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

Goal state before:
pp: case refine_1.inr
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}
x : α
xs : x ∈ s
ε : ℝ
εpos : ε > 0
a : Set α
ha : a ∈ f x
h'a : a ⊆ closedBall x ε
a_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
a_int : (interior a).Nonempty
r : ℝ
ar : a ⊆ closedBall x r
μr : μ (closedBall x (3 * r)) ≤ ↑C * μ a
h' : min r ε = ε ∧ ε < r
⊢ (min r ε, x, a).2.2 ⊆ closedBall (min r ε, x, a).2.1 (min r ε, x, a).1

Tactic:
rwa [h'.1]

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
c d : ℝ≥0
hcd : c < d
s : Set α
hc : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a
hd : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a
x : α
x✝ : x ∈ s
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : μ o < ∞
s' : Set α := s ∩ o
h : ¬μ (s ∩ o) = 0
⊢ ↑c * μ s' < ↑d * μ s'

Tactic:
by gcongr; exact measure_ne_top_of_subset inter_subset_right μo.ne

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

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this✝ : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
hξε : (ξ + E.indicator fun x ↦ ↑ε) ∈ measurableLE ν μ
this : ∫⁻ (a : α), ξ a + E.indicator (fun x ↦ ↑ε) a ∂ν ≤ sSup (measurableLEEval ν μ)
⊢ False

Tactic:
refine not_lt.2 this ?_

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this✝ : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
hξε : (ξ + E.indicator fun x ↦ ↑ε) ∈ measurableLE ν μ
this : ∫⁻ (a : α), ξ a + E.indicator (fun x ↦ ↑ε) a ∂ν ≤ sSup (measurableLEEval ν μ)
⊢ sSup (measurableLEEval ν μ) < ∫⁻ (a : α), ξ a + E.indicator (fun x ↦ ↑ε) a ∂ν
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
s : Set ℝ
hf : MonotoneOn f s
hs : MeasurableSet s
⊢ ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

Tactic:
exact hf.ae_differentiableWithinAt_of_mem

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 c b

Tactic:
by
  rw [← variationOnFromTo.add hf ha hc hb, add_sub_cancel_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
⊢ eVariationOn f s = ⨆ p, ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))

Tactic:
apply le_antisymm

Goal state after:
case a
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))

case a
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
⊢ ⨆ p, ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i)) ≤ eVariationOn f s
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:
measurability

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

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 y x)).toReal = variationOnFromTo f (φ '' t) (φ x) (φ y)

Tactic:
variationOnFromTo.eq_of_ge _ _ (hφ hy hx h),

Goal state after:
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 y x)).toReal = -(eVariationOn f (φ '' t ∩ Icc (φ y) (φ x))).toReal
Module: Mathlib.Analysis.Complex.Spectrum

Goal state before:
pp: A : Type u_1
inst✝³ : NonUnitalRing A
inst✝² : Module ℂ A
inst✝¹ : IsScalarTower ℂ A A
inst✝ : SMulCommClass ℂ A A
a : A
⊢ SpectrumRestricts ↑a ⇑Complex.reCLM ↔ ∀ x ∈ σₙ ℂ a, x = ↑x.re

Tactic:
Unitization.quasispectrum_eq_spectrum_inr' _ ℂ,

Goal state after:
A : Type u_1
inst✝³ : NonUnitalRing A
inst✝² : Module ℂ A
inst✝¹ : IsScalarTower ℂ A A
inst✝ : SMulCommClass ℂ A A
a : A
⊢ SpectrumRestricts ↑a ⇑Complex.reCLM ↔ ∀ x ∈ spectrum ℂ ↑a, x = ↑x.re
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : PartialOrder R
inst✝⁶ : NonUnitalSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : StarOrderedRing R
inst✝³ : TopologicalSpace R
inst✝² : ContinuousStar R
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousSqrt R
⊢ ∀ (x y : C(α, R)), x ≤ y ↔ ∃ s, y = x + star s * s

Tactic:
intro f g

Goal state after:
α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : PartialOrder R
inst✝⁶ : NonUnitalSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : StarOrderedRing R
inst✝³ : TopologicalSpace R
inst✝² : ContinuousStar R
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousSqrt R
f g : C(α, R)
⊢ f ≤ g ↔ ∃ s, g = f + star s * 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
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))

Tactic:
have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)

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)
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case mp
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
u : ℕ → α := ⋯
x✝ : α
⊢ (∃ y, (s.orderEmbOfFin ⋯) y = x✝) → ∃ x ≤ s.card - 1, (s.orderEmbOfFin ⋯) ⟨min x (s.card - 1), ⋯⟩ = x✝

pp: case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
u : ℕ → α := ⋯
x✝ : α
⊢ (∃ x ≤ s.card - 1, (s.orderEmbOfFin ⋯) ⟨min x (s.card - 1), ⋯⟩ = x✝) → ∃ y, (s.orderEmbOfFin ⋯) y = x✝

Tactic:
· rintro ⟨i, rfl⟩; exact ⟨i.val, by grind⟩

Goal state after:
case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
this : 0 < s.card
u : ℕ → α := ⋯
x✝ : α
⊢ (∃ x ≤ s.card - 1, (s.orderEmbOfFin ⋯) ⟨min x (s.card - 1), ⋯⟩ = x✝) → ∃ y, (s.orderEmbOfFin ⋯) y = x✝
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case e'_3
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 : x ≤ y
⊢ φ '' t ∩ Icc (φ x) (φ y) = φ '' (t ∩ Icc x y)

Tactic:
apply le_antisymm

Goal state after:
case e'_3.a
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 : x ≤ y
⊢ φ '' t ∩ Icc (φ x) (φ y) ⊆ φ '' (t ∩ Icc x y)

case a
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 : x ≤ y
⊢ φ '' (t ∩ Icc x y) ⊆ φ '' t ∩ Icc (φ x) (φ y)
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
a : A
g : R → R
ha : p a
f : C(↑(spectrum R a), R)
h : ⇑f = (spectrum R a).domRestrict (Function.extend Subtype.val (⇑f) g)
hg : ContinuousOn (Function.extend Subtype.val (⇑f) g) (spectrum R a)
⊢ (cfcHom ha) f =
    (cfcHom ha) { toFun := (spectrum R a).domRestrict (Function.extend Subtype.val (⇑f) g), continuous_toFun := ⋯ }

Tactic:
congr!

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
x : α
hx :
  ∀ (i : ℕ),
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u i).indicator f y - (u i).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
n : ℕ
hn : x ∈ u n
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)

Tactic:
apply Tendsto.congr' _ (hx n)

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
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
x : α
hx :
  ∀ (i : ℕ),
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u i).indicator f y - (u i).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
n : ℕ
hn : x ∈ u n
⊢ (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) =ᶠ[v.filterAt x] fun a ↦
    (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a
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
⊢ Commute (cfc f a) (cfc g a)

Tactic:
refine cfc_cases (fun x ↦ Commute x (cfc g a)) a f (by simp) fun hf ha ↦ ?_

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
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ Commute ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }) (cfc g a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
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)
B : ∀ y ∈ s, ∃ y' ∈ s ∩ Ici y, δ ≤ eVariationOn f (s ∩ Icc y y')
⊢ False

Tactic:
choose! y y_mem le_y using B

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 : 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))
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : MetricSpace R
inst✝¹¹ : IsTopologicalSemiring R
inst✝¹⁰ : ContinuousStar R
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝⁵ : UniqueHom R A
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 ((fun x ↦ s • x) '' spectrum R a)
ha : p a
⊢ cfc (fun x ↦ f (s • x)) a = cfc f (s • a)

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

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

Goal state before:
pp: case pos
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) ∧ g 0 = 0
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a

Tactic:
obtain ⟨hg, hg0⟩ := hg

Goal state after:
case pos
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
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
f' : R → R
h : ⇑f = (σₙ R a).domRestrict f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
⊢ (cfcₙHom ha) f = (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f', continuous_toFun := ⋯, map_zero' := ⋯ }

Tactic:
congr!

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

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)

Tactic:
rw [cfcₙ_apply_of_not_map_zero a f_zero, mkD_of_not_zero, map_zero]

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ (σₙ R a).domRestrict f 0 ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁹ : CommSemiring R
inst✝¹⁸ : PartialOrder R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : ContinuousSqrt R
inst✝¹¹ : StarOrderedRing R
inst✝¹⁰ : NoZeroDivisors R
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (∀ (a_1 : R) (b : a_1 ∈ σₙ R a),
      0 ⟨a_1, b⟩ ≤ { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } ⟨a_1, b⟩) ↔
    ∀ x ∈ σₙ R a, 0 ≤ f x

Tactic:
congr!

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

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ σₙ R a, 0 ≤ f x
h₁ : ContinuousOn f (σₙ R a)
h₂ : f 0 = 0
⊢ 0 ≤ cfcₙ f a

Tactic:
simpa using cfcₙ_mono h

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

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
h₁ : ContinuousOn f (σₙ R a)
h₂ : f 0 = 0
⊢ cfcₙ f a ≤ 0

Tactic:
simpa using cfcₙ_mono h

Goal state after: