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

Goal state before:

Tactic:
μ + (ν - ξ)
  _ = μ.restrict s + μ.restrict sᶜ + (ν - ξ).restrict s + (ν - ξ).restrict sᶜ := by
    rw [restrict_add_restrict_compl hs, add_assoc, restrict_add_restrict_compl hs]
  _ = μ.restrict s + (ν - ξ).restrict s + (μ.restrict sᶜ + (ν - ξ).restrict sᶜ) := by abel
  _ = (μ.restrict s + ν.restrict s - ξ.restrict s) +
      (μ.restrict sᶜ + ν.restrict sᶜ - ξ.restrict sᶜ) := by rw [h_le_s, h_le_s_compl]
  _ = (μ + ν - ξ).restrict s + (μ + ν - ξ).restrict sᶜ := by
      simp [restrict_sub_eq_restrict_sub_restrict hs,
        restrict_sub_eq_restrict_sub_restrict hs.compl]
  _ = μ + ν - ξ := by rw [restrict_add_restrict_compl hs]

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
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 : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})
B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := ⋯
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ ↑t ^ (n + 1) = ↑(t ^ (n + 1))

Tactic:
ENNReal.coe_zpow t_ne_zero'

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
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 : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})
B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ ↑t ^ (n + 1) = ↑t ^ (n + 1)
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
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
Bx : t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝
Cx : Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝))
a : Set α
| ρ a / μ a

Tactic:
rw [eq_add]

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)
B : ∀ᵐ (x : α) ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x))
a✝ : α
Ax : Tendsto (fun a ↦ (ρ.singularPart μ) a / μ a) (v.filterAt a✝) (𝓝 0)
Bx : t.rnDeriv μ a✝ = ρ.rnDeriv μ a✝
Cx : Tendsto (fun a ↦ t a / μ a) (v.filterAt a✝) (𝓝 (t.rnDeriv μ a✝))
a : Set α
| (ρ.singularPart μ + t) a / μ a
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_3
inst✝ : PseudoMetricSpace E
f : α → E
s : Finset α
hne : s.Nonempty
⊢ BoundedVariationOn f ↑s

Tactic:
have := s.card_pos.2 hne

Goal state after:
case inr
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
⊢ BoundedVariationOn f ↑s
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 : 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 := fun i ↦ if i ≤ n then u i else if i = n + 1 then y else a
A :
  ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) + edist (f a) (f y) ≤
    ∑ i ∈ Finset.range (n + 2), edist (f (v (i + 1))) (f (v i))
⊢ MonotoneOn v (Iic (n + 2))

Tactic:
by grind [MonotoneOn, StrictMonoOn]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝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)
⊢ (cfcHom ha) f = cfc (Function.extend Subtype.val (⇑f) g) a

Tactic:
have h : f = (spectrum R a).domRestrict (Function.extend Subtype.val f g) := by
    ext; simp

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
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)
⊢ (cfcHom ha) f = cfc (Function.extend Subtype.val (⇑f) 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)
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)

Tactic:
inter_assoc,

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.MeasureTheory.Covering.Differentiation

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

Tactic:
coe_nnnorm

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
this : Tendsto (ENNReal.toReal ∘ fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
a : Set α
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ IntegrableOn (fun y ↦ ‖f y - f x‖) a μ
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 : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x y : α
⊢ eVariationOn f (s ∩ Ioc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ Ico (toDual y) (toDual x))

Tactic:
Ico_toDual,

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x y : α
⊢ eVariationOn f (s ∩ Ioc x y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Ioc x y)
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✝²³ : Semifield R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : Field S
inst✝¹⁷ : StarRing S
inst✝¹⁶ : MetricSpace S
inst✝¹⁵ : IsTopologicalRing S
inst✝¹⁴ : ContinuousStar S
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : Algebra R S
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus S A q
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
⊢ (nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯)
      { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } =
    a

Tactic:
exact ((h a).mp ha).2.nonUnitalStarAlgHom_id <| cfcₙHom_id ((h a).mp ha).1

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
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
⊢ (cfcₙHom ha) f = cfcₙ (Function.extend Subtype.val (⇑f) g) a

Tactic:
have h : f = (σₙ R a).domRestrict (Function.extend Subtype.val f g) := by
    ext; simp
  have hg : ContinuousOn (Function.extend Subtype.val f g) (σₙ R a) :=
    continuousOn_iff_continuous_domRestrict.mpr <| h ▸ map_continuous f
  have hg0 : (Function.extend Subtype.val f g) 0 = 0 := by
    rw [← quasispectrum.coe_zero (R := R) a, Subtype.val_injective.extend_apply]
    exact map_zero f
  generalize Function.extend Subtype.val f g = f' at *
  rw [cfcₙ_apply ..]
  congr!

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)

Tactic:
by_cases! h : ν ≤ n

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ν ≤ n
⊢ eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)

case neg
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : n < ν
⊢ eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)
Module: Mathlib.Topology.ContinuousMap.Polynomial

Goal state before:
pp: case refine_2
R : Type u_1
α : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
g : R[X]
f : C(α, R)
x : α
⊢ ∀ (n : ℕ) (a : R), ((aeval f) ((monomial n) a)) x = eval (f x) ((monomial n) a)

Tactic:
intro n a
    simp

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
hf : ContinuousOn f✝ (σₙ R a✝)
hf0 : f✝ 0 = 0
hg : ContinuousOn g (σₙ R a✝)
hg0 : g 0 = 0
ha : p a✝
f : R → R
a : A
⊢ Commute (star (cfcₙ f a)) (cfcₙ f a)

Tactic:
refine cfcₙ_cases (fun x ↦ Commute (star x) x) _ _ (Commute.zero_right _) fun _ _ _ ↦ ?_
    simp only [Commute, SemiconjBy]
    rw [← cfcₙ_apply f a, ← cfcₙ_star, ← cfcₙ_mul .., ← cfcₙ_mul ..]
    congr! 2
    exact mul_comm _ _

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

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
A : Subalgebra ℝ C(X, ℝ)
hA : A.SeparatesPoints
f : C(X, ℝ)
K : Set X
hK : IsCompact K
ε : ℝ
pos : 0 < ε
restrict_on_K : C(X, ℝ) →⋆ₐ[ℝ] C(↑K, ℝ) := compStarAlgHom' ℝ ℝ { toFun := Subtype.val, continuous_toFun := ⋯ }
AK : Subalgebra ℝ C(↑K, ℝ) := Subalgebra.map (↑restrict_on_K) A
hsep : AK.SeparatesPoints
gK : C(↑K, ℝ)
hgKAK✝ : gK ∈ AK
hgapprox : ∀ (x : ↑K), ‖↑⟨gK, hgKAK✝⟩ x - K.domRestrict (⇑f) x‖ < ε
g : C(X, ℝ)
hgA : g ∈ A
hgKAK : ↑restrict_on_K g = gK
x : X
hxK : x ∈ K
⊢ g x = gK ⟨x, hxK⟩

Tactic:
← hgKAK

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
A : Subalgebra ℝ C(X, ℝ)
hA : A.SeparatesPoints
f : C(X, ℝ)
K : Set X
hK : IsCompact K
ε : ℝ
pos : 0 < ε
restrict_on_K : C(X, ℝ) →⋆ₐ[ℝ] C(↑K, ℝ) := compStarAlgHom' ℝ ℝ { toFun := Subtype.val, continuous_toFun := ⋯ }
AK : Subalgebra ℝ C(↑K, ℝ) := Subalgebra.map (↑restrict_on_K) A
hsep : AK.SeparatesPoints
gK : C(↑K, ℝ)
hgKAK✝ : gK ∈ AK
hgapprox : ∀ (x : ↑K), ‖↑⟨gK, hgKAK✝⟩ x - K.domRestrict (⇑f) x‖ < ε
g : C(X, ℝ)
hgA : g ∈ A
hgKAK : ↑restrict_on_K g = gK
x : X
hxK : x ∈ K
⊢ g x = (↑restrict_on_K g) ⟨x, hxK⟩
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ.realContinuousMapZeroOfNNReal ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f) = φ f

Tactic:
by
  simp only [realContinuousMapZeroOfNNReal_apply]
  convert_to φ f - φ 0 = φ f using 2
  on_goal -1 => rw [map_zero, sub_zero]
  all_goals
    congr
    ext x
    simp

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

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
⊢ Function.Injective realContinuousMapZeroOfNNReal

Tactic:
intro φ ψ h
  ext f
  simpa using congr($(h) ((ContinuousMapZero.mk ⟨toReal, continuous_coe⟩ rfl).comp f))

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
⊢ ‖((splitMul 𝕜 E) x).2‖ ^ 2 ≤ ‖((splitMul 𝕜 E) (x⋆ * x)).2‖

Tactic:
refine (Real.le_sqrt (norm_nonneg _) (norm_nonneg _)).mp ?_
  simp only [Unitization.splitMul_apply]
  rw [← sSup_unitClosedBall_eq_norm]
  refine csSup_le ((Metric.nonempty_closedBall.2 zero_le_one).image _) ?_
  rintro - ⟨b, hb, rfl⟩
  simp only
  -- rewrite to a more convenient form; this is where we use the C⋆-property
  rw [← Real.sqrt_sq (norm_nonneg _), Real.sqrt_le_sqrt_iff (norm_nonneg _), sq,
    ← CStarRing.norm_star_mul_self, add_apply, star_add, mul_apply',
    Algebra.algebraMap_eq_smul_one, smul_apply,
    one_apply_eq_self, star_mul, star_smul, add_mul, smul_mul_assoc, ← mul_smul_comm,
    mul_assoc, ← mul_add, ← sSup_unitClosedBall_eq_norm]
  refine (norm_mul_le _ _).trans ?_
  calc
    _ ≤ ‖star x.fst • (x.fst • b + x.snd * b) + star x.snd * (x.fst • b + x.snd * b)‖ := by
      nth_rewrite 2 [← one_mul ‖_ + _‖]
      gcongr
      exact (norm_star b).symm ▸ mem_closedBall_zero_iff.1 hb
    _ ≤ sSup (_ '' Metric.closedBall 0 1) := le_csSup ?_ ⟨b, hb, ?_⟩
  -- now we just check the side conditions for `le_csSup`. There is nothing of interest here.
  · refine ⟨‖(star x * x).fst‖ + ‖(star x * x).snd‖, ?_⟩
    rintro _ ⟨y, hy, rfl⟩
    refine (norm_add_le _ _).trans ?_
    gcongr
    · rw [Algebra.algebraMap_eq_smul_one]
      refine (norm_smul _ _).trans_le ?_
      simpa only [mul_one] using!
        mul_le_mul_of_nonneg_left (mem_closedBall_zero_iff.1 hy) (norm_nonneg (star x * x).fst)
    · exact (unit_le_opNorm _ y <| mem_closedBall_zero_iff.1 hy).trans (opNorm_mul_apply_le _ _ _)
  · simp only [add_apply, mul_apply', Unitization.snd_star, Unitization.snd_mul,
      Unitization.fst_mul, Unitization.fst_star, Algebra.algebraMap_eq_smul_one, smul_apply,
      one_apply_eq_self, smul_add, mul_add, add_mul]
    simp only [smul_smul, smul_mul_assoc, ← add_assoc, ← mul_assoc, mul_smul_comm]

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

Goal state before:
pp: case h₁
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
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖

pp: case h₂
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
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖

Tactic:
· simp [norm_smul]

Goal state after:
case h₂
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
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
a b : A
⊢ { toFun := fun φ ↦ φ (a + b), continuous_toFun := ⋯ } =
    { toFun := fun φ ↦ φ a, continuous_toFun := ⋯ } + { toFun := fun φ ↦ φ b, continuous_toFun := ⋯ }

Tactic:
by ext; simp only [map_add, coe_mk, coe_add, Pi.add_apply]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
⊢ y₀ ∈ f x₀

Tactic:
by
  by_contra
  obtain ⟨s, hs, t, ht, hst⟩ := Filter.disjoint_iff.mp <| RegularSpace.regular hf_closed this
  suffices ∃ᶠ n in l, y n ∈ s by
    apply this
    filter_upwards [hy₀ ht] with n hn hyn
    exact hst.notMem_of_mem_left hyn hn
  apply hy.mp
  filter_upwards [hx (hf s hs)] with n hn hyn
  simp only [← subset_interior_iff_mem_nhdsSet, preimage_ofPred_eq, mem_ofPred_eq] at hn
  exact interior_subset <| hn hyn

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
U : SetRel α α
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)
t : Set α
ht₁ : t.Finite
ht₂ : ↑K ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
⊢ ∃ t, t.Finite ∧ U.preimage t ∈ l

Tactic:
← SetRel.preimage_eq_biUnion

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
U : SetRel α α
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage (Prod.swap ⁻¹' V)
t : Set α
ht₁ : t.Finite
ht₂ : ↑K ⊆ V.preimage t
⊢ ∃ t, t.Finite ∧ U.preimage t ∈ l
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ U ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hKT : K \ U ⊆ ⋃ i ∈ T, i
⊢ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ insert (𝒫 U) ((fun V ↦ {s | (s ∩ V).Nonempty}) '' T)

Tactic:
sUnion_insert,

Goal state after:
case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ U ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hKT : K \ U ⊆ ⋃ i ∈ T, i
⊢ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ 𝒫 U ∪ ⋃₀ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' T)