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

Goal state before:
pp: case convert_2
a : ℕ
z : ℂ
hz : z ≠ 0
this✝ :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) ((z⁻¹ • 1) 0) 1
H : 1 / ‖z⁻¹ • 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹)
    ((FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)).compContinuousLinearMap (z⁻¹ • 1)) 0 ‖z‖ₑ
⊢ Set.EqOn ((z ^ (a + 1))⁻¹ • fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹) (fun x ↦ 1 / (z - x) ^ (a + 1))
    (Metric.eball 0 ‖z‖ₑ)

Tactic:
intro w hw

Goal state after:
case convert_2
a : ℕ
z : ℂ
hz : z ≠ 0
this✝ :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) ((z⁻¹ • 1) 0) 1
H : 1 / ‖z⁻¹ • 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹)
    ((FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)).compContinuousLinearMap (z⁻¹ • 1)) 0 ‖z‖ₑ
w : ℂ
hw : w ∈ Metric.eball 0 ‖z‖ₑ
⊢ ((z ^ (a + 1))⁻¹ • fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹) w = (fun x ↦ 1 / (z - x) ^ (a + 1)) w
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: a : ℕ
z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) 0 ‖z‖ₑ

Tactic:
by
  have := one_div_one_sub_pow_hasFPowerSeriesOnBall_zero a
  rw [← map_zero (z⁻¹ • 1 : ℂ →L[ℂ] ℂ)] at this
  have := this.compContinuousLinearMap
  have H : 1 / ‖(z⁻¹ • 1 : ℂ →L[ℂ] ℂ)‖ₑ = ‖z‖ₑ := by simp [enorm_smul, enorm_inv, hz]
  simp only [one_div, FunLike.coe_smul, H, Function.comp_def] at this
  convert (this.const_smul (c := (z ^ (a + 1))⁻¹)).congr ?_
  · ext n
    simp only [FormalMultilinearSeries.smul_apply, smul_apply,
      FormalMultilinearSeries.compContinuousLinearMap_apply]
    simp [add_assoc, pow_add _ _ (a + 1), mul_assoc]
  · intro w hw
    simp [← mul_inv_rev, ← mul_pow, sub_mul, mul_right_comm _ w, hz]

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

Goal state before:
pp: a : ℝ
H : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)
this : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1
⊢ HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ a) (binomialSeries ℝ a) 0 1

Tactic:
convert! (Complex.reCLM.comp_hasFPowerSeriesOnBall this.compContinuousLinearMap).congr ?_

Goal state after:
case e'_10
a : ℝ
H : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)
this : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1
⊢ binomialSeries ℝ a =
    Complex.reCLM.compFormalMultilinearSeries ((binomialSeries ℂ a).compContinuousLinearMap Complex.ofRealCLM)

case e'_12
a : ℝ
H : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)
this : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1
⊢ 1 = 1 / ‖Complex.ofRealCLM‖ₑ

case convert_2
a : ℝ
H : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)
this : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1
⊢ Set.EqOn (⇑Complex.reCLM ∘ (fun x ↦ (1 + x) ^ ↑a) ∘ ⇑Complex.ofRealCLM) (fun x ↦ (1 + x) ^ a)
    (Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ))
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ linearGrowthInf v + 0 = linearGrowthInf v

Tactic:
exact add_zero (linearGrowthInf v)

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 g : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hf : ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)
hfg : abs ∘ g ≤ᶠ[l] abs ∘ f
z : ℕ
x : α
hx : (abs ∘ g) x ≤ (abs ∘ f) x
⊢ |k x ^ z| * |g x| ≤ |k x ^ z| * |f x|

Tactic:
gcongr _ * ?_; exact hx

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

Goal state before:
pp: case inr
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
hx : x ∈ s
ε : ℝ
εpos : ε > 0
t : Set X
tf : t ∈ f x
ht : t ⊆ closedBall x (min ε δ)
h't : t ∈ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
⊢ ∃ t ∈ g x, t ⊆ closedBall x ε

Tactic:
refine False.elim (h't.2.2 ?_)
        exact ht.trans (closedBall_subset_closedBall (min_le_right _ _))

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

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
x : X
a✝ : Set X
⊢ a✝ ∈ v.setsAt x → MeasurableSet a✝

Tactic:
ha

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
x : X
a✝ : Set X
ha : a✝ ∈ v.setsAt x
⊢ MeasurableSet a✝
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: case insert
α : Type u_1
u : α → ℕ → ℝ≥0∞
a : α
t : Finset α
a_t : a ∉ t
ha : expGrowthSup (∑ x ∈ t, u x) = ⨆ x ∈ t, expGrowthSup (u x)
⊢ max (expGrowthSup (u a)) (expGrowthSup (∑ x ∈ t, u x)) = max (expGrowthSup (u a)) (⨆ x ∈ t, expGrowthSup (u x))

Tactic:
ha

Goal state after:
case insert
α : Type u_1
u : α → ℕ → ℝ≥0∞
a : α
t : Finset α
a_t : a ∉ t
ha : expGrowthSup (∑ x ∈ t, u x) = ⨆ x ∈ t, expGrowthSup (u x)
⊢ max (expGrowthSup (u a)) (⨆ x ∈ t, expGrowthSup (u x)) = max (expGrowthSup (u a)) (⨆ x ∈ t, expGrowthSup (u x))
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
⊢ a ∈ t' → ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

Tactic:
ha

Goal state after:
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
⊢ a ∈ t' → ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
ha

Goal state after:
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n ↦ {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
this : Tendsto (fun n ↦ g x + ↑(u n)) atTop (𝓝 (g x + ↑0))
⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x

Tactic:
simp only [ENNReal.coe_zero, add_zero] at this

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n ↦ {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
hx : g x < f x
this : Tendsto (fun n ↦ g x + ↑(u n)) atTop (𝓝 (g x))
⊢ ∀ᶠ (n : ℕ) in atTop, g x + ↑(u n) ≤ f x
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : IsFiniteMeasure (μ.restrict s)
hμ0 : μ s = 0
⊢ ⨍⁻ (x : α) in s, f x ∂μ ≤ essSup f μ

Tactic:
rw [laverage, ← setLIntegral_univ]
    exact le_of_eq_of_le (setLIntegral_measure_zero univ f <| by simp [hμ0]) zero_le

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
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
a : ι
⊢ a ∈ t' → r a ≤ 1

Tactic:
ha

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)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
a : ι
ha : a ∈ t'
⊢ r a ≤ 1
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ 0 ≤ᵐ[μ.restrict s] fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ

Tactic:
refine measure_mono_null (fun x hx ↦ ?_) H

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
x : α
hx : x ∈ {x | (fun x ↦ 0 x ≤ (fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) x) x}ᶜ
⊢ x ∈ {x | f x ≤ ⨍ (a : α) in s, f a ∂μ}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ofMeasurable (fun t x ↦ μ t - ν t) ⋯ ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ μ - ν = measure_sub

Tactic:
rw [MeasureTheory.Measure.sub_def]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ofMeasurable (fun t x ↦ μ t - ν t) ⋯ ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ sInf {d | μ ≤ d + ν} = measure_sub
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝ : NormedAddCommGroup E
x : α
f : α → E
hf : LocallyIntegrable f μ
w : Set α
w_nhds : w ∈ 𝓝 x
hw : IntegrableOn f w μ
a : Set α
⊢ a ⊆ w → IntegrableOn f a μ

Tactic:
ha

Goal state after:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝ : NormedAddCommGroup E
x : α
f : α → E
hf : LocallyIntegrable f μ
w : Set α
w_nhds : w ∈ 𝓝 x
hw : IntegrableOn f w μ
a : Set α
ha : a ⊆ w
⊢ IntegrableOn f a μ
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: case pos.inl.left
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 y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := ⋯
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ closedBall y r ⊆ closedBall x ((K + 1) * r)

pp: case pos.inl.right
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 y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := ⋯
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ μ (closedBall x (3 * ((K + 1) * r))) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall y r)

Tactic:
· apply closedBall_subset_closedBall'
      rw [dist_comm]
      linarith

Goal state after:
case pos.inl.right
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 y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := ⋯
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ μ (closedBall x (3 * ((K + 1) * r))) ≤ ↑(scalingConstantOf μ (max (4 * K + 3) 3)) * μ (closedBall y r)
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 ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < ↑w
w_lt : ↑w < z
n : ℕ
hn : u n < w
a : Set α
⊢ ρ a < ↑(u n) * μ a → 0 < μ a → μ a < ∞ → ρ a / μ a < z

Tactic:
ha

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 ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < ↑w
w_lt : ↑w < z
n : ℕ
hn : u n < w
a : Set α
ha : ρ a < ↑(u n) * μ a
⊢ 0 < μ a → μ a < ∞ → ρ a / μ a < z
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) ∧
    Tendsto (fun b ↦ (f b - f x) / (b - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)

Tactic:
exact ⟨L2, L1⟩

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

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 : LocallyBoundedVariationOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
ba : b ≤ a
⊢ variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Icc b a → ∀ ⦃y : α⦄, y ∈ s ∩ Icc b a → edist (f x) (f y) = 0

Tactic:
by
  rw [variationOnFromTo.eq_of_ge _ _ ba, neg_eq_zero, ENNReal.toReal_eq_zero_iff,
    or_iff_left (hf b a hb ha), eVariationOn.eq_zero_iff]

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
s : Set α
f : α →ᵤ[{s}] E
this : ∀ s_1 ∈ {s}, TendstoUniformlyOn (⇑(UniformOnFun.toFun {s}) ∘ id) ((UniformOnFun.toFun {s}) f) (𝓝 f) s_1
⊢ ∀ x ∈ s, Tendsto (fun i ↦ id i x) (𝓝 f) (𝓝 (f x))

Tactic:
simp_rw [← tendstoUniformlyOn_singleton_iff_tendsto]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
s : Set α
f : α →ᵤ[{s}] E
this : ∀ s_1 ∈ {s}, TendstoUniformlyOn (⇑(UniformOnFun.toFun {s}) ∘ id) ((UniformOnFun.toFun {s}) f) (𝓝 f) s_1
⊢ ∀ x ∈ s, TendstoUniformlyOn id f (𝓝 f) {x}
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
h : ν₂ ⟂ₘ μ
x : α
hx_add : (ν₁ + ν₂).rnDeriv μ x = (ν₁.rnDeriv μ + ν₂.rnDeriv μ) x
hx_zero : ν₂.rnDeriv μ x = 0 x
⊢ (ν₁ + ν₂).rnDeriv μ x = ν₁.rnDeriv μ x

Tactic:
simp [hx_add, hx_zero]

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

Goal state before:
pp: case inr.refine_4
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
h : LocallyBoundedVariationOn f s
c : α
cs : c ∈ s
⊢ ∀ x ∈ s,
    ∀ y ∈ s,
      (fun x ↦ (variationOnFromTo f s c x + f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x + f x) / 2) x +
          ((fun x ↦ (variationOnFromTo f s c x - f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x - f x) / 2) x) =
        variationOnFromTo f s x y

Tactic:
intro x hx y hy
    rw [← variationOnFromTo.add h hx cs hy, variationOnFromTo.eq_neg_swap]
    ring

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
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑t ^ 2 * ρ s

Tactic:
have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne] using 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
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑t ^ 2 * ρ s
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: 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
y : α
hy : y ∈ s
⊢ ‖g y‖ₑ ≤ ‖g x‖ₑ + eVariationOn g s

Tactic:
show g y = g x + (g y - g x) by abel,

Goal state after:
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
y : α
hy : y ∈ s
⊢ ‖g x + (g y - g x)‖ₑ ≤ ‖g x‖ₑ + eVariationOn g s
Module: Mathlib.Analysis.BoundedVariation

Goal state before:
pp: s : Set ℝ
p q : ℝ → ℝ
hp : MonotoneOn p s
hq : MonotoneOn q s
h : LocallyBoundedVariationOn (p - q) s
x : ℝ
⊢ (x ∈ s → DifferentiableWithinAt ℝ p s x) →
    (x ∈ s → DifferentiableWithinAt ℝ q s x) → x ∈ s → DifferentiableWithinAt ℝ (p - q) s x

Tactic:
hxp

Goal state after:
s : Set ℝ
p q : ℝ → ℝ
hp : MonotoneOn p s
hq : MonotoneOn q s
h : LocallyBoundedVariationOn (p - q) s
x : ℝ
hxp : x ∈ s → DifferentiableWithinAt ℝ p s x
⊢ (x ∈ s → DifferentiableWithinAt ℝ q s x) → x ∈ s → DifferentiableWithinAt ℝ (p - q) s x
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
⊢ ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s

Tactic:
have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne] using 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
⊢ ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:

Tactic:
ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
        rw [← ENNReal.coe_zpow t_ne_zero']
        apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ
        intro x hx
        apply hx.2.2.trans_le (le_of_eq _)
        rw [ENNReal.coe_zpow t_ne_zero']
      _ = ∫⁻ _ in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by
        simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
      _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by
        apply lintegral_mono_ae ((ae_restrict_iff' M).2 (Eventually.of_forall fun x hx => ?_))
        grw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one, hx.2.1]
      _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas

Goal state after: