mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ

Tactic:
by
  rw [← withDensity_eq_iff_of_sigmaFinite]
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    suffices (r • ν).singularPart μ + withDensity μ (rnDeriv (r • ν) μ)
        = (r • ν).singularPart μ + r • withDensity μ (rnDeriv ν μ) by
      rwa [Measure.add_right_inj] at this
    rw [← (r • ν).haveLebesgueDecomposition_add μ, singularPart_smul, ← smul_add,
      ← ν.haveLebesgueDecomposition_add μ]
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _

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

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
h : ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
⊢ ∀ (i : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }),
    ∑ i_1 ∈ Finset.range i.1, edist (f (↑i.2 (i_1 + 1))) (f (↑i.2 i_1)) = 0

Tactic:
rintro ⟨n, u, um, us⟩

Goal state after:
case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
h : ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
n : ℕ
u : ℕ → α
um : Monotone u
us : ∀ (i : ℕ), u i ∈ s
⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) = 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ

Tactic:
refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ∞
⊢ ¬r.toNNReal = 0

Tactic:
ENNReal.toNNReal_eq_zero_iff

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ∞
⊢ ¬(r = 0 ∨ r = ∞)
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 := ⋯
ρs : ρ s ≠ ∞
A : ∀ (q : ℝ≥0), 1 ≤ q → μ s ≤ (↑q)⁻¹ * ρ s
B : Tendsto (fun q ↦ (↑q)⁻¹ * ρ s) atTop (𝓝 0)
⊢ ∀ᶠ (c : ℝ≥0) in atTop, μ s ≤ (↑c)⁻¹ * ρ s

Tactic:
exact eventually_atTop.2 ⟨1, A⟩

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

Goal state before:
pp: case mp
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
f g : C(α, R)₀
⊢ f ≤ g → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p

Tactic:
le_def,

Goal state after:
case mp
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
f g : C(α, R)₀
⊢ (∀ (x : α), f x ≤ g x) → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case refine_1
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
this : Tendsto (fun t ↦ ↑t ^ 2 * ρ s) (𝓝[>] 1) (𝓝 (ρ s))
⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 1, (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑c ^ 2 * ρ s

Tactic:
filter_upwards [self_mem_nhdsWithin] with _ ht

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
this : Tendsto (fun t ↦ ↑t ^ 2 * ρ s) (𝓝[>] 1) (𝓝 (ρ s))
a✝ : ℝ≥0
ht : a✝ ∈ Ioi 1
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑a✝ ^ 2 * ρ s
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case refine_2
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
this : Tendsto (fun t ↦ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s) (𝓝[>] 1) (𝓝 ((μ.withDensity (v.limRatioMeas hρ)) s))
⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 1, ρ s ≤ ↑c * (μ.withDensity (v.limRatioMeas hρ)) s

Tactic:
filter_upwards [self_mem_nhdsWithin] with _ ht

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
this : Tendsto (fun t ↦ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s) (𝓝[>] 1) (𝓝 ((μ.withDensity (v.limRatioMeas hρ)) s))
a✝ : ℝ≥0
ht : a✝ ∈ Ioi 1
⊢ ρ s ≤ ↑a✝ * (μ.withDensity (v.limRatioMeas hρ)) s
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
t : Set β
φ : β → α
hφ : AntitoneOn φ t
⊢ eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)

Tactic:
by
  apply le_antisymm (comp_le_of_antitoneOn f φ hφ (mapsTo_image φ t))
  cases isEmpty_or_nonempty β
  · simp [Set.eq_empty_of_isEmpty]
  let ψ := φ.invFunOn t
  have ψφs : EqOn (φ ∘ ψ) id (φ '' t) := (surjOn_image φ t).rightInvOn_invFunOn
  have ψts := (surjOn_image φ t).mapsTo_invFunOn
  have hψ : AntitoneOn ψ (φ '' t) := Function.antitoneOn_of_rightInvOn_of_mapsTo hφ ψφs ψts
  change eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t
  rw [← eq_of_eqOn (ψφs.comp_left : EqOn (f ∘ φ ∘ ψ) (f ∘ id) (φ '' t))]
  exact comp_le_of_antitoneOn _ ψ hψ ψts

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

Goal state before:
pp: case h₂
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 : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} := ⋯
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
c : E
ct : c ∈ t
xc : ‖f x - c‖ₑ < ε / 2
a : Set α
ha : (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a < ε / 2
h'a : 0 < μ a
h''a : μ a < ∞
⊢ ∫⁻ (x_1 : α) in a, ‖f x - c‖ₑ ∂μ ≤ ε / 2 * μ a

Tactic:
simp only [lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter]
        gcongr

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)
b : α
hb : b ∈ s ∩ Iio a
this✝¹ : Nonempty ((n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s ∩ Iio a })
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s ∩ Iio a
this✝ :
  Tendsto (fun y ↦ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y)) (𝓝[s ∩ Iio a] a)
    (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) l))
this : s ∩ Ioo (u n) a ∈ 𝓝[s ∩ Iio a] a
y : α
hy : y ∈ s ∩ Ioo (u n) a
h'y : y ∈ s ∩ Iio a
v : ℕ → α := ⋯
⊢ edist (f a) (f y) ≤ edist (f (v (n + 1))) (f (v n)) + edist (f (v (n + (1 + 1)))) (f (v (n + 1)))

Tactic:
exact le_add_left (by grind)

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

Goal state before:
pp: case e_a
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
a : Set α
ha : a ⊆ u n
h'a : MeasurableSet a
y : α
hy : y ∈ a
⊢ ‖(u n).indicator f y - (u n).indicator f x‖ₑ = ‖f y - f x‖ₑ

Tactic:
indicator_of_mem (ha hy) f,

Goal state after:
case e_a
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
a : Set α
ha : a ⊆ u n
h'a : MeasurableSet a
y : α
hy : y ∈ a
⊢ ‖f y - (u n).indicator f x‖ₑ = ‖f y - f x‖ₑ
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)
⊢ Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)

Tactic:
have := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hx

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 (ENNReal.toReal 0))
⊢ Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: 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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (spectrum R a)
⊢ cfc (∑ i, f ↑i) a = ∑ i ∈ s, cfc (f i) a

Tactic:
← sum_coe_sort s

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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (spectrum R a)
⊢ cfc (∑ i, f ↑i) a = ∑ i, cfc (f ↑i) a
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 : CompleteSpace E
inst✝ : DenselyOrdered α
f : α → E
a : α
hf : BoundedVariationOn f univ
this✝¹ : TopologicalSpace α := Preorder.topology α
this✝ : OrderTopology α
ha : ¬IsBot a
this : (𝓝[<] a).NeBot
⊢ (𝓝[univ ∩ Iio a] a).NeBot

Tactic:
by simpa

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
ha : p a
inst✝ : UniqueHom R A
φ : C(↑(σₙ R a), R)₀ →⋆ₙₐ[R] A
hφ₁ : Continuous[_, inst✝⁴] ⇑φ
hφ₂ : φ (ContinuousMapZero.id (σₙ R a)) = a
⊢ (cfcₙHom ha) (ContinuousMapZero.id (σₙ R a)) = φ (ContinuousMapZero.id (σₙ R a))

Tactic:
rw [cfcₙHom_id ha, hφ₂]

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

Goal state before:
pp: case neg
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 | hg) := hg

Goal state after:
case neg.inl
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)
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a

case neg.inr
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 : ¬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: case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (σₙ R a)
⊢ cfcₙ (∑ i, f ↑i) a = ∑ i ∈ s, cfcₙ (f i) a

Tactic:
← sum_coe_sort s

Goal state after:
case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (σₙ R a)
⊢ cfcₙ (∑ i, f ↑i) a = ∑ i, cfcₙ (f ↑i) a
Module: Mathlib.Topology.ContinuousMap.Polynomial

Goal state before:
pp: case mp
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), (↑(toContinuousMapOnAlgHom I) p) x = (↑(compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) f) x
⊢ f ∈ polynomialFunctions (Set.Icc a b)

Tactic:
dsimp at w

Goal state after:
case mp
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
⊢ f ∈ polynomialFunctions (Set.Icc a b)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
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 g : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ g x
hf : ContinuousOn f (σₙ R a)
hg : ContinuousOn g (σₙ R a)
hf0 : f 0 = 0
hg0 : g 0 = 0
ha : ¬p a
⊢ cfcₙ f a ≤ cfcₙ g a

Tactic:
· simp only [cfcₙ_apply_of_not_predicate _ ha, le_rfl]

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

Goal state before:
pp: case mpr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : Semifield R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
x : ↑(σₙ R a)
hx : ↑x = 0
⊢ f x = 0

Tactic:
rw [show x = 0 from Subtype.val_injective hx, map_zero]

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
k : 𝕜
x : Unitization 𝕜 A
⊢ ‖(splitMul 𝕜 A) (k • x)‖ ≤ ‖k‖ * ‖x‖

Tactic:
map_smul,

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
k : 𝕜
x : Unitization 𝕜 A
⊢ ‖k • (splitMul 𝕜 A) x‖ ≤ ‖k‖ * ‖x‖
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case pos.right
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := fun x y ↦ {z | f z - ε < (g x y) z}
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := fun x ↦ ⋯.choose
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ :=
  fun x ↦ Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhds_y x))
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := fun x ↦ ⟨(ys x).sup' ⋯ fun y ↦ g x y, ⋯⟩
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := fun x ↦ {z | ↑(h x) z < f z + ε}
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯.choose
xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhds)
xs_nonempty : xs.Nonempty
k : ↑L := ⟨xs.inf' xs_nonempty fun x ↦ ↑(h x), ⋯⟩
z x : X
⊢ f z - ε < ↑(h x) z

Tactic:
apply lt_h

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
r : 𝕜
x : WithLp 1 (Unitization 𝕜 A)
⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖

Tactic:
by
    simp_rw [unitization_norm_def, ofLp_smul, fst_smul, snd_smul, norm_smul, mul_add]
    exact le_rfl

Goal state after: