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

Goal state before:
pp: case refine_2
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
A : Set α
hA : MeasurableSet A
⊢ μ A = (⇑(μ - ν.withDensity ξ, ξ).1 + ⇑(ν.withDensity (μ - ν.withDensity ξ, ξ).2)) A

Tactic:
Pi.add_apply,

Goal state after:
case refine_2
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
A : Set α
hA : MeasurableSet A
⊢ μ A = (μ - ν.withDensity ξ, ξ).1 A + (ν.withDensity (μ - ν.withDensity ξ, ξ).2) A
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ uIcc a b → ∀ ⦃y : α⦄, y ∈ s ∩ uIcc a b → edist (f x) (f y) = 0

Tactic:
rcases le_total a b with (ab | ba)
  · rw [uIcc_of_le ab]
    exact variationOnFromTo.eq_zero_iff_of_le hf ha hb ab
  · rw [uIcc_of_ge ba]
    exact variationOnFromTo.eq_zero_iff_of_ge hf ha hb ba

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
b : α
bs : b ∈ s
c : α
cs : c ∈ s
bc : b ≤ c
⊢ |f b - f c| ≤ |f c - f b|

Tactic:
abs_sub_comm (f b) (f c)

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
b : α
bs : b ∈ s
c : α
cs : c ∈ s
bc : b ≤ c
⊢ |f c - f b| ≤ |f c - f b|
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
⊢ variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y)

Tactic:
· rw [variationOnFromTo.eq_of_ge _ _ h, variationOnFromTo.eq_of_ge _ _ (hφ hy hx h),
      eVariationOn.comp_inter_Icc_eq_of_monotoneOn f φ hφ hy hx]

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
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
by
  wlog hxy : y ≤ x generalizing x y
  · rw [edist_comm]
    exact this hy hx (le_of_not_ge hxy)
  let u : ℕ → α := fun n => if n = 0 then y else x
  have hu : Monotone u := monotone_nat_of_le_succ fun
  | 0 => hxy
  | (_ + 1) => le_rfl
  have us : ∀ i, u i ∈ s := fun
  | 0 => hy
  | (_ + 1) => hx
  simpa only [Finset.sum_range_one] using! sum_le (n := 1) hu us

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

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

Tactic:
exact (measurable_rnDeriv _ _).aemeasurable

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

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

Tactic:
exact (measurable_rnDeriv _ _).aemeasurable

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Set α
h : BoundedVariationOn f s
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
exact edist_le f hx hy

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

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : u n ≤ x
⊢ ∃ v m,
    Monotone v ∧
      (∀ (i : ℕ), v i ∈ s) ∧
        x ∈ v '' Iio m ∧
          ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))

Tactic:
let v i := if i ≤ n then u i else x

Goal state after:
case inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : u n ≤ x
v : ℕ → α := fun i ↦ if i ≤ n then u i else x
⊢ ∃ v m,
    Monotone v ∧
      (∀ (i : ℕ), v i ∈ s) ∧
        x ∈ v '' Iio m ∧
          ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable ((ν₁ + ν₂).rnDeriv μ) μ

Tactic:
exact (measurable_rnDeriv _ _).aemeasurable

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

Goal state before:
pp: case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ

Tactic:
exact ((measurable_rnDeriv _ _).add (measurable_rnDeriv _ _)).aemeasurable

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

Goal state before:
pp: case inl
α : Type u_1
inst✝² : LinearOrder α
E : Type u_3
inst✝¹ : PseudoMetricSpace E
inst✝ : CompleteSpace E
f : α → E
a b : α
hf : BoundedVariationOn f univ
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
hb : 𝓝[<] b = ⊥
⊢ Function.leftLim (variationOnFromTo f univ a) b = variationOnFromTo f univ a b - dist (f b) (Function.leftLim f b)

pp: case inr
α : Type u_1
inst✝² : LinearOrder α
E : Type u_3
inst✝¹ : PseudoMetricSpace E
inst✝ : CompleteSpace E
f : α → E
a b : α
hf : BoundedVariationOn f univ
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
hb : (𝓝[<] b).NeBot
⊢ Function.leftLim (variationOnFromTo f univ a) b = variationOnFromTo f univ a b - dist (f b) (Function.leftLim f b)

Tactic:
· simp [leftLim_eq_of_eq_bot _ hb]

Goal state after:
case inr
α : Type u_1
inst✝² : LinearOrder α
E : Type u_3
inst✝¹ : PseudoMetricSpace E
inst✝ : CompleteSpace E
f : α → E
a b : α
hf : BoundedVariationOn f univ
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
hb : (𝓝[<] b).NeBot
⊢ Function.leftLim (variationOnFromTo f univ a) b = variationOnFromTo f univ a b - dist (f b) (Function.leftLim f b)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
⊢ ∃ v m,
    Monotone v ∧
      (∀ (i : ℕ), v i ∈ s) ∧
        x ∈ v '' Iio m ∧
          ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))

Tactic:
have hw : Monotone w := by
    apply monotone_nat_of_le_succ fun i => ?_
    rcases lt_trichotomy (i + 1) N with (hi | hi | hi)
    · grind [Monotone]
    · have A : i < N := hi ▸ i.lt_succ_self
      have := Nat.find_min exists_N A
      grind
    · grind [Monotone]

Goal state after:
case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
⊢ ∃ v m,
    Monotone v ∧
      (∀ (i : ℕ), v i ∈ s) ∧
        x ∈ v '' Iio m ∧
          ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
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
f : C(α, R)
⊢ ∀ (y : C(α, R)), f ≤ y ↔ ∃ s, y = f + star s * s

Tactic:
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.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:
rw [le_def, ← ContinuousMap.coe_coe, ← ContinuousMap.coe_coe g, ← ContinuousMap.le_def,
        StarOrderedRing.le_iff]
      rintro ⟨p, hp_mem, hp⟩
      induction hp_mem using AddSubmonoid.closure_induction_left generalizing f g with
      | zero => exact ⟨0, zero_mem _, by ext x; congrm($(hp) x)⟩
      | add_left s s_mem p p_mem hp' =>
        obtain ⟨s, rfl⟩ := s_mem
        simp only at *
        have h₀ : (star s * s + p) 0 = 0 := by simpa using congr($(hp) 0).symm
        rw [← add_assoc] at hp
        have p'₀ : 0 ≤ p 0 := by rw [← StarOrderedRing.nonneg_iff] at p_mem; exact p_mem 0
        have s₉ : (star s * s) 0 = 0 := le_antisymm ((le_add_of_nonneg_right p'₀).trans_eq h₀)
          (star_mul_self_nonneg (s 0))
        have s₀' : s 0 = 0 := by aesop
        let s' : C(α, R)₀ := ⟨s, s₀'⟩
        obtain ⟨p', hp'_mem, rfl⟩ := hp' (f + star s' * s') g hp
        refine ⟨star s' * s' + p', ?_, by rw [add_assoc]⟩
        exact add_mem (AddSubmonoid.subset_closure ⟨s', rfl⟩) hp'_mem

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 : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
n : ℕ
c : E
hi✝ : c ∈ t
⊢ Integrable ((A.set n).indicator fun x ↦ c) μ

Tactic:
by
            simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),
              integrableOn_const_iff (C := c), A.finite n, or_true]

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
⊢ eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)

Tactic:
apply le_antisymm (comp_le_of_monotoneOn f φ hφ (mapsTo_image φ t))

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
t : Set β
φ : β → α
hφ : MonotoneOn φ t
⊢ eVariationOn f (φ '' t) ≤ eVariationOn (f ∘ φ) t
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 ∈ s, 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 ∈ s, cfc (f i) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
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
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ -f x) a = -(cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }

Tactic:
← map_neg,

Goal state after:
case pos
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
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ -f x) a = (cfcHom ha) (-{ toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
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 : NonUnitalClosedEmbeddingContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : UniqueHom R A
inst✝ : CompleteSpace R
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
this : NonUnitalContinuousFunctionalCalculus R A p
⊢ IsClosedEmbedding ⇑(cfcₙHom ha)

Tactic:
cfcₙHom_eq_restrict f ha ((h a).mp ha).1 ((h a).mp ha).2

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 : NonUnitalClosedEmbeddingContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : UniqueHom R A
inst✝ : CompleteSpace R
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
this : NonUnitalContinuousFunctionalCalculus R A p
⊢ IsClosedEmbedding ⇑(nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯)
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
r : R
a : A
h : ∀ x ∈ spectrum R a, r ≤ x
ha : p a
⊢ (algebraMap R A) r ≤ cfc id a

Tactic:
exact algebraMap_le_cfc id r a h

Goal state after:
Module: Mathlib.Topology.Algebra.StarSubalgebra

Goal state before:
pp: case mem.inr
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
x y✝ : A
hy✝ : y✝ ∈ elemental R x
P : (u : A) → u ∈ elemental R x → Prop
self : P x ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u + v) ⋯
mul : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R x)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
y u : A
hu✝ : u ∈ {x} ∪ star {x}
hy : u ∈ ↑(adjoin R {x})
hu : star u = x
star_self : P (star (star u)) ⋯
⊢ P u ⋯

Tactic:
star_star

Goal state after:
case mem.inr
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
x y✝ : A
hy✝ : y✝ ∈ elemental R x
P : (u : A) → u ∈ elemental R x → Prop
self : P x ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u + v) ⋯
mul : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R x)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
y u : A
hu✝ : u ∈ {x} ∪ star {x}
hy : u ∈ ↑(adjoin R {x})
hu : star u = x
star_self : P u ⋯
⊢ P u ⋯
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
f g : R → R
a : A
⊢ Commute (cfcₙ f a) (cfcₙ g a)

Tactic:
by
  refine cfcₙ_cases (fun x ↦ Commute x (cfcₙ g a)) a f (by simp) fun hf hf0 ha ↦ ?_
  refine cfcₙ_cases (fun x ↦ Commute _ x) a g (by simp) fun hg hg0 _ ↦ ?_
  exact Commute.all _ _ |>.map _

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✝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 ∈ s, 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 ∈ s, cfcₙ (f i) 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✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing 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
ha : p a
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -(cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }

Tactic:
← map_neg,

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing 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
ha : p a
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = (cfcₙHom ha) (-{ toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: case h
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := ⋯
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := ⋯
⊢ S ⊆ Finset.univ

Tactic:
exact S.subset_univ

Goal state after: