mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : AEMeasurable f μ
hN : μ N = 0
this : 0 < μ {x | f x ≤ ⨍⁻ (a : α), f a ∂μ}
⊢ ∃ x ∉ N, f x ≤ ⨍⁻ (a : α), f a ∂μ

Tactic:
rw [← measure_sdiff_null hN] at this

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : AEMeasurable f μ
hN : μ N = 0
this : 0 < μ ({x | f x ≤ ⨍⁻ (a : α), f a ∂μ} \ N)
⊢ ∃ x ∉ N, f x ≤ ⨍⁻ (a : α), f a ∂μ
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
i : ι
hif : IntegrableOn f (a i) μ
hig : Integrable (g i) μ
hisupp : support (g i) ⊆ a i
hibound : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
⊢ (support fun y ↦ g i y • f y) ⊆ a i

Tactic:
apply Subset.trans _ hisupp

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
i : ι
hif : IntegrableOn f (a i) μ
hig : Integrable (g i) μ
hisupp : support (g i) ⊆ a i
hibound : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
⊢ (support fun y ↦ g i y • f y) ⊆ support (g i)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
⊢ ∀ᵐ (x : α) ∂ν, μ.rnDeriv ν x < ∞

Tactic:
by
  suffices ∀ n, ∀ᵐ x ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞ by
    filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)
  intro n
  rw [← ae_restrict_iff' (measurableSet_spanningSets _ _)]
  apply ae_lt_top (measurable_rnDeriv _ _)
  refine (lintegral_rnDeriv_lt_top_of_measure_ne_top _ ?_).ne
  exact (measure_spanningSets_lt_top _ _).ne

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
x : α
hf : ∃ᶠ (ε : ℝ) in 𝓝[>] 0, ∃ a ∈ t, r a = ε ∧ c a = x
hx : x ∈ s
ε : ℝ
hε : ε > 0
⊢ ∃ a ∈ t, r a ≤ ε ∧ c a = x

Tactic:
frequently_nhdsWithin_iff,

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
x : α
hf : ∃ᶠ (x_1 : ℝ) in 𝓝 0, (∃ a ∈ t, r a = x_1 ∧ c a = x) ∧ x_1 ∈ Ioi 0
hx : x ∈ s
ε : ℝ
hε : ε > 0
⊢ ∃ a ∈ t, r a ≤ ε ∧ c a = x
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν.singularPart (r • μ) + (r • μ).withDensity (ν.rnDeriv (r • μ)) =
    ν.singularPart (r • μ) + r⁻¹ • (r • μ).withDensity (ν.rnDeriv μ)

Tactic:
rw [← ν.haveLebesgueDecomposition_add (r • μ), singularPart_smul_right _ _ _ hr,
      ENNReal.smul_def r, withDensity_smul_measure, ← ENNReal.smul_def, ← smul_assoc,
      smul_eq_mul, inv_mul_cancel₀ hr, one_smul]

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν = ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)
Module: Mathlib.Order.Monotone.Extension

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
⊢ g x ≤ g y

Tactic:
by_cases hx : Disjoint (Iic x) s <;> by_cases hy : Disjoint (Iic y) s

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : Disjoint (Iic x) s
hy : Disjoint (Iic y) s
⊢ g x ≤ g y

case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ g x ≤ g y

case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : Disjoint (Iic y) s
⊢ g x ≤ g y

case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x ↦ if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ g x ≤ g y
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Sort u_2
f : ℕ → α → ℝ≥0∞
m : ℕ
a : α
c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
d : ℝ≥0∞ := max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
hd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
⊢ c ≤ d ∧ d ≤ c

Tactic:
hc,

Goal state after:
α : Sort u_2
f : ℕ → α → ℝ≥0∞
m : ℕ
a : α
c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
d : ℝ≥0∞ := max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
hd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)
⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ d ∧ d ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: 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
⊢ Measurable ξ

Tactic:
convert! Measurable.iSup fun n ↦ (iSup_mem_measurableLE _ hf₁ n).1

Goal state after:
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
x✝ : α
⊢ ξ x✝ = ⨆ i, ⨆ k, ⨆ (_ : k ≤ i), f k x✝
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: case h2
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[>] x, x_1 + -1 * (x_1 - x) ^ 2 ∈ Ioi x

Tactic:
· filter_upwards [Ioo_mem_nhdsGT <| show x < x + 1 by simp]
        rintro y ⟨hy : x < y, h'y : y < x + 1⟩
        rw [mem_Ioi]
        nlinarith

Goal state after:
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.singularPart (r • μ) + (r • μ).withDensity (ν.rnDeriv (r • μ)) =
    ν.singularPart (r • μ) + r⁻¹ • (r • μ).withDensity (ν.rnDeriv μ)

Tactic:
rw [← ν.haveLebesgueDecomposition_add (r • μ), singularPart_smul_right _ _ _ hr,
      ENNReal.smul_def r, withDensity_smul_measure, ← ENNReal.smul_def, ← smul_assoc,
      smul_eq_mul, inv_mul_cancel₀ hr, one_smul]

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 = ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)
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
⊢ ENNReal.ofReal (dist (f x) (f y)) ≤ ENNReal.ofReal (eVariationOn f s).toReal

Tactic:
ENNReal.ofReal_toReal h,

Goal state after:
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
⊢ ENNReal.ofReal (dist (f x) (f y)) ≤ eVariationOn f s
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
hab : a ≤ b
n : ℕ
u : ℕ → α
hmono : StrictMonoOn u (Iic n)
hi : ∀ i ∈ Iic n, u i ∈ {a, b}
hn : 2 ≤ n
this : u 0 < u 1
⊢ ∑ i ∈ Finset.range ⟨n, ⟨u, ⋯⟩⟩.fst, edist (f (↑⟨n, ⟨u, ⋯⟩⟩.snd (i + 1))) (f (↑⟨n, ⟨u, ⋯⟩⟩.snd i)) ≤ edist (f a) (f b)

Tactic:
have := hmono (by grind) (by grind) one_lt_two

Goal state after:
case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
a b : α
hab : a ≤ b
n : ℕ
u : ℕ → α
hmono : StrictMonoOn u (Iic n)
hi : ∀ i ∈ Iic n, u i ∈ {a, b}
hn : 2 ≤ n
this✝ : u 0 < u 1
this : u 1 < u 2
⊢ ∑ i ∈ Finset.range ⟨n, ⟨u, ⋯⟩⟩.fst, edist (f (↑⟨n, ⟨u, ⋯⟩⟩.snd (i + 1))) (f (↑⟨n, ⟨u, ⋯⟩⟩.snd i)) ≤ edist (f a) (f b)
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✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
⊢ cfc g a = cfc (fun x ↦ (algebraMap R S) (g (f x))) a

Tactic:
by_cases hg : ContinuousOn g (spectrum R a)

Goal state after:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
hg : ContinuousOn g (spectrum R a)
⊢ cfc g a = cfc (fun x ↦ (algebraMap R S) (g (f x))) a

case neg
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc g a = cfc (fun x ↦ (algebraMap R S) (g (f x))) a
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
⊢ CompactSpace ↑(σₙ R a)

Tactic:
have := NonUnitalContinuousFunctionalCalculus.compactSpace_quasispectrum (R := S) a
    rw [← isCompact_iff_compactSpace] at this ⊢
    simpa using halg.isCompact_preimage this

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

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

Tactic:
mkD_of_not_continuousOn f_cont,

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
C : ℝ
hf : MonotoneOn f s
h : ∀ x ∈ s, |f x| ≤ C
this : eVariationOn f s ≤ ENNReal.ofReal (2 * C)
⊢ ENNReal.ofReal (2 * C) < ∞

Tactic:
simp [mul_lt_top]

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

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

Tactic:
rw [flip' _ _ _ h]
  simp only [iterate_derivative_comp_one_sub_X, eval_mul, eval_pow, eval_neg, eval_one, eval_comp,
    eval_sub, eval_X, sub_self, iterate_derivative_at_0]
  obtain rfl | h' := h.eq_or_lt
  · simp
  · norm_cast
    congr
    lia

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ IsTopologicalBasis ((fun u ↦ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}) '' {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U})

Tactic:
refine isTopologicalBasis_of_subbasis rfl |>.isTopologicalBasis_of_exists_subset ?_ ?_ <;>
    rw [forall_mem_image]
  · intro u ⟨hu₁, hu₂⟩
    simp_rw [ofPred_and, ofPred_forall]
    exact (isOpen_sUnion hu₂).powerset_vietoris.inter <|
      hu₁.isOpen_biInter fun U hU => isOpen_inter_nonempty_of_isOpen (hu₂ U hU)
  · intro t ⟨ht₁, ht₂⟩ s hs
    rw [exists_mem_image]
    rcases eq_empty_or_nonempty s with rfl | _
    · exists ∅; simpa
    obtain ⟨u, v, hu, hv, rfl⟩ : ∃ u v, u ⊆ powerset '' {U | IsOpen U} ∧
        v ⊆ (fun V => {s | (s ∩ V).Nonempty}) '' {U | IsOpen U} ∧ u ∪ v = t := by
      rw [← inter_eq_left, inter_union_distrib_left] at ht₂
      exact ⟨_, _, inter_subset_right, inter_subset_right, ht₂⟩
    rw [finite_union] at ht₁
    obtain ⟨u, hu₁, hu₂, rfl⟩ := exists_subset_image_finite_and.mp ⟨u, hu, ht₁.1, rfl⟩
    obtain ⟨v, hv₁, hv₂, rfl⟩ := exists_subset_image_finite_and.mp ⟨v, hv, ht₁.2, rfl⟩
    let U := ⋂₀ u
    have hU : IsOpen U := hu₂.isOpen_sInter hu₁
    simp_rw [sInter_union, show ⋂₀ (powerset '' u) = U.powerset by ext; simp [U], sInter_image]
      at hs ⊢
    rw [mem_inter_iff, mem_powerset_iff, mem_iInter₂] at hs
    exists insert U ((U ∩ ·) '' v)
    simp_rw [show ⋃₀ (insert U ((U ∩ ·) '' v)) = U by simp, mem_ofPred, finite_insert,
      forall_mem_insert, forall_mem_image, ← inter_assoc, inter_eq_left.mpr hs.1]
    refine ⟨⟨hv₂.image _, hU, fun V hV => hU.inter (hv₁ hV)⟩, by grind,
      fun t ⟨htU, _, ht⟩ => ⟨htU, mem_iInter₂_of_mem ?_⟩⟩
    simpa only [inter_eq_left.mpr htU] using! ht

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∀ᶠ (a : (Set α × Set α) × Set α × Set α) in 𝓤 (Set α × Set α),
    ((fun x ↦ x.1 ∪ x.2) a.1, (fun x ↦ x.1 ∪ x.2) a.2) ∈ hausdorffEntourage U

Tactic:
filter_upwards [entourageProd_mem_uniformity (Filter.mem_lift' hU) (Filter.mem_lift' hU)]
    with _ ⟨h₁, h₂⟩ using union_mem_hausdorffEntourage U h₁ h₂

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), UpperHemicontinuousOn (F n) s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
hx₀u : f x₀ ⊆ u
W : Set (β × β)
hW : W ∈ uniformity β
left✝ : IsOpen[instTopologicalSpaceProd] W
hWu : ∀ x ∈ f x₀, ball x W ⊆ u
V : Set (β × β)
hV : V ∈ uniformity β
hVsym : SetRel.IsSymm V
hVcomp : SetRel.comp V V ⊆ W
⊢ ∀ᶠ (x' : α) in 𝓝[s] x₀, f x' ⊆ u

Tactic:
obtain ⟨U, ⟨hUmem, hUopen⟩, hUsub⟩ := uniformity_hasBasis_open.mem_iff.mp hV

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), UpperHemicontinuousOn (F n) s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
hx₀u : f x₀ ⊆ u
W : Set (β × β)
hW : W ∈ uniformity β
left✝ : IsOpen[instTopologicalSpaceProd] W
hWu : ∀ x ∈ f x₀, ball x W ⊆ u
V : Set (β × β)
hV : V ∈ uniformity β
hVsym : SetRel.IsSymm V
hVcomp : SetRel.comp V V ⊆ W
U : SetRel β β
hUsub : id U ⊆ V
hUmem : U ∈ uniformity β
hUopen : IsOpen[instTopologicalSpaceProd] U
⊢ ∀ᶠ (x' : α) in 𝓝[s] x₀, f x' ⊆ u
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
s : Set α
hs : s ∈ (range fun x ↦ {x})ᶜ
h : s.Nonempty
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 s

Tactic:
obtain ⟨U, V, hU, hV, hxU, hyV, hUV⟩ := t2_separation hxy

Goal state after:
case inr.inr
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
t : TopologicalSpace (Set α)
h₁ : IsOpen[t] {∅}
h₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}
s : Set α
hs : s ∈ (range fun x ↦ {x})ᶜ
h : s.Nonempty
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
U V : Set α
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
hxU : x ∈ U
hyV : y ∈ V
hUV : Disjoint U V
⊢ (range fun x ↦ {x})ᶜ ∈ 𝓝 s
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ TotallyBounded (𝒫 t)

Tactic:
simp_rw [(𝓤 α).basis_sets.uniformity_hausdorff.totallyBounded_iff, Function.comp_id,
    Set.powerset, Set.ofPred_subset, Set.mem_iUnion]
  intro (U : SetRel α α) hU
  obtain ⟨u, hu, ht⟩ := ht U hU
  refine ⟨u.powerset, hu.powerset, fun s hs => ⟨u ∩ U.image s, by grind, fun x hx => ?_,
    fun x ⟨_, hx⟩ => hx⟩⟩
  obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht (hs hx))
  exact ⟨y, ⟨hy, ⟨x, hx, hxy⟩⟩, hxy⟩

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case pos
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}
L : Set α
hL : L ∈ s
hLu : L ⊆ ⋃₀ u
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
sUnion_eq_biUnion

Goal state after:
case pos
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}
L : Set α
hL : L ∈ s
hLu : L ⊆ ⋃ i ∈ u, i
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q