mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → μ K + ↑ε < μ x
lower_bound_iInf : μ K + ↑ε ≤ μ K
⊢ False

Tactic:
]

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → μ K + ↑ε < μ x
lower_bound_iInf : μ K + ↑ε ≤ μ K
⊢ False
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
⊢ ∫ (x : X) in univ, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
⊢ ∫ (x : X) in univ, f x ∂μ = ∫ (x : X) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: 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.outerMeasure (interior ↑K') ≤ μ K'

Tactic:
]

Goal state after:
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.outerMeasure (interior ↑K') ≤ μ K'
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ ∫⁻ (x : α) in t, ‖f x‖ₑ ∂μ < ∞ ↔ ∫⁻ (a : α) in t, ‖f a‖ₑ ∂μ < ∞

pp: case hf
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ ∀ (g : G) (x : α), ‖f (g • x)‖ₑ = ‖f x‖ₑ

Tactic:
]

Goal state after:
case hf
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ ∀ (g : G) (x : α), ‖f (g • x)‖ₑ = ‖f x‖ₑ
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: 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
⊢ μ { carrier := closure ↑K, isCompact' := ⋯ } ≤ μ.outerMeasure (closure ↑K)

Tactic:
]

Goal state after:
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
⊢ μ { carrier := closure ↑K, isCompact' := ⋯ } ≤ μ.outerMeasure (closure ↑K)
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: 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
⊢ μ { carrier := closure ↑K, isCompact' := ⋯ } ≤ μ.measure (closure ↑K)

Tactic:
by
      rw [μ.measure_apply (isClosed_closure.measurableSet)]
      exact μ.le_outerMeasure_compacts _

Goal state after:
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
μ : Measure X
inst✝ : PartialOrder E
f : X → E
hf : AEStronglyMeasurable f μ
h_union : {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0}
⊢ ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x < 0} ∪ {x | f x = 0}, f x ∂μ

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure X
inst✝ : PartialOrder E
f : X → E
hf : AEStronglyMeasurable f μ
h_union : {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0}
⊢ ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x < 0} ∪ {x | f x = 0}, f x ∂μ
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case hf
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
x : α
⊢ ‖f x‖ₑ = ‖f x‖ₑ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
ν : Measure α
inst✝ : NormedSpace ℝ E
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
f : α → E
hf : Integrable f ν
⊢ ∫ (x : α), f x ∂ν = ∫ (x : α), f x ∂ν

pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
ν : Measure α
inst✝ : NormedSpace ℝ E
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
f : α → E
hf : Integrable f ν
⊢ Integrable f (sum fun g ↦ ν.restrict (g • s))

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
ν : Measure α
inst✝ : NormedSpace ℝ E
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
f : α → E
hf : Integrable f ν
⊢ Integrable f (sum fun g ↦ ν.restrict (g • s))
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
ν : Measure α
inst✝ : NormedSpace ℝ E
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
f : α → E
hf : Integrable f ν
⊢ Integrable f ν

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
ν : Measure α
inst✝ : NormedSpace ℝ E
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
f : α → E
hf : Integrable f ν
⊢ Integrable f ν
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
⊢ ∫ (x : X) in {x | 0 ≤ f x}, ‖f x‖ ∂μ + ∫ (x : X) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ =
    ∫ (x : X) in {x | 0 ≤ f x}, ‖f x‖ ∂μ + ∫ (x : X) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ

Tactic:
]

Goal state after:
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
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
this :
  ∃ x ∈ {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V},
    x.card = sInf (Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V})
⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V ∧ t.card = index K V

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
this :
  ∃ x ∈ {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V},
    x.card = sInf (Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V})
⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V ∧ t.card = index K V
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
⊢ index (↑K) V ≤ index ↑K ↑K₀ * index (↑K₀) V

Tactic:
obtain ⟨t, h1t, h2t⟩ := index_elim K₀.isCompact hV

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
⊢ index (↑K) V ≤ index ↑K ↑K₀ * index (↑K₀) V
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case e_a
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}
⊢ f x = f x

pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}
⊢ 0 ≤ f x

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}
⊢ 0 ≤ f 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
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
⊢ index (↑K) V ≤ t.card * s.card

Tactic:
]

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
⊢ index (↑K) V ≤ t.card * s.card
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : IntegrableOn f s μ
⊢ IntegrableOn f s μ

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : IntegrableOn f s μ
⊢ IntegrableOn f s μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case e_a
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
⊢ ∫ (x : X) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ = ∫ (a : X) in {x | 0 ≤ f x}ᶜ