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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ m, ⋂ n, f m n
this : Tendsto (fun m ↦ γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0))
⊢ Tendsto (fun m ↦ γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)

Tactic:
simpa only [mul_zero, tsub_zero]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
r : ℝ≥0
hmeas : Measurable (μ.rnDeriv ν)
hsing : μ.singularPart ν ⟂ₘ ν
hadd : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
hr : ¬r = 0
⊢ r⁻¹ • μ.rnDeriv ν = ↑r⁻¹ • μ.rnDeriv ν

Tactic:
simp [ENNReal.smul_def]

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

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

Tactic:
←
      hadd,

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

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
s : Set α
E : Type u_3
inst✝² : PseudoMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
l : E
a b : α
ha : a ∈ s
hb : b ∈ s
hf : LocallyBoundedVariationOn f s
h'f : Tendsto f (𝓝[s ∩ Iio b] b) (𝓝 l)
H :
  Tendsto (fun x ↦ variationOnFromTo f s a b - variationOnFromTo f s x b) (𝓝[s ∩ Iio b] b)
    (𝓝 (variationOnFromTo f s a b - dist (f b) l))
⊢ Tendsto (variationOnFromTo f s a) (𝓝[s ∩ Iio b] b) (𝓝 (variationOnFromTo f s a b - dist (f b) l))

Tactic:
by
    apply Tendsto.congr' _ H
    filter_upwards [self_mem_nhdsWithin] with x hx
    rw [variationOnFromTo.sub_left hf ha hb hx.1]

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

Goal state before:

Tactic:
∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))
  _ ≤ eVariationOn f (s ∩ Icc (u 0) (u n)) :=
      sum_le_of_monotoneOn_Iic (hu.monotoneOn _) (by grind [Monotone])
  _ ≤ ⨆ p ∈ {p : α × α | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Icc p.1 p.2) := by
    apply le_biSup (f := fun (p : α × α) ↦ eVariationOn f (s ∩ Icc p.1 p.2)) (i := (u 0, u n))
    grind [Monotone]

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_3
inst✝¹ : PseudoMetricSpace E
inst✝ : CompleteSpace E
f : α → E
a b : α
hf : BoundedVariationOn f univ
this✝¹ : TopologicalSpace α := Preorder.topology α
this✝ : OrderTopology α
hb : (𝓝[<] b).NeBot
this :
  Tendsto f (𝓝[<] b) (𝓝 (Function.leftLim f b)) →
    Tendsto (variationOnFromTo f univ a) (𝓝[<] b) (𝓝 (variationOnFromTo f univ a b - dist (f b) (Function.leftLim f b)))
⊢ Tendsto (variationOnFromTo f univ a) (𝓝[<] b) (𝓝 (variationOnFromTo f univ a b - dist (f b) (Function.leftLim f b)))

Tactic:
exact this (hf.tendsto_leftLim _)

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
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
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := ⋯
f : α → ℝ≥0∞ := ⋯
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ν (s ∩ f ⁻¹' {∞}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := ⋯
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ (n - 1) * μ (s ∩ f ⁻¹' I) ≤ ρ (s ∩ f ⁻¹' I)

Tactic:
← ENNReal.coe_zpow (zero_lt_one.trans ht).ne'

Goal state after:
case h₂
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
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ν (s ∩ f ⁻¹' {∞}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑(t ^ (n - 1)) * μ (s ∩ f ⁻¹' I) ≤ ρ (s ∩ f ⁻¹' I)
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
a b : α
hab : a ≤ b
⊢ eVariationOn f {a, b} ≤ edist (f a) (f b)

Tactic:
simp only [eVariationOn_eq_strictMonoOn, iSup_le_iff]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
hab : a ≤ b
⊢ ∀ (i : (n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ {a, b} }),
    ∑ i_1 ∈ Finset.range i.fst, edist (f (↑i.snd (i_1 + 1))) (f (↑i.snd i_1)) ≤ edist (f a) (f b)
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
x y : α
hs : IsGreatest s x
ht : IsLeast t y
hxy : x ≤ y
⊢ eVariationOn f (s ∪ t) = eVariationOn f s + edist (f x) (f y) + eVariationOn f t

Tactic:
rw [(by grind [hs.1, ht.1] : s ∪ t = (s ∪ {x, y}) ∪ t), union f _ ht, union f hs]

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
x y : α
hs : IsGreatest s x
ht : IsLeast t y
hxy : x ≤ y
⊢ eVariationOn f s + eVariationOn f {x, y} + eVariationOn f t = eVariationOn f s + edist (f x) (f y) + eVariationOn f t

α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
x y : α
hs : IsGreatest s x
ht : IsLeast t y
hxy : x ≤ y
⊢ IsLeast {x, y} x

α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
x y : α
hs : IsGreatest s x
ht : IsLeast t y
hxy : x ≤ y
⊢ IsGreatest (s ∪ {x, y}) y
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)
y : α
hy : y ∈ s
y' : α
hy' : ε ≤ eVariationOn f (s ∩ Ici y')
y'_mem : y' ∈ s ∩ Ici y
a : α
ha : a ∈ s ∩ Ici y'
b : α
hb : b ∈ s ∩ Ici y'
hab : a < b
h : δ < eVariationOn f (s ∩ Ici y' ∩ Icc a b)
⊢ δ ≤ eVariationOn f (s ∩ Icc y b)

Tactic:
have : Ici y' ∩ Icc a b = Icc a b := by grind

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
hy : y ∈ s
y' : α
hy' : ε ≤ eVariationOn f (s ∩ Ici y')
y'_mem : y' ∈ s ∩ Ici y
a : α
ha : a ∈ s ∩ Ici y'
b : α
hb : b ∈ s ∩ Ici y'
hab : a < b
h : δ < eVariationOn f (s ∩ Ici y' ∩ Icc a b)
this : Ici y' ∩ Icc a b = Icc a b
⊢ δ ≤ eVariationOn f (s ∩ Icc y b)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case hom_map_spectrum
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), spectrum R ((starAlgHom (cfcHom ⋯) ⋯) f_1) = range ⇑f_1

pp: case predicate_hom
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)

Tactic:
case hom_map_spectrum =>
      simp only [SpectrumRestricts.starAlgHom_apply, ← @spectrum.preimage_algebraMap (R := R) S,
        cfcHom_map_spectrum, Set.ext_iff, Set.mem_preimage, Set.mem_range, ContinuousMap.comp_apply,
        ContinuousMap.coe_mk, StarAlgHom.ofId_apply, halg.injective.eq_iff]
      exact fun _ _ ↦ ((h a).mp ha).2.homeomorph.exists_congr fun _ ↦ Iff.rfl

Goal state after:
case predicate_hom
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
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 f (cfc (HSMul.hSMul s) a) = cfc f (s • a)

Tactic:
cfc_smul_id ..

Goal state after:
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 f (s • a) = cfc f (s • 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✝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
h : ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x) * f x) a = cfc f a * cfc (fun x ↦ star (f x)) a

Tactic:
← cfc_mul ..

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✝ 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
h : ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x) * f x) a = cfc (fun x ↦ f x * star (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✝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:
by
    rw [cfcₙHom_id ha, hφ₂]

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✝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
h : cfcₙ f a = cfcₙ g a
ha : p a
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
⊢ Set.EqOn f g (σₙ R a)

Tactic:
rw [cfcₙ_apply f a, cfcₙ_apply g a] at h

Goal state after:
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
ha : p a
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
h :
  (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) { toFun := (σₙ R a).domRestrict g, continuous_toFun := ⋯, map_zero' := ⋯ }
⊢ Set.EqOn f g (σₙ R a)
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
inst✝ : UniqueHom R A
r : R
f : R → R
a : A
hf : ContinuousOn f ((fun x ↦ r * x) '' σₙ R a)
hf0 : f 0 = 0
ha : p a
⊢ cfcₙ f (cfcₙ (fun x ↦ r * x) a) = cfcₙ f (r • a)

Tactic:
cfcₙ_const_mul_id r a

Goal state after:
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
inst✝ : UniqueHom R A
r : R
f : R → R
a : A
hf : ContinuousOn f ((fun x ↦ r * x) '' σₙ R a)
hf0 : f 0 = 0
ha : p a
⊢ cfcₙ f (r • a) = cfcₙ f (r • 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
x : R
hx : x ∈ σₙ R (cfcₙ 0 0)
⊢ 0 0 = 0

Tactic:
cfc_zero_tac

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✝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
⊢ cfcₙ f a ≤ cfcₙ g a

Tactic:
by
  by_cases ha : p a
  · rw [cfcₙ_apply f a, cfcₙ_apply g a]
    exact cfcₙHom_mono ha fun x ↦ h x.1 x.2
  · simp only [cfcₙ_apply_of_not_predicate _ ha, le_rfl]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: 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 := IsTopologicalAddGroup.rightUniformSpace 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)
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (x : ↑I), gauge U ((bernsteinApproximation n f) x - f x) < 1

Tactic:
filter_upwards [nhds_zero.eventually_lt_const (half_pos one_pos), eventually_ne_atTop 0]
    with n nh hn₀ x

Goal state after:
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 := IsTopologicalAddGroup.rightUniformSpace 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
⊢ gauge U ((bernsteinApproximation n f) x - f x) < 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
⊢ Function.Injective realContinuousMapOfNNReal

Tactic:
intro φ ψ h
  ext f
  simpa using congr($(h) ((ContinuousMap.mk toReal continuous_coe).comp f))

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
x : Unitization 𝕜 A
⊢ ‖x‖ = max ‖x.toProd.1‖ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.toProd.1 + (mul 𝕜 A) x.toProd.2‖

Tactic:
norm_def,

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
x : Unitization 𝕜 A
⊢ ‖(splitMul 𝕜 A) x‖ = max ‖x.toProd.1‖ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.toProd.1 + (mul 𝕜 A) x.toProd.2‖
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
⊢ ‖k • x‖ ≤ ‖k‖ * ‖x‖

Tactic:
norm_def,

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
⊢ ‖(splitMul 𝕜 A) (k • x)‖ ≤ ‖k‖ * ‖x‖
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
a : A
⊢ ‖↑a‖ = ‖a‖

Tactic:
simp [norm_eq_sup]

Goal state after: