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

Goal state before:
pp: 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
⊢ 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:
convert (this.const_smul (c := (z ^ (a + 1))⁻¹)).congr ?_

Goal state after:
case e'_10
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
⊢ (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) =
    (z ^ (a + 1))⁻¹ •
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)).compContinuousLinearMap (z⁻¹ • 1)

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)
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 : AnalyticAt 𝕜 f z
p : A[X]
⊢ AnalyticAt 𝕜 (fun x ↦ (aeval (f x)) p) z

Tactic:
← analyticWithinAt_univ

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 : 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
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case refine_2.convert_11
w x : ℂ
hw : w ≠ x
⊢ 0 < ‖w - x‖ₑ

Tactic:
simpa [sub_eq_zero]

Goal state after:
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:
· rw [log_pos_real' y_real, ENNReal.top_mul', EReal.top_add_coe, log_eq_top_iff]
      simp only [ite_eq_right_iff, zero_ne_top, imp_false]
      exact (ENNReal.toReal_pos_iff.1 y_real).1.ne'

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
⊢ (x * ∞).log = ↑(Real.log x.toReal) + ∞.log

pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log

Tactic:
· simp [(ENNReal.toReal_pos_iff.1 x_real).1.ne']

Goal state after:
case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
⊢ (x ^ y).log = ↑y * x.log

Tactic:
rcases ENNReal.trichotomy x with (rfl | rfl | x_real)
    · rw [ENNReal.zero_rpow_of_pos y_pos, log_zero, EReal.mul_bot_of_pos]; norm_cast
    · rw [ENNReal.top_rpow_of_pos y_pos, log_top, EReal.mul_top_of_pos]; norm_cast
    · have x_ne_zero := (ENNReal.toReal_pos_iff.1 x_real).1.ne'
      have x_ne_top := (ENNReal.toReal_pos_iff.1 x_real).2.ne
      simp only [log, rpow_eq_zero_iff, x_ne_zero, false_and, x_ne_top, or_self, ↓reduceIte,
        rpow_eq_top_iff]
      norm_cast
      exact ENNReal.toReal_rpow x y ▸ Real.log_rpow x_real y

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

Goal state before:
pp: a✝ b✝ : EReal
a b : ℝ
hb : 0 ≤ ↑b
h : ↑a < ↑b
x : ℝ
hx : ∀ (b_1 : ℝ), x ≤ b_1 → ∃ n, a * b_1 ≤ ↑n ∧ ↑n ≤ b * b_1
n : ℕ
x_n : x ≤ ↑n
⊢ ∀ᶠ (n : ℕ) in atTop, ∃ m, ↑a * ↑n ≤ ↑m ∧ ↑m ≤ ↑b * ↑n

Tactic:
refine eventually_atTop.2 ⟨n, fun k n_k ↦ ?_⟩

Goal state after:
a✝ b✝ : EReal
a b : ℝ
hb : 0 ≤ ↑b
h : ↑a < ↑b
x : ℝ
hx : ∀ (b_1 : ℝ), x ≤ b_1 → ∃ n, a * b_1 ≤ ↑n ∧ ↑n ≤ b * b_1
n : ℕ
x_n : x ≤ ↑n
k : ℕ
n_k : n ≤ k
⊢ ∃ m, ↑a * ↑k ≤ ↑m ∧ ↑m ≤ ↑b * ↑k
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case neg.inr
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
u : Set ι
hu : Maximal (fun x ↦ x ∈ T) u
A : Set ι := {a' | a' ∈ t ∧ ∀ c ∈ u, Disjoint (B a') (B c)}
Anonempty : A.Nonempty
m : ℝ := sSup (δ '' A)
bddA : BddAbove (δ '' A)
a' : ι
a'A : a' ∈ A
ha' : m / τ ≤ δ a'
a'_ne_u : a' ∉ u
c : ι
ct : c ∈ t
b : ι
ba'u : b ∈ insert a' u
hcb : (B c ∩ B b).Nonempty
H : ∀ d ∈ u, Disjoint (B c) (B d)
H' : b ∈ u
⊢ ∃ c_1 ∈ insert a' u, (B c ∩ B c_1).Nonempty ∧ δ c ≤ τ * δ c_1

