mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Polynomial

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
inst✝² : NormedCommRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
σ : Type u_5
f : E → σ → B
hf : ∀ (i : σ), AnalyticAt 𝕜 (fun x ↦ f x i) z
p✝ p : MvPolynomial σ A
i : σ
hp : AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) p) z
⊢ AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) (p * X i)) z

Tactic:
map_mul,

Goal state after:
𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
inst✝² : NormedCommRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
σ : Type u_5
f : E → σ → B
hf : ∀ (i : σ), AnalyticAt 𝕜 (fun x ↦ f x i) z
p✝ p : MvPolynomial σ A
i : σ
hp : AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) p) z
⊢ AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) p * (aeval (f x)) (X i)) z
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case h'
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ Metric.eball x r) x
⊢ s ∩ Metric.eball x r ⊆ insert x s ∩ Metric.eball x r

Tactic:
insert_eq_of_mem hx

Goal state after:
case h'
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ Metric.eball x r) x
⊢ s ∩ Metric.eball x r ⊆ s ∩ Metric.eball x r
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case h'
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ Metric.eball x r) x
⊢ s ∩ Metric.eball x r ⊆ insert x s ∩ Metric.eball x r

Tactic:
rw [insert_eq_of_mem hx]

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
x : E
h : ContDiffWithinAt 𝕜 ω f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
σ : Perm (Fin n)
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 ω f (insert x s ∩ u)
⊢ (iteratedFDerivWithin 𝕜 n f s x) (v ∘ ⇑σ) = (iteratedFDerivWithin 𝕜 n f s x) v

Tactic:
insert_eq_of_mem hx

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
x : E
h : ContDiffWithinAt 𝕜 ω f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
σ : Perm (Fin n)
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 ω f (s ∩ u)
⊢ (iteratedFDerivWithin 𝕜 n f s x) (v ∘ ⇑σ) = (iteratedFDerivWithin 𝕜 n f s x) v
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ (∞ * y).log = ⊤ + y.log

Tactic:
log_pos_real' y_real,

Goal state after:
case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ (∞ * y).log = ⊤ + ↑(Real.log y.toReal)
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z
z : ℤ
⊢ (fun a ↦ k a ^ z * f a) =O[l] fun a ↦ (k a)⁻¹

Tactic:
refine ((isBigO_refl (fun a => k a ^ z) l).mul (h (-(z + 1)))).trans ?_

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z
z : ℤ
⊢ (fun x ↦ k x ^ z * k x ^ (-(z + 1))) =O[l] fun a ↦ (k a)⁻¹
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
⊢ expGrowthSup u⁻¹ = -expGrowthInf u

Tactic:
rw [expGrowthInf, ← limsup_neg]
  refine limsup_congr (Eventually.of_forall fun n ↦ ?_)
  rw [Pi.neg_apply, Pi.inv_apply, div_eq_mul_inv, div_eq_mul_inv, ← EReal.neg_mul, log_inv]

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this✝ : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ p, ⋃ q, u' ↑p ∩ v ↑p ↑q
μt : μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
x : α
hx : x ∉ t
i : ↑s
H : x ∈ u' ↑i
⊢ f x ≤ (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x

Tactic:
simp only [H, piecewise_eq_of_mem]

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this✝ : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ p, ⋃ q, u' ↑p ∩ v ↑p ↑q
μt : μ t ≤ 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
x : α
hx : x ∉ t
i : ↑s
H : x ∈ u' ↑i
⊢ f x ≤ ↑i
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg.refine_1
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
a' : EReal
a_a' : a < a'
a_top' : a' ≠ ⊤
a_0' : 0 < a'
a_a_inv' : a'⁻¹ < a⁻¹
a_v' : ∀ᶠ (n : ℕ) in atTop, a' * ↑n ≤ ↑(v n)
b_u : ∃ᶠ (x : ℕ) in atTop, (b * ↑x ≤ u x ∧ 0 < x) ∧ ∃ m, a'⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a⁻¹ * ↑x
M M' : ℕ
aM_M' : ↑M ≤ a'⁻¹ * ↑M'
n : ℕ
n_M' : M' ≤ n
bn_un : b * ↑n ≤ u n
right✝ : 0 < n
k : ℕ
an_k' : a'⁻¹ * ↑n ≤ ↑k
k_an : ↑k ≤ a⁻¹ * ↑n
⊢ M ≤ k

Tactic:
apply Nat.cast_le.1 <| aM_M'.trans <| an_k'.trans' _

Goal state after:
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
a' : EReal
a_a' : a < a'
a_top' : a' ≠ ⊤
a_0' : 0 < a'
a_a_inv' : a'⁻¹ < a⁻¹
a_v' : ∀ᶠ (n : ℕ) in atTop, a' * ↑n ≤ ↑(v n)
b_u : ∃ᶠ (x : ℕ) in atTop, (b * ↑x ≤ u x ∧ 0 < x) ∧ ∃ m, a'⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a⁻¹ * ↑x
M M' : ℕ
aM_M' : ↑M ≤ a'⁻¹ * ↑M'
n : ℕ
n_M' : M' ≤ n
bn_un : b * ↑n ≤ u n
right✝ : 0 < n
k : ℕ
an_k' : a'⁻¹ * ↑n ≤ ↑k
k_an : ↑k ≤ a⁻¹ * ↑n
⊢ a'⁻¹ * ↑M' ≤ a'⁻¹ * ↑n
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set (Set α)
f g : α → ℝ≥0∞
h_eq : m0 = MeasurableSpace.generateFrom s
h_inter : IsPiSystem s
basic : ∀ t ∈ s, ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
h_univ : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), g x ∂μ
hf_int : ∫⁻ (x : α), f x ∂μ ≠ ∞
⊢ ∀ (t : Set α), MeasurableSet t → ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ

Tactic:
refine MeasurableSpace.induction_on_inter h_eq h_inter ?_ basic ?_ ?_
  · simp
  · intro t ht h_eq
    rw [setLIntegral_compl ht, setLIntegral_compl ht, h_eq, h_univ]
    · refine ne_of_lt ?_
      calc ∫⁻ x in t, g x ∂μ
      _ ≤ ∫⁻ x, g x ∂μ := setLIntegral_le_lintegral t _
      _ < ∞ := by rw [← h_univ]; exact hf_int.lt_top
    · refine ne_of_lt ?_
      calc ∫⁻ x in t, f x ∂μ
      _ ≤ ∫⁻ x, f x ∂μ := setLIntegral_le_lintegral t _
      _ < ∞ := hf_int.lt_top
  · intro t htd htm h
    simp_rw [lintegral_iUnion htm htd, h]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
h : μ.withDensity (μ.rnDeriv μ) + μ.singularPart μ = μ
⊢ μ.rnDeriv μ =ᵐ[μ] fun x ↦ 1

Tactic:
singularPart_self,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
h : μ.withDensity (μ.rnDeriv μ) + 0 = μ
⊢ μ.rnDeriv μ =ᵐ[μ] fun x ↦ 1
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
ρ ν : Measure α
inst✝ : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a
ε : ℝ≥0
εpos : 0 < ε
x✝ : ν s < ∞
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : ν U ≤ ν s + ↑ε
⊢ ρ s ≤ ν s + ↑ε

Tactic:
let f : α → Set (Set α) := fun _ => {a | ρ a ≤ ν a ∧ a ⊆ U}

Goal state after:
α : Type u_1
inst✝³ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
ρ ν : Measure α
inst✝ : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ν a
ε : ℝ≥0
εpos : 0 < ε
x✝ : ν s < ∞
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : ν U ≤ ν s + ↑ε
f : α → Set (Set α) := fun x ↦ {a | ρ a ≤ ν a ∧ a ⊆ U}
⊢ ρ s ≤ ν s + ↑ε
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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ μ s = 0

Tactic:
apply le_antisymm _ bot_le

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
ε : ℝ≥0
εpos : ε > 0
s : Set α := ⋯
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ μ s ≤ ⊥
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x : α
ι : Type u_2
l : Filter ι
w : ι → α
δ : ι → ℝ
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
ε : ℝ
hε : ε > 0
h : l.NeBot
δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0
δlim : Tendsto δ l (𝓝 0)
hK : 0 < K + 1
⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε

Tactic:
apply (((Metric.tendsto_nhds.mp δlim _ (div_pos hε hK)).and δpos).and xmem).mono

Goal state after:
case refine_2.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x : α
ι : Type u_2
l : Filter ι
w : ι → α
δ : ι → ℝ
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
ε : ℝ
hε : ε > 0
h : l.NeBot
δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0
δlim : Tendsto δ l (𝓝 0)
hK : 0 < K + 1
⊢ ∀ (x_1 : ι),
    (dist (δ x_1) 0 < ε / (K + 1) ∧ δ x_1 ∈ Ioi 0) ∧ x ∈ closedBall (w x_1) (K * δ x_1) →
      closedBall (w x_1) (δ x_1) ⊆ closedBall x ε
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: α : Type u_1
inst✝⁶ : PseudoMetricSpace α
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : IsUnifLocDoublingMeasure μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
E : Type u_2
inst✝ : NormedAddCommGroup E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
⊢ ∀ᵐ (x : α) ∂μ,
    ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      Tendsto δ l (𝓝[>] 0) →
        (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
          Tendsto (fun j ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)

Tactic:
filter_upwards [(vitaliFamily μ K).ae_tendsto_average_norm_sub hf] with x hx ι l w δ δlim
    xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

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

Goal state before:
pp: case refine_3
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
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n ↦ s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n ↦ s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) (u i) ∩ toMeasurable (ρ + μ) (w i_1) ⊆
    toMeasurable μ sᶜ ∪ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) (u i) ∩ toMeasurable (ρ + μ) (w i_1)

Tactic:
exact subset_union_right

Goal state after:
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
s : Set ℝ
hf : MonotoneOn f s
hs : MeasurableSet s
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict s, DifferentiableWithinAt ℝ f s x

Tactic:
rw [ae_restrict_iff' hs]
  exact hf.ae_differentiableWithinAt_of_mem

Goal state after:
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: case inr
α : Type u_2
inst✝⁶ : LinearOrder α
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set α
f : α → E
g : α → F
hf : BoundedVariationOn f s
hg : BoundedVariationOn g s
B : E →L[ℝ] F →L[ℝ] G
x : α
hx : x ∈ s
A : ∀ y ∈ s, ‖f y‖ₑ ≤ ‖f x‖ₑ + eVariationOn f s
A' : ∀ y ∈ s, ‖g y‖ₑ ≤ ‖g x‖ₑ + eVariationOn g s
⊢ eVariationOn (fun x ↦ (B (f x)) (g x)) s < ∞

Tactic:
eVariationOn_bilinear_comp_le A A'

Goal state after:
case inr
α : Type u_2
inst✝⁶ : LinearOrder α
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set α
f : α → E
g : α → F
hf : BoundedVariationOn f s
hg : BoundedVariationOn g s
B : E →L[ℝ] F →L[ℝ] G
x : α
hx : x ∈ s
A : ∀ y ∈ s, ‖f y‖ₑ ≤ ‖f x‖ₑ + eVariationOn f s
A' : ∀ y ∈ s, ‖g y‖ₑ ≤ ‖g x‖ₑ + eVariationOn g s
⊢ ‖B‖ₑ * ((‖f x‖ₑ + eVariationOn f s) * eVariationOn g s + (‖g x‖ₑ + eVariationOn g s) * eVariationOn f s) < ∞
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
h : LocallyBoundedVariationOn f univ
⊢ ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x

Tactic:
filter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
h : LocallyBoundedVariationOn f univ
x : ℝ
hx : x ∈ univ → DifferentiableWithinAt ℝ f univ x
⊢ DifferentiableAt ℝ f x
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: case mp.add_left
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)
p✝ p : C(α, R)
p_mem : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hp' : ∀ (f g : C(α, R)₀), ↑g = ↑f + p → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p
f : C(α, R)₀
s : C(α, R)
h₀ : (star s * s + p) 0 = 0
p'₀ : 0 ≤ p 0
s₉ : (star s * s) 0 = 0
s₀' : s 0 = 0
s' : C(α, R)₀ := { toContinuousMap := s, map_zero' := s₀' }
p' : C(α, R)₀
hp'_mem : p' ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hp : ↑(f + star s' * s' + p') = ↑f + star s * s + p
⊢ star s' * s' + p' ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)

Tactic:
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 : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)

Tactic:
have : ∀ n, ∀ᵐ x ∂μ,
      Tendsto (fun a => (∫⁻ y in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a)
      (v.filterAt x) (𝓝 0) := by
    intro n
    apply ae_tendsto_lintegral_enorm_sub_div_of_integrable
    exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n)

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
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
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
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
⊢ 𝓝[s ∩ Iio x ∩ Ici y] x = 𝓝[s ∩ Iio x] x

Tactic:
nhdsWithin_inter_of_mem'

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
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
⊢ 𝓝[s ∩ Iio x] x = 𝓝[s ∩ Iio x] x

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
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
⊢ Ici y ∈ 𝓝[s ∩ Iio x] x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
q : R[X]
f : R → R
a : A
hf : ContinuousOn f ((fun x ↦ Polynomial.eval x q) '' spectrum R a)
ha : p a
⊢ cfc f (cfc (fun x ↦ Polynomial.eval x q) a) = cfc f ((aeval a) q)

Tactic:
cfc_polynomial ..

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
q : R[X]
f : R → R
a : A
hf : ContinuousOn f ((fun x ↦ Polynomial.eval x q) '' spectrum R a)
ha : p a
⊢ cfc f ((aeval a) q) = cfc f ((aeval a) q)
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 : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : 𝓝[s ∩ Iio x] x = ⊥
this : s = s ∩ Iio x ∪ s ∩ Ici x
y : α
hy : y ∈ s ∩ Ici x
⊢ 0 = eVariationOn f (s ∩ Icc y x)

Tactic:
apply (eVariationOn.subsingleton _ (by grind [Set.Subsingleton])).symm

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

Goal state before:
pp: case inr.refine_1
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 : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
⊢ (𝓝[s ∩ Iio x ∩ Ici y] x).NeBot

Tactic:
nhdsWithin_inter_of_mem'

Goal state after:
case inr.refine_1
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 : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
⊢ (𝓝[s ∩ Iio x] x).NeBot

case inr.refine_1
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 : α
h : ContinuousWithinAt f (s ∩ Iic x) x
h' : (𝓝[s ∩ Iio x] x).NeBot
y : α
hy : y < x
⊢ Ici y ∈ 𝓝[s ∩ Iio x] x