mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case h.h₁
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E x).toNNReal ≤ 1

pp: case h₂
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E y).toNNReal ≤ 1

Tactic:
<;>

Goal state after:
case h.h₁
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E x).toNNReal ≤ 1

case h₂
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E y).toNNReal ≤ 1
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case pos
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

Tactic:
<;>

Goal state after:
case pos
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
hposbdd : ∫ (ω : α), max (f ω) 0 ∂μ ≤ μ.real univ • ↑r
⊢ (∫ (a : α), ‖f a‖ ^ ENNReal.toReal 1 ∂μ) ^ (ENNReal.toReal 1)⁻¹ ≤ (2 * μ univ * ↑r).toReal

Tactic:
ENNReal.toReal_one,

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
hposbdd : ∫ (ω : α), max (f ω) 0 ∂μ ≤ μ.real univ • ↑r
⊢ (∫ (a : α), ‖f a‖ ^ 1 ∂μ) ^ 1⁻¹ ≤ (2 * μ univ * ↑r).toReal
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case pos
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : IsTop b
⊢ Ioc a b ⊆ Ioc a b

pp: case neg
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : ¬IsTop b
⊢ Ioo a b ⊆ Ioc a b

Tactic:
<;>

Goal state after:
case pos
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : IsTop b
⊢ Ioc a b ⊆ Ioc a b

case neg
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : ¬IsTop b
⊢ Ioo a b ⊆ Ioc a b
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case pos
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : IsTop b
⊢ Ioo a b ⊆ Ioc a b

pp: case neg
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : ¬IsTop b
⊢ Ioo a b ⊆ Ioo a b

Tactic:
<;>

Goal state after:
case pos
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : IsTop b
⊢ Ioo a b ⊆ Ioc a b

case neg
R : Type u_1
inst✝ : LinearOrder R
a b : R
h : ¬IsTop b
⊢ Ioo a b ⊆ Ioo a b
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ
ε0 : 0 < ε
⊢ ∀ᶠ (x : ι) in atTop, ∫ (x : X) in s x, f x ∂μ ∈ Metric.closedBall (∫ (x : X) in S, f x ∂μ) ε

Tactic:
lift ε to ℝ≥0 using ε0.le

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
⊢ ∀ᶠ (x : ι) in atTop, ∫ (x : X) in s x, f x ∂μ ∈ Metric.closedBall (∫ (x : X) in S, f x ∂μ) ↑ε
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
⊢ ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)

Tactic:
intro t K

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
⊢ ∃ p, Icc a' b ∩ s i ⊆ Iotop p.1 p.2 ∧ ofReal (↑f p.2 - ↑f p.1) < f.length (s i) + ↑(ε' i)

Tactic:
have A : Icc a' b ∩ s i ⊆ Ioc p q' := by
      rintro x ⟨hx, h'x⟩
      apply spq
      simp [h'x, notMem_botSet_of_lt (aa'.trans_le hx.1)]

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
⊢ ∃ p, Icc a' b ∩ s i ⊆ Iotop p.1 p.2 ∧ ofReal (↑f p.2 - ↑f p.1) < f.length (s i) + ↑(ε' i)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this✝ : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
htq' : ¬IsTop q'
this : (𝓝[>] q').NeBot
⊢ ContinuousWithinAt (fun r ↦ ofReal (↑f r - ↑f p)) (Ioi q') q'

Tactic:
apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ⋯
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this✝ : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
htq' : ¬IsTop q'
this : (𝓝[>] q').NeBot
⊢ ContinuousWithinAt (fun r ↦ ↑f r - ↑f p) (Ioi q') q'
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : a ≤ c
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

pp: case inr
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : c ≤ a
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

Tactic:
<;>

Goal state after:
case inl
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : a ≤ c
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

case inr
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : c ≤ a
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : μ K < ∞
⊢ μ.measure ↑K ≤ μ K + ↑ε

Tactic:
obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)

Goal state after:
case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : μ K < ∞
K' : Compacts G
K'_hyp : K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ↑ε
⊢ μ.measure ↑K ≤ μ K + ↑ε
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ ↑(index (↑K) U)

pp: case hb
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ ↑(index (↑K₀) U)

Tactic:
<;>

Goal state after:
case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ ↑(index (↑K) U)

case hb
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ ↑(index (↑K₀) U)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ index (↑K) U

pp: case hb
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ index (↑K₀) U

Tactic:
<;>

Goal state after:
case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ index (↑K) U

case hb
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ index (↑K₀) U
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
⊢ ∀ (a : G), g₁ * a ∈ ↑K₀ → a ∈ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V

Tactic:
intro g₂ hg₂

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
⊢ g₂ ∈ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹

pp: case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₂.carrier * V⁻¹

Tactic:
<;>

Goal state after:
case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹

case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₂.carrier * V⁻¹
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
h0 : ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ
h1 : ∫ (x : X) in sᶜ, ‖f x‖ ∂μ ≤ ∫ (x : X) in sᶜ, C ∂μ
h2 : ∫ (x : X) in sᶜ, C ∂μ = μ.real sᶜ * C
⊢ ‖∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ‖ ≤ μ.real sᶜ * C

Tactic:
h0,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
h0 : ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ
h1 : ∫ (x : X) in sᶜ, ‖f x‖ ∂μ ≤ ∫ (x : X) in sᶜ, C ∂μ
h2 : ∫ (x : X) in sᶜ, C ∂μ = μ.real sᶜ * C
⊢ ‖∫ (x : X) in sᶜ, f x ∂μ‖ ≤ μ.real sᶜ * C
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 < ∫ (a : X) in {x | R < f x}, f a - R ∂μ

Tactic:
setIntegral_pos_iff_support_of_nonneg_ae

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 < μ ((support fun a ↦ f a - R) ∩ {x | R < f x})

case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 ≤ᵐ[μ.restrict {x | R < f x}] fun a ↦ f a - R

case hfi
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ IntegrableOn (fun a ↦ f a - R) {x | R < f x} μ
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
this : Quotient.mk α_mod_G ⁻¹' Quotient.mk α_mod_G '' A n = ⋃ g, (fun x ↦ g • x) '' A n
⊢ MeasurableSet (Quotient.mk α_mod_G '' A n)

Tactic:
measurableSet_quotient,

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
this : Quotient.mk α_mod_G ⁻¹' Quotient.mk α_mod_G '' A n = ⋃ g, (fun x ↦ g • x) '' A n
⊢ MeasurableSet (Quotient.mk'' ⁻¹' Quotient.mk α_mod_G '' A n)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ ofReal (↑f x - l - (↑f x - leftLim (↑f) x)) = ofReal (leftLim (↑f) x - l)

pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ f.measure {x} ≠ ∞

pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ {x} ⊆ Iic x

Tactic:
<;>

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ ofReal (↑f x - l - (↑f x - leftLim (↑f) x)) = ofReal (leftLim (↑f) x - l)

R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ f.measure {x} ≠ ∞

R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ {x} ⊆ Iic x
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ ↑(index (↑K) U) ≤ ↑(index ↑K ↑K₀) * ↑(index (↑K₀) U)

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ 0 < ↑(index (↑K₀) U)

Tactic:
<;>

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ ↑(index (↑K) U) ≤ ↑(index ↑K ↑K₀) * ↑(index (↑K₀) U)

G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ 0 < ↑(index (↑K₀) U)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case ha
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
h1K : IsCompact K
h2K : (interior K).Nonempty
⊢ 0 < ↑(index (↑{ carrier := K, isCompact' := h1K }) U)

pp: case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
h1K : IsCompact K
h2K : (interior K).Nonempty
⊢ 0 < ↑(index (↑K₀) U)

Tactic:
<;>

Goal state after:
case ha
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
h1K : IsCompact K
h2K : (interior K).Nonempty
⊢ 0 < ↑(index (↑{ carrier := K, isCompact' := h1K }) U)

case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
h1K : IsCompact K
h2K : (interior K).Nonempty
⊢ 0 < ↑(index (↑K₀) U)