mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
inst✝ : MetricSpace X
D : DeloneSet X
r : ℝ≥0
hr : r ≤ D.packingRadius / 2
x y z : X
hx : x ∈ D
hy : y ∈ D
hxz : x ∈ ball z ↑r
hyz : y ∈ ball z ↑r
hne : ¬x = y
⊢ ↑r + ↑r ≤ ↑D.packingRadius

Tactic:
by rw [← add_halves D.packingRadius, NNReal.coe_add]; gcongr

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

Goal state before:
pp: x y : ℝ≥0∞
h : x < y
⊢ (if x = 0 then ⊥ else if x = ∞ then ⊤ else ↑(Real.log x.toReal)) <
    if y = 0 then ⊥ else if y = ∞ then ⊤ else ↑(Real.log y.toReal)

Tactic:
split_ifs

Goal state after:
case pos
x y : ℝ≥0∞
h : x < y
h✝¹ : x = 0
h✝ : y = 0
⊢ ⊥ < ⊥

case pos
x y : ℝ≥0∞
h : x < y
h✝² : x = 0
h✝¹ : ¬y = 0
h✝ : y = ∞
⊢ ⊥ < ⊤

case neg
x y : ℝ≥0∞
h : x < y
h✝² : x = 0
h✝¹ : ¬y = 0
h✝ : ¬y = ∞
⊢ ⊥ < ↑(Real.log y.toReal)

case pos
x y : ℝ≥0∞
h : x < y
h✝² : ¬x = 0
h✝¹ : x = ∞
h✝ : y = 0
⊢ ⊤ < ⊥

case pos
x y : ℝ≥0∞
h : x < y
h✝³ : ¬x = 0
h✝² : x = ∞
h✝¹ : ¬y = 0
h✝ : y = ∞
⊢ ⊤ < ⊤

case neg
x y : ℝ≥0∞
h : x < y
h✝³ : ¬x = 0
h✝² : x = ∞
h✝¹ : ¬y = 0
h✝ : ¬y = ∞
⊢ ⊤ < ↑(Real.log y.toReal)

case pos
x y : ℝ≥0∞
h : x < y
h✝² : ¬x = 0
h✝¹ : ¬x = ∞
h✝ : y = 0
⊢ ↑(Real.log x.toReal) < ⊥

case pos
x y : ℝ≥0∞
h : x < y
h✝³ : ¬x = 0
h✝² : ¬x = ∞
h✝¹ : ¬y = 0
h✝ : y = ∞
⊢ ↑(Real.log x.toReal) < ⊤

case neg
x y : ℝ≥0∞
h : x < y
h✝³ : ¬x = 0
h✝² : ¬x = ∞
h✝¹ : ¬y = 0
h✝ : ¬y = ∞
⊢ ↑(Real.log x.toReal) < ↑(Real.log y.toReal)
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
⊢ linearGrowthSup v + 0 = linearGrowthSup v

Tactic:
exact add_zero (linearGrowthSup v)

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

Goal state before:
pp: v : ℕ → ℕ
h : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
M : ℕ
this : Tendsto (fun n ↦ ↑(v n)) atTop (𝓝 ⊤)
n : ℕ
⊢ ↑↑M < ↑(v n) → M ≤ v n

Tactic:
by exact_mod_cast le_of_lt

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
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
this : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ g x, t ⊆ closedBall x ε
t : Set (X × Set X)
ts : ∀ p ∈ t, p.1 ∈ s
tdisj : t.PairwiseDisjoint fun p ↦ p.2
tg : ∀ p ∈ t, p.2 ∈ g p.1
μt : μ (s \ ⋃ p ∈ t, p.2) = 0
⊢ ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0

Tactic:
exact ⟨t, ts, tdisj, fun p hp => (tg p hp).1, μt⟩

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hs : NullMeasurableSet s μ
hs₀ : μ s ≠ 0
hsc₀ : μ sᶜ ≠ 0
⊢ ⨍⁻ (x : α), f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in sᶜ, f x ∂μ)

Tactic:
simpa only [union_compl_self, restrict_univ] using
    laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
      (measure_ne_top _ _)

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hgi : ∫⁻ (x : α), g x ∂μ ≠ ∞
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this : f =ᵐ[μ.restrict (s ∪ t)] g
h1 : ∀ᵐ (x : α) ∂μ, x ∈ hf'.sigmaFiniteSetᶜ → f x = 0 x
h2 : ∀ᵐ (x : α) ∂μ, x ∈ hg'.sigmaFiniteSetᶜ → g x = 0 x
x : α
⊢ (x ∈ hf'.sigmaFiniteSetᶜ → f x = 0 x) →
    (x ∈ hg'.sigmaFiniteSetᶜ → g x = 0 x) → x ∈ hf'.sigmaFiniteSetᶜ ∩ hg'.sigmaFiniteSetᶜ → f x = g x

Tactic:
h1

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hgi : ∫⁻ (x : α), g x ∂μ ≠ ∞
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this : f =ᵐ[μ.restrict (s ∪ t)] g
h1✝ : ∀ᵐ (x : α) ∂μ, x ∈ hf'.sigmaFiniteSetᶜ → f x = 0 x
h2 : ∀ᵐ (x : α) ∂μ, x ∈ hg'.sigmaFiniteSetᶜ → g x = 0 x
x : α
h1 : x ∈ hf'.sigmaFiniteSetᶜ → f x = 0 x
⊢ (x ∈ hg'.sigmaFiniteSetᶜ → g x = 0 x) → x ∈ hf'.sigmaFiniteSetᶜ ∩ hg'.sigmaFiniteSetᶜ → f x = g x
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hf : IntegrableOn f s μ
hf₀ : 0 ≤ᵐ[μ.restrict s] f
⊢ ENNReal.ofReal (⨍ (x : α) in s, f x ∂μ) = (∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ) / μ s

Tactic:
by
  simpa using ofReal_average hf hf₀

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

Goal state before:
pp: case inr
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
⊢ ∃ K, μ (closedBall x K) < ∞ ∧ ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K

Tactic:
obtain ⟨a, hav, R0a⟩ : ∃ a ∈ v, R0 / 2 < r a := by
        obtain ⟨r', r'mem, hr'⟩ : ∃ r' ∈ r '' v, R0 / 2 < r' :=
          exists_lt_of_lt_csSup (vnonempty.image _) (half_lt_self R0pos)
        rcases (mem_image _ _ _).1 r'mem with ⟨a, hav, rfl⟩
        exact ⟨a, hav, hr'⟩

Goal state after:
case inr
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
⊢ ∃ K, μ (closedBall x K) < ∞ ∧ ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
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, μ (closedBall x K) < ∞ ∧ ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K

Tactic:
have Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x := by
      intro a hav
      apply dist_le_add_of_nonempty_closedBall_inter_closedBall
      refine hav.2.mono ?_
      gcongr
      exacts [hB a (ut (vu hav)), ball_subset_closedBall]
    set R0 := sSup (r '' v) with R0_def
    have R0_bdd : BddAbove (r '' v) := by
      refine ⟨1, fun r' hr' => ?_⟩
      rcases (mem_image _ _ _).1 hr' with ⟨b, hb, rfl⟩
      exact le_trans (ut' (vu hb)).2 (hR1 (c b))
    rcases le_total R0 (R x) with (H | H)
    · refine ⟨20 * R x, hRμ x, fun a au hax => ?_⟩
      refine (hB a (ut au)).trans ?_
      apply closedBall_subset_closedBall'
      have : r a ≤ R0 := le_csSup R0_bdd (mem_image_of_mem _ ⟨au, hax⟩)
      linarith [Idist_v a ⟨au, hax⟩, hR0 x]
    · have R0pos : 0 < R0 := (hR0 x).trans_le H
      have vnonempty : v.Nonempty := by
        by_contra h
        rw [nonempty_iff_ne_empty, Classical.not_not] at h
        rw [h, image_empty, Real.sSup_empty] at R0_def
        exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def))
      obtain ⟨a, hav, R0a⟩ : ∃ a ∈ v, R0 / 2 < r a := by
        obtain ⟨r', r'mem, hr'⟩ : ∃ r' ∈ r '' v, R0 / 2 < r' :=
          exists_lt_of_lt_csSup (vnonempty.image _) (half_lt_self R0pos)
        rcases (mem_image _ _ _).1 r'mem with ⟨a, hav, rfl⟩
        exact ⟨a, hav, hr'⟩
      refine ⟨8 * R0, ?_, ?_⟩
      · apply lt_of_le_of_lt (measure_mono _) (hRμ (c a))
        apply closedBall_subset_closedBall'
        rw [dist_comm]
        linarith [Idist_v a hav, (ut' (vu hav)).2]
      · intro b bu hbx
        refine (hB b (ut bu)).trans ?_
        apply closedBall_subset_closedBall'
        have : r b ≤ R0 := le_csSup R0_bdd (mem_image_of_mem _ ⟨bu, hbx⟩)
        linarith [Idist_v b ⟨bu, hbx⟩]

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)}
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
I : ∑' (a : ↑v), μ (B✝ ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B✝ ↑↑a) < ε / ↑C
k : Set α := ⋃ a ∈ w, B✝ ↑a
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B✝ a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B✝ a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := ⟨b, bv⟩
b'_notmem_w : b' ∉ w
b'' : { a // a ∉ w } := ⟨b', b'_notmem_w⟩
e : α
ea : e ∈ B✝ a
eb : e ∈ B✝ b
A : dist (c a) e ≤ r a
B : dist e (c b) ≤ r b
⊢ dist (c a) (c b) ≤ 3 * r b

Tactic:
linarith only [dist_triangle (c a) e (c b), A, B, bdiam]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν : Measure α
inst✝ : SigmaFinite ν
s : Set α
hs : MeasurableSet s
⊢ (ν.withDensity (s.indicator 1)).rnDeriv ν =ᵐ[ν] s.indicator 1

Tactic:
exact rnDeriv_withDensity _ (measurable_one.indicator hs)

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

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

Tactic:
ENNReal.coe_one,

Goal state after:
case pos.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 y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ (max (4 * K + 3) 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ μ (closedBall x (3 * (R / 4))) ≤ 1 * μ (closedBall y r)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case refine_3
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
C : ℝ≥0
h : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)
s : Set α
f : α → Set (Set α)
fsubset :
  ∀ x ∈ s,
    f x ⊆
      {a |
        IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧
          (interior a).Nonempty ∧ ∃ r, a ⊆ closedBall x r ∧ μ (closedBall x (3 * r)) ≤ ↑C * μ a}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
t : Set (ℝ × α × Set α) :=
  {p |
    p.2.2 ⊆ closedBall p.2.1 p.1 ∧
      μ (closedBall p.2.1 (3 * p.1)) ≤ ↑C * μ p.2.2 ∧
        (interior p.2.2).Nonempty ∧
          IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] p.2.2 ∧ p.2.2 ∈ f p.2.1 ∧ p.2.1 ∈ s}
A : ∀ x ∈ s, ∀ ε > 0, ∃ p ∈ t, p.1 ≤ ε ∧ p.2.1 = x
t' : Set (ℝ × α × Set α)
t't : t' ⊆ t
left✝ : t'.Countable
t'_disj : t'.PairwiseDisjoint fun p ↦ p.2.2
μt' : μ (s \ ⋃ a ∈ t', a.2.2) = 0
⊢ ∀ p ∈ (fun p ↦ p.2) '' t', p.2 ∈ f p.1

Tactic:
rintro - ⟨q, hq, rfl⟩
      exact (t't hq).2.2.2.2.1

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 ⟂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
μa_pos : 0 < μ a
μa_lt_top : μ a < ∞
⊢ ρ a / μ a < z

Tactic:
ENNReal.div_lt_iff (.inl μa_pos.ne') (.inl μa_lt_top.ne),

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
μa_pos : 0 < μ a
μa_lt_top : μ a < ∞
⊢ ρ a < z * μ a
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA νA : ℝ≥0
hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : νA ≤ 0
⊢ μA = 0

Tactic:
· rw [le_zero_iff] at hb
      simpa [hb] using hA₃ 0

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

Goal state before:
pp: case zero.right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (x : α) in A, (fun x ↦ ⨆ k, ⨆ (_ : k ≤ 0), f k x) x ∂μ ≤ ν A

Tactic:
simp [(hf 0).2 A hA]

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

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
a b : α
hab : a ≤ b
⊢ |variationOnFromTo f s a b| ≤ (eVariationOn f s).toReal

pp: case neg
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
a b : α
hab : ¬a ≤ b
⊢ |variationOnFromTo f s a b| ≤ (eVariationOn f s).toReal

Tactic:
· simp only [variationOnFromTo, hab, ↓reduceIte, ENNReal.abs_toReal]
    exact ENNReal.toReal_mono hf (eVariationOn.mono _ inter_subset_left)

Goal state after:
case neg
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
a b : α
hab : ¬a ≤ b
⊢ |variationOnFromTo f s a b| ≤ (eVariationOn f s).toReal
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
b : α
bs : b ∈ s
c : α
cs : c ∈ s
bc : b ≤ c
⊢ (variationOnFromTo f s a + f) b ≤ (variationOnFromTo f s a + f) c

Tactic:
suffices f b - f c ≤ variationOnFromTo f s a c - variationOnFromTo f s a b by simp; linarith

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
hf : LocallyBoundedVariationOn f s
a : α
as : a ∈ s
b : α
bs : b ∈ s
c : α
cs : c ∈ s
bc : b ≤ c
⊢ f b - f c ≤ variationOnFromTo f s a c - variationOnFromTo f s a b
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 : α
hx : x ∈ s
y : α
hy : 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:
← variationOnFromTo.add h hx cs hy,

Goal state after:
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 : α
hx : x ∈ s
y : α
hy : 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 c + variationOnFromTo f s c y
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
I : ℕ → α
hI : Monotone I
n : ℕ
⊢ ∑ i ∈ Finset.range n, eVariationOn f (Icc (I i) (I (i + 1))) = eVariationOn f (Icc (I 0) (I n))

Tactic:
by
  convert!
      sum f hI (s := Icc (I 0) (I n)) (n := n)
        (hn := by intros; rw [mem_Icc]; constructor <;> (apply hI; lia))
    with i hi
  · simp only [right_eq_inter]
    gcongr <;> (apply hI; rw [Finset.mem_range] at hi; lia)
  · simp

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 : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
⊢ ∀ᶠ (x_1 : Set α) in v.filterAt x, (∫⁻ (y : α) in x_1, ‖f y - f x‖ₑ ∂μ) / μ x_1 ≤ ε

Tactic:
obtain ⟨c, ct, xc⟩ : ∃ c ∈ t, ‖f x - c‖ₑ < ε / 2 := by
    simp_rw [← edist_eq_enorm_sub]
    have : f x ∈ closure t := ht (mem_range_self _)
    exact EMetric.mem_closure_iff.1 this (ε / 2) (ENNReal.half_pos (ne_of_gt εpos))

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 : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K} :=
  μ.finiteSpanningSetsInOpen'
t : Set E
t_count : t.Countable
ht : range f ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
main :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ c ∈ t,
        Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
          (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
x : α
hx :
  ∀ (n : ℕ),
    ∀ c ∈ t,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - (A.set n).indicator (fun x ↦ c) y‖ₑ ∂μ) / μ a) (v.filterAt x)
        (𝓝 ‖f x - (A.set n).indicator (fun x ↦ c) x‖ₑ)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
M : ∀ c ∈ t, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ)
ε : ℝ≥0∞
εpos : ε > 0
c : E
ct : c ∈ t
xc : ‖f x - c‖ₑ < ε / 2
⊢ ∀ᶠ (x_1 : Set α) in v.filterAt x, (∫⁻ (y : α) in x_1, ‖f y - f x‖ₑ ∂μ) / μ x_1 ≤ ε