularPart, singularPart, dif_neg hl, dif_neg, smul_zero]
    refine fun hl' ↦ hl ?_
    rw [← inv_smul_smul₀ hr μ]
    infer_instance

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 : μ.HaveLebesgueDecomposition ν
inst✝ : IsFiniteMeasure μ
⊢ μ - μ.singularPart ν = ν.withDensity (μ.rnDeriv ν)

Tactic:
nth_rw 1 [← rnDeriv_add_singularPart μ ν]
  exact Measure.add_sub_cancel

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

Goal state before:
pp: α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
R : ℝ := scalingScaleOf μ (max (4 * K + 3) 3)
Rpos : 0 < R
A :
  ∀ (x : α),
    ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall x r)
⊢ VitaliFamily μ

Tactic:
exact (Vitali.vitaliFamily μ (scalingConstantOf μ (max (4 * K + 3) 3)) A).enlarge (R / 4)
    (by linarith)

Goal state after:
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
C : ℝ≥0
h : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)
s : Set α
f : α → Set (Set α)
fsubset :
  ∀ x ∈ s,
    f x ⊆
      {a |
        IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧
          (interior a).Nonempty ∧ ∃ r, a ⊆ closedBall x r ∧ μ (closedBall x (3 * r)) ≤ ↑C * μ a}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
t : Set (ℝ × α × Set α) :=
  {p |
    p.2.2 ⊆ closedBall p.2.1 p.1 ∧
      μ (closedBall p.2.1 (3 * p.1)) ≤ ↑C * μ p.2.2 ∧
        (interior p.2.2).Nonempty ∧
          IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] p.2.2 ∧ p.2.2 ∈ f p.2.1 ∧ p.2.1 ∈ s}
A : ∀ x ∈ s, ∀ ε > 0, ∃ p ∈ t, p.1 ≤ ε ∧ p.2.1 = x
t' : Set (ℝ × α × Set α)
t't : t' ⊆ t
left✝ : t'.Countable
t'_disj : t'.PairwiseDisjoint fun p ↦ p.2.2
μt' : μ (s \ ⋃ a ∈ t', a.2.2) = 0
⊢ ⋃ p ∈ (fun p ↦ p.2) '' t', p.2 = ⋃ a ∈ t', a.2.2

Tactic:
biUnion_image

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
C : ℝ≥0
h : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)
s : Set α
f : α → Set (Set α)
fsubset :
  ∀ x ∈ s,
    f x ⊆
      {a |
        IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧
          (interior a).Nonempty ∧ ∃ r, a ⊆ closedBall x r ∧ μ (closedBall x (3 * r)) ≤ ↑C * μ a}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
t : Set (ℝ × α × Set α) :=
  {p |
    p.2.2 ⊆ closedBall p.2.1 p.1 ∧
      μ (closedBall p.2.1 (3 * p.1)) ≤ ↑C * μ p.2.2 ∧
        (interior p.2.2).Nonempty ∧
          IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] p.2.2 ∧ p.2.2 ∈ f p.2.1 ∧ p.2.1 ∈ s}
A : ∀ x ∈ s, ∀ ε > 0, ∃ p ∈ t, p.1 ≤ ε ∧ p.2.1 = x
t' : Set (ℝ × α × Set α)
t't : t' ⊆ t
left✝ : t'.Countable
t'_disj : t'.PairwiseDisjoint fun p ↦ p.2.2
μt' : μ (s \ ⋃ a ∈ t', a.2.2) = 0
⊢ ⋃ y ∈ t', y.2.2 = ⋃ a ∈ t', a.2.2
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
x : α
x✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : ρ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = ∞} ∩ o
⊢ μ s ≤ 0

Tactic:
have ρs : ρ s ≠ ∞ := measure_ne_top_of_subset inter_subset_right μo.ne

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
x : α
x✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : ρ o < ∞
s : Set α := {x | v.limRatioMeas hρ x = ∞} ∩ o
ρs : ρ s ≠ ∞
⊢ μ s ≤ 0
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:
wlog hμ : IsFiniteMeasure μ generalizing μ

Goal state after:
case inr
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SFinite μ
inst✝ : SigmaFinite ν
this : ∀ (μ : Measure α) [SFinite μ], IsFiniteMeasure μ → μ.HaveLebesgueDecomposition ν
hμ : ¬IsFiniteMeasure μ
⊢ μ.HaveLebesgueDecomposition ν

α : Type u_1
m : MeasurableSpace α
ν : Measure α
inst✝¹ : SigmaFinite ν
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
⊢ μ.HaveLebesgueDecomposition ν
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: case inr.refine_4
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
⊢ ∀ x ∈ s,
    ∀ y ∈ s,
      (fun x ↦ (variationOnFromTo f s c x + f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x + f x) / 2) x +
          ((fun x ↦ (variationOnFromTo f s c x - f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x - f x) / 2) x) =
        variationOnFromTo f s x y

Tactic:
· intro x hx y hy
    rw [← variationOnFromTo.add h hx cs hy, variationOnFromTo.eq_neg_swap]
    ring

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ ∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i)) +
      ∑ i ∈ Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) ≤
    ∑ i ∈ Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i))

Tactic:
← Finset.sum_union

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ ∑ x ∈ Finset.range n ∪ Finset.Ico (n + 1) (n + 1 + m), edist (f (w (x + 1))) (f (w x)) ≤
    ∑ i ∈ Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i))

α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ Disjoint (Finset.range n) (Finset.Ico (n + 1) (n + 1 + m))
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case e'_5
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✝
Cx : Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝))
⊢ 𝓝 (ρ.rnDeriv μ a✝) = 𝓝 (0 + t.rnDeriv μ a✝)

Tactic:
· simp only [Bx, zero_add]

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
s : Set α
t : Set α := toMeasurable μ s
A : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
B : ∀ᵐ (x : α) ∂μ.restrict s, t.indicator 1 x = 1
⊢ ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 1)

Tactic:
filter_upwards [A, B] with x hx h'x

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
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set α := toMeasurable μ s
A : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
B : ∀ᵐ (x : α) ∂μ.restrict s, t.indicator 1 x = 1
x : α
hx : Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
h'x : t.indicator 1 x = 1
⊢ Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 1)
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
⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 (i + 1))) ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 i)) ≤
    eVariationOn f s

Tactic:
← Finset.sum_range_reflect

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
⊢ ∑ j ∈ Finset.range (n, ⟨u, ⋯⟩).1,
      edist ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 ((n, ⟨u, ⋯⟩).1 - 1 - j + 1))) ((f ∘ φ) (↑(n, ⟨u, ⋯⟩).2 ((n, ⟨u, ⋯⟩).1 - 1 - j))) ≤
    eVariationOn f s
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
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
h : y ≤ x
⊢ eVariationOn (f ∘ φ) (t ∩ Icc x y) = eVariationOn f (φ '' t ∩ Icc (φ x) (φ y))

Tactic:
rw [eVariationOn.subsingleton, eVariationOn.subsingleton]

Goal state after:
case inr.hs
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
h : y ≤ x
⊢ (φ '' t ∩ Icc (φ x) (φ y)).Subsingleton

case inr.hs
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
h : y ≤ x
⊢ (t ∩ Icc x y).Subsingleton
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
ha : p a
⊢ cfc f a = (cfcHom ha) (mkD ((spectrum R a).domRestrict f) 0)

Tactic:
by_cases hf : 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
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ cfc f a = (cfcHom ha) (mkD ((spectrum R a).domRestrict f) 0)

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
ha : p a
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a = (cfcHom ha) (mkD ((spectrum R a).domRestrict f) 0)
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ∞
h''a : IntegrableOn f a μ
A : IntegrableOn (fun y ↦ ↑‖f y - f x‖₊) a μ
⊢ (μ a).toReal⁻¹ * (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ).toReal = (μ.real a)⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ

Tactic:
dsimp [enorm]

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ∞
h''a : IntegrableOn f a μ
A : IntegrableOn (fun y ↦ ↑‖f y - f x‖₊) a μ
⊢ (μ a).toReal⁻¹ * (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ).toReal = (μ.real a)⁻¹ * ∫ (y : α) in a, ‖f y - f x‖ ∂μ
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✝ g : R → R
a✝ : A
ha : p a✝
hf : ContinuousOn f✝ (spectrum R a✝)
hg : ContinuousOn g (spectrum R a✝)
f : R → R
a : A
⊢ SemiconjBy (star (cfc f a)) (cfc f a) (cfc f a)

Tactic:
SemiconjBy

Goal state after:
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
ha : p a✝
hf : ContinuousOn f✝ (spectrum R a✝)
hg : ContinuousOn g (spectrum R a✝)
f : R → R
a : A
⊢ star (cfc f a) * cfc f a = cfc f a * star (cfc f a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: 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
s : Set α
hf : BoundedVariationOn f s
x : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : x ≤ y
⊢ eVariationOn f (s ∩ Ico y x) = eVariationOn f (s ∩ Icc y x)

Tactic:
rw [eVariationOn.subsingleton, eVariationOn.subsingleton]

Goal state after:
case inl.hs
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
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : x ≤ y
⊢ (s ∩ Icc y x).Subsingleton

case inl.hs
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
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : x ≤ y
⊢ (s ∩ Ico y x).Subsingleton
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ -f x) a = -cfc f a

Tactic:
obtain (ha | hf) := not_and_or.mp h
    · simp [cfc_apply_of_not_predicate a ha]
    · rw [cfc_apply_of_not_continuousOn a hf, cfc_apply_of_not_continuousOn, neg_zero]
      exact fun hf_neg ↦ hf <| by simpa using hf_neg.fun_neg

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✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
⊢ 0 ≤ cfc f a

Tactic:
by_cases hf : ContinuousOn f (spectrum R a)

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ 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✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
⊢ cfc f a ≤ 0

Tactic:
by_cases hf : ContinuousOn f (spectrum R a)

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
a b : α
hf : MonotoneOn f s
as : a ∈ s
bs : b ∈ s
⊢ eVariationOn f (s ∩ Set.Icc a b) = ENNReal.ofReal (f b - f a)

Tactic:
by
  rcases le_or_gt a b with hab | hab
  · have hle : eVariationOn f (s ∩ Icc a b) ≤ .ofReal (f b - f a) := by
      apply iSup_le _
      rintro ⟨n, ⟨u, hu, us⟩⟩
      calc
        _ = ∑ i ∈ range n, .ofReal (f (u (i + 1)) - f (u i)) := by
          refine sum_congr rfl fun i hi => ?_
          simp only [Finset.mem_range] at hi
          rw [edist_dist, Real.dist_eq, abs_of_nonneg]
          exact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))
        _ = .ofReal (∑ i ∈ range n, (f (u (i + 1)) - f (u i))) := by
          rw [ofReal_sum_of_nonneg]
          exact fun i _ ↦ sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))
        _ = .ofReal (f (u n) - f (u 0)) := by rw [sum_range_sub (f <| u ·)]
        _ ≤ _ :=
          ofReal_le_ofReal (sub_le_sub (hf (us n).1 bs (us n).2.2) (hf as (us 0).1 (us 0).2.1))
    have h : BoundedVariationOn f (s ∩ Icc a b) := (hle.trans_lt ofReal_lt_top).ne
    apply eq_of_le_of_ge hle (ofReal_le_of_le_toReal _)
    grw [← h.dist_le (x := a) (y := b)] <;> grind [Real.dist_eq]
  · simp [hab, hf bs as hab.le]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Polynomial

Goal state before:
pp: a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I) ↔
    f ∈ polynomialFunctions (Set.Icc a b)

Tactic:
fconstructor

Goal state after:
case mp
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I) →
    f ∈ polynomialFunctions (Set.Icc a b)

case mpr
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ polynomialFunctions (Set.Icc a b) →
    f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I)
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ
n k : ℕ
ih : k ≤ n + 1 → LinearIndependent ℚ fun ν ↦ bernsteinPolynomial ℚ n ↑ν
h : k + 1 ≤ n + 1
⊢ LinearIndependent ℚ (Fin.init fun ν ↦ bernsteinPolynomial ℚ n ↑ν) ∧
    bernsteinPolynomial ℚ n ↑(Fin.last k) ∉ span ℚ (Set.range (Fin.init fun ν ↦ bernsteinPolynomial ℚ n ↑ν))

Tactic:
fconstructor

Goal state after:
case succ.left
n k : ℕ
ih : k ≤ n + 1 → LinearIndependent ℚ fun ν ↦ bernsteinPolynomial ℚ n ↑ν
h : k + 1 ≤ n + 1
⊢ LinearIndependent ℚ (Fin.init fun ν ↦ bernsteinPolynomial ℚ n ↑ν)

case succ.right
n k : ℕ
ih : k ≤ n + 1 → LinearIndependent ℚ fun ν ↦ bernsteinPolynomial ℚ n ↑ν
h : k + 1 ≤ n + 1
⊢ bernsteinPolynomial ℚ n ↑(Fin.last k) ∉ span ℚ (Set.range (Fin.init fun ν ↦ bernsteinPolynomial ℚ n ↑ν))