mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case e'_12
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2) * cos (x.2 * π)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedCommRing.toCommRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ (NormedAlgebra.toNormedSpace ℝ).toModule
⊢ (p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
      (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ =
    rexp (log p.1 * p.2) • -sin (p.2 * π) • π • ContinuousLinearMap.snd ℝ ℝ ℝ +
      cos (p.2 * π) •
        rexp (log p.1 * p.2) • (log p.1 • ContinuousLinearMap.snd ℝ ℝ ℝ + p.2 • p.1⁻¹ • ContinuousLinearMap.fst ℝ ℝ ℝ)

Tactic:
simp_rw [rpow_sub_one hp.ne, smul_add, ← add_assoc, smul_smul, ← add_smul, ← mul_assoc,
    mul_comm (cos _), ← rpow_def_of_neg hp]

Goal state after:
case e'_12
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2) * cos (x.2 * π)
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedCommRing.toCommRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ (NormedAlgebra.toNormedSpace ℝ).toModule
⊢ (p.2 * (p.1 ^ p.2 / p.1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
      (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ =
    (rexp (log p.1 * p.2) * -sin (p.2 * π) * π + p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ +
      (p.1 ^ p.2 * p.2 * p.1⁻¹) • ContinuousLinearMap.fst ℝ ℝ ℝ
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 : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
f : E → B
hf : AnalyticWithinAt 𝕜 f Set.univ z
p : A[X]
⊢ AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) p) Set.univ z

Tactic:
exact hf.aeval_polynomial p

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

Goal state before:
pp: w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall ((fun z ↦ 1 / (z - w) ^ 2) - fun z ↦ 1 / w ^ 2)
    (FormalMultilinearSeries.ofScalars ℂ
      ((fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2))) - fun i ↦ Nat.casesOn i (w ^ (-2)) 0))
    x ‖w - x‖ₑ

Tactic:
FormalMultilinearSeries.ofScalars_sub

Goal state after:
w x : ℂ
hw : w ≠ x
⊢ HasFPowerSeriesOnBall ((fun z ↦ 1 / (z - w) ^ 2) - fun z ↦ 1 / w ^ 2)
    ((FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2))) -
      FormalMultilinearSeries.ofScalars ℂ fun i ↦ Nat.casesOn i (w ^ (-2)) 0)
    x ‖w - x‖ₑ
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: p : ℝ
⊢ (deriv fun x ↦ x ^ p) =O[atTop] fun x ↦ x ^ (p - 1)

Tactic:
rcases eq_or_ne p 0 with rfl | hp

Goal state after:
case inl
⊢ (deriv fun x ↦ x ^ 0) =O[atTop] fun x ↦ x ^ (0 - 1)

case inr
p : ℝ
hp : p ≠ 0
⊢ (deriv fun x ↦ x ^ p) =O[atTop] fun x ↦ x ^ (p - 1)
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
inst✝ : MetricSpace X
D : DeloneSet X
x y : X
hx : x ∈ D
hy : y ∈ D
hne : x ≠ y
⊢ ENNReal.ofReal ↑D.packingRadius < ENNReal.ofReal (dist x y)

Tactic:
simpa [edist_dist] using D.isSeparated_packingRadius hx hy hne

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : z < 0
this : Tendsto (fun a ↦ k a ^ z) l (𝓝 0)
⊢ Tendsto f l (𝓝 0)

Tactic:
simpa using! h 0

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: 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
h : SuperpolynomialDecay l k f
z : ℤ
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
⊢ Tendsto (fun x ↦ 1 / k x) l (𝓝 0)

Tactic:
by simpa using! hk.inv_tendsto_atTop

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: 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
h : SuperpolynomialDecay l k f
z : ℤ
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
this : (fun x ↦ 1) =o[l] k
⊢ f =o[l] fun a ↦ k a ^ z

Tactic:
have : f =o[l] fun x : α => k x * k x ^ (z - 1) := by
    simpa using! this.mul_isBigO ((superpolynomialDecay_iff_isBigO f hk).1 h <| z - 1)

Goal state after:
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
h : SuperpolynomialDecay l k f
z : ℤ
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
this✝ : (fun x ↦ 1) =o[l] k
this : f =o[l] fun x ↦ k x * k x ^ (z - 1)
⊢ f =o[l] fun a ↦ k a ^ z
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
h' : ∀ (n : ℕ), u n ≤ 0
⊢ linearGrowthInf u ≤ 0

Tactic:
exact (linearGrowthInf_monotone h').trans_eq (linearGrowthInf_const zero_ne_bot zero_ne_top)

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)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
x : α
⊢ Continuous (HMul.hMul 20)

Tactic:
fun_prop

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

Goal state before:
pp: case refine_2
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)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1))

Tactic:
calc
        γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) =
            γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by
          rw [pow_succ, mul_one_div, _root_.sub_half]
        _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by
          simp only [sub_eq_add_neg]; abel
        _ ≤ d (e (n + 1)) + d (f m n) := add_le_add (le_of_lt <| he₂ _) ih
        _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by
          rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (he₁ _), add_assoc]
        _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by
          rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_sdiff_left, union_inter_cancel_left]
          · abel
          · exact he₁ _
        _ ≤ γ + d (f m (n + 1)) := by grw [d_le_γ _ <| (he₁ _).union (hf _ _)]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ μ ≤ ν' + ⊤.restrict sᶜ + ν

Tactic:
add_right_comm,

Goal state after:
case a.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ μ ≤ ν' + ν + ⊤.restrict sᶜ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ✝ ν : Measure α
ι : Type u_2
inst✝¹ : Countable ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν
⊢ Measurable (∑' (i : ι), (μ i).rnDeriv ν)

Tactic:
fun_prop

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : ν.HaveLebesgueDecomposition μ
hμν : μ ≪ ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)
⊢ μ ≪ μ.withDensity (ν.rnDeriv μ)

Tactic:
refine AbsolutelyContinuous.mk (fun s _ hνs ↦ ?_)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : ν.HaveLebesgueDecomposition μ
hμν : μ ≪ ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)
s : Set α
x✝ : MeasurableSet s
hνs : (μ.withDensity (ν.rnDeriv μ)) s = 0
⊢ μ s = 0
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
s : Set α
hs : MeasurableSet s
h₁ : ∀ (t : Set α), MeasurableSet t → t ⊆ s → μ t ≤ ν t
h₂ : ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ν t ≤ μ t
⊢ ∃ s, IsHahnDecomposition μ ν s

Tactic:
refine ⟨s, hs, ?_, ?_⟩

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
s : Set α
hs : MeasurableSet s
h₁ : ∀ (t : Set α), MeasurableSet t → t ⊆ s → μ t ≤ ν t
h₂ : ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ν t ≤ μ t
⊢ μ.restrict s ≤ ν.restrict s

case refine_2
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
s : Set α
hs : MeasurableSet s
h₁ : ∀ (t : Set α), MeasurableSet t → t ⊆ s → μ t ≤ ν t
h₂ : ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ν t ≤ μ t
⊢ ν.restrict sᶜ ≤ μ.restrict sᶜ
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : TopologicalSpace α
f : α → ℝ
hs : IsConnected s
hf : ContinuousOn f s
hint : IntegrableOn f s μ
hμfin : μ s ≠ ∞
hμ0 : μ s ≠ 0
ave : ℝ := ⨍ (x : α) in s, f x ∂μ
⊢ ∃ c ∈ s, f c = ⨍ (x : α) in s, f x ∂μ

Tactic:
let S₁ : Set α := {x | x ∈ s ∧ f x ≤ ave}

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : TopologicalSpace α
f : α → ℝ
hs : IsConnected s
hf : ContinuousOn f s
hint : IntegrableOn f s μ
hμfin : μ s ≠ ∞
hμ0 : μ s ≠ 0
ave : ℝ := ⨍ (x : α) in s, f x ∂μ
S₁ : Set α := {x | x ∈ s ∧ f x ≤ ave}
⊢ ∃ c ∈ s, f c = ⨍ (x : α) in s, f x ∂μ
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
c d : ℝ≥0
hcd : c < d
s : Set α
hc : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a
hd : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a
⊢ μ s = 0

Tactic:
by
  apply measure_null_of_locally_null s fun x _ => ?_
  obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top μ x
  refine ⟨s ∩ o, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), ?_⟩
  let s' := s ∩ o
  by_contra h
  apply lt_irrefl (ρ s')
  calc
    ρ s' ≤ c * μ s' := v.measure_le_of_frequently_le (c • μ) hρ s' fun x hx => hc x hx.1
    _ < d * μ s' := by gcongr; exact measure_ne_top_of_subset inter_subset_right μo.ne
    _ ≤ ρ s' := v.measure_le_of_frequently_le ρ smul_absolutelyContinuous s' fun x hx ↦ hd x hx.1

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

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y ↦ (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
L4 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b ↦ (↑f b - ↑f x) / (b - x)) (𝓝[≠] x) (𝓝 (f.measure.rnDeriv volume x).toReal)

Tactic:
← nhdsLT_sup_nhdsGT,

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y ↦ (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
L4 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b ↦ (↑f b - ↑f x) / (b - x)) (𝓝[<] x ⊔ 𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: 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
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b ↦ (f b - f x) / (b - x)) (𝓝[≠] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)

Tactic:
← nhdsLT_sup_nhdsGT,

Goal state after:
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
L1 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b ↦ (f b - f x) / (b - x)) (𝓝[<] x ⊔ 𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case a
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
p : (n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s }
hp :
  ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
    ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))
⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) ≤
    ⨆ p, ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))

Tactic:
rw [hp]

Goal state after:
case a
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
n : ℕ
u : ℕ → α
u_mono : Monotone u
u_mem : ∀ (i : ℕ), u i ∈ s
p : (n : ℕ) × { u // StrictMonoOn u (Iic n) ∧ ∀ i ∈ Iic n, u i ∈ s }
hp :
  ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) =
    ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))
⊢ ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i)) ≤
    ⨆ p, ∑ i ∈ Finset.range p.fst, edist (f (↑p.snd (i + 1))) (f (↑p.snd i))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν : Measure α
inst✝¹ : SigmaFinite ν
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : ℕ → Set α := disjointed (spanningSets ν)
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : ∀ (n : ℕ), Fact (ν (s n) < ∞)
ξ : ℕ → Measure α := fun n ↦ (μ.restrict (s n)).singularPart (ν.restrict (s n))
f : ℕ → α → ℝ≥0∞ := fun n ↦ (s n).indicator ((μ.restrict (s n)).rnDeriv (ν.restrict (s n)))
hfm : ∀ (n : ℕ), Measurable (f n)
hξ : sum ξ ⟂ₘ ν
hadd : sum ξ + ν.withDensity (∑' (n : ℕ), f n) = μ
⊢ Measurable (sum ξ, ∑' (n : ℕ), f n).2

Tactic:
fun_prop

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x y : α
hx : x ∈ s
hy : y ∈ s
hxy : y ≤ x
u : ℕ → α := fun n ↦ if n = 0 then y else x
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
⊢ edist (f x) (f y) ≤ eVariationOn f s

Tactic:
simpa only [Finset.sum_range_one] using! sum_le (n := 1) hu us

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

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : 0 < N
A : N - 1 + 1 = N
B : N - 1 + 1 < N + 1
⊢ edist (f (w (N + 1))) (f (w (N - 1))) ≤ ∑ i ∈ Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))

Tactic:
have C : N - 1 < N + 1 := lt_of_le_of_lt N.pred_le N.lt_succ_self

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i ↦ if i < N then u i else if i = N then x else u (i - 1)
hw : Monotone w
Npos : 0 < N
A : N - 1 + 1 = N
B : N - 1 + 1 < N + 1
C : N - 1 < N + 1
⊢ edist (f (w (N + 1))) (f (w (N - 1))) ≤ ∑ i ∈ Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

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

Tactic:
have := hmono (by simp) (by simp) zero_lt_one
      simp [(by grind : u 0 = a), (by grind : u 1 = b), edist_comm]

Goal state after:
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
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set α := ⋯
A : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
B : ∀ᵐ (x : α) ∂μ.restrict s, t.indicator 1 x = 1
x : α
hx : Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 1)
h'x : t.indicator 1 x = 1
⊢ (fun a ↦ μ (t ∩ a) / μ a) =ᶠ[v.filterAt x] fun a ↦ μ (s ∩ a) / μ a

Tactic:
filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha

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
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set α := toMeasurable μ s
A : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
B : ∀ᵐ (x : α) ∂μ.restrict s, t.indicator 1 x = 1
x : α
hx : Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 1)
h'x : t.indicator 1 x = 1
a✝ : Set α
ha : MeasurableSet a✝
⊢ μ (t ∩ a✝) / μ a✝ = μ (s ∩ a✝) / μ a✝
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case e'_3.a.inl.inr
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
h : x ≤ y
u : β
us : u ∈ t
vφx : φ x ≤ φ u
vφy : φ u ≤ φ y
xu : x ≤ u
yu : y ≤ u
⊢ φ u ∈ φ '' (t ∩ Icc x y)

Tactic:
rw [le_antisymm vφy (hφ hy us yu)]

Goal state after:
case e'_3.a.inl.inr
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
h : x ≤ y
u : β
us : u ∈ t
vφx : φ x ≤ φ u
vφy : φ u ≤ φ y
xu : x ≤ u
yu : y ≤ u
⊢ φ y ∈ φ '' (t ∩ Icc x y)
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
f : R → R
a : A
ha : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ (if h : p a ∧ ContinuousOn f (spectrum R a) then
      (cfcHom ⋯) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }
    else 0) =
    0

Tactic:
dif_neg ha

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
f : R → R
a : A
ha : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ 0 = 0