Tactic:
rw [← not_disjoint_iff_nonempty_inter] at hcb
        exact (hcb (H _ H')).elim

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hs₀ : μ s ≠ 0
ht₀ : μ t ≠ 0
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
⊢ μ s / (μ s + μ t) + μ t / (μ s + μ t) = 1

Tactic:
← ENNReal.add_div,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hs₀ : μ s ≠ 0
ht₀ : μ t ≠ 0
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
⊢ (μ s + μ t) / (μ s + μ t) = 1
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hs₀ : ¬μ s = 0
⊢ μ s / (μ s + μ t) + μ t / (μ s + μ t) = 1

Tactic:
← ENNReal.add_div,

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hs₀ : ¬μ s = 0
⊢ (μ s + μ t) / (μ s + μ t) = 1
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case e'_6
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
hf : ¬Integrable f μ
h : Integrable (fun x ↦ f x - ⨍ (a : α), f a ∂μ) μ
⊢ f = (fun x ↦ f x - ⨍ (a : α), f a ∂μ) + fun x ↦ ⨍ (a : α), f a ∂μ

Tactic:
exact (sub_add_cancel _ _).symm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : AEMeasurable f (μ.restrict s)
h : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞
this : 0 < μ {x | x ∈ s ∧ (f x).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ}
⊢ 0 < μ {x | x ∈ s ∧ f x ≤ ⨍⁻ (a : α) in s, f a ∂μ}

Tactic:
← ofPred_inter_eq_sep,

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : AEMeasurable f (μ.restrict s)
h : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞
this : 0 < μ {x | x ∈ s ∧ (f x).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ}
⊢ 0 < μ ({a | f a ≤ ⨍⁻ (a : α) in s, f a ∂μ} ∩ s)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : AEMeasurable f (μ.restrict s)
h : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞
this : 0 < μ {x | x ∈ s ∧ (f x).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ}
⊢ 0 < (μ.restrict s) {a | f a ≤ ⨍⁻ (a : α) in s, f a ∂μ}

Tactic:
← ofPred_inter_eq_sep,

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : AEMeasurable f (μ.restrict s)
h : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞
this : 0 < μ ({a | (f a).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ} ∩ s)
⊢ 0 < (μ.restrict s) {a | f a ≤ ⨍⁻ (a : α) in s, f a ∂μ}
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case inr.refine_1
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)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ μ (closedBall x (8 * R0)) < ∞

Tactic:
apply lt_of_le_of_lt (measure_mono _) (hRμ (c a))

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 := ⋯
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := ⋯
vu : v ⊆ u
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := ⋯
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ closedBall x (8 * R0) ⊆ closedBall (c a) (20 * R (c a))
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < μ {x | x ∈ s ∧ ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g x).toReal}
⊢ 0 < μ {x | x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f x}

Tactic:
← ofPred_inter_eq_sep,

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < μ {x | x ∈ s ∧ ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g x).toReal}
⊢ 0 < μ ({a | ⨍⁻ (a : α) in s, f a ∂μ ≤ f a} ∩ s)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < μ {x | x ∈ s ∧ ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g x).toReal}
⊢ 0 < (μ.restrict s) {a | ⨍⁻ (a : α) in s, g a ∂μ ≤ f a}

Tactic:
← ofPred_inter_eq_sep,

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < μ ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} ∩ s)
⊢ 0 < (μ.restrict s) {a | ⨍⁻ (a : α) in s, g a ∂μ ≤ f a}
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)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ ∑' (a : ↑v), μ (B ↑a) = μ (⋃ a ∈ v, B a)

Tactic:
by
        rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]
        exact u_disj.subset vu

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
inst✝ : IsProbabilityMeasure μ
hf : AEMeasurable f μ
⊢ ∃ x, f x ≤ ∫⁻ (a : α), f a ∂μ

Tactic:
simpa only [laverage_eq_lintegral] using exists_le_laverage (IsProbabilityMeasure.ne_zero μ) hf

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

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ (r • μ).singularPart ν = r • μ.singularPart ν

Tactic:
rw [singularPart, singularPart, dif_neg hl, dif_neg, smul_zero]

Goal state after:
case neg.hnc
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ ¬(r • μ).HaveLebesgueDecomposition ν
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f ν
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)

Tactic:
rw [withDensity_congr_ae hf.ae_eq_mk] at hadd ⊢
  exact eq_withDensity_rnDeriv hf.measurable_mk hs hadd

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 : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
⊢ r • ν = (r • ν).singularPart μ + r • μ.withDensity (ν.rnDeriv μ)

Tactic:
singularPart_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
⊢ r • ν = r • ν.singularPart μ + r • μ.withDensity (ν.rnDeriv μ)
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
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:
have I1 : closedBall x (3 * ((K + 1) * r)) ⊆ closedBall y ((4 * K + 3) * r) := by
        apply closedBall_subset_closedBall'
        linarith
      have I2 : closedBall y ((4 * K + 3) * r) ⊆ closedBall y (max (4 * K + 3) 3 * r) := by
        gcongr
        exact le_max_left ..
      apply (measure_mono (I1.trans I2)).trans
      exact measure_mul_le_scalingConstantOf_mul _
        ⟨zero_lt_three.trans_le (le_max_right _ _), le_rfl⟩ hr

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (slope (↑hf.stieltjesFunction) x) (𝓝[<] x ⊔ 𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
⊢ HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x

Tactic:
tendsto_sup,

Goal state after:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (slope (↑hf.stieltjesFunction) x) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (slope (↑hf.stieltjesFunction) x) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
⊢ HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv volume x).toReal 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
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
⊢ ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ w n)

Tactic:
by
        conv_rhs => rw [inter_comm]
        rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J]

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✝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
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
A : Set α
hE₃ : ↑ε * ν (A ∩ E) ≤ μ (A ∩ E) - ∫⁻ (a : α) in A ∩ E, ξ a ∂ν
hA : MeasurableSet A
⊢ ↑ε * ν (A ∩ E) + ∫⁻ (a : α) in A ∩ E, ξ a ∂ν ≤ μ (A ∩ E)

Tactic:
refine add_le_of_le_tsub_right_of_le (hξle _ (hA.inter hE₁)) hE₃

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✝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
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (x : α) in A, (ξ + E.indicator fun x ↦ ↑ε) x ∂ν ≤ μ A

Tactic:
have : ∫⁻ a in A, (ξ + E.indicator fun _ ↦ (ε : ℝ≥0∞)) a ∂ν =
            ∫⁻ a in A ∩ E, ε + ξ a ∂ν + ∫⁻ a in A \ E, ξ a ∂ν := by
          simp only [lintegral_add_left measurable_const, lintegral_add_left hξm,
            setLIntegral_const, add_assoc, lintegral_inter_add_sdiff _ _ hE₁, Pi.add_apply,
            lintegral_indicator hE₁, restrict_apply hE₁]
          rw [inter_comm, add_comm]

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
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this✝ : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
A : Set α
hA : MeasurableSet A
this :
  ∫⁻ (a : α) in A, (ξ + E.indicator fun x ↦ ↑ε) a ∂ν = ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν + ∫⁻ (a : α) in A \ E, ξ a ∂ν
⊢ ∫⁻ (x : α) in A, (ξ + E.indicator fun x ↦ ↑ε) x ∂ν ≤ μ A
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
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
m n : ℕ
I : (ρ + μ) (u m) ≠ ∞
J : (ρ + μ) (w n) ≠ ∞
A :
  ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B :
  ↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
H :
  ¬ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ∞ ∧
    ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ∞
⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ∞

Tactic:
exact H.2

Goal state after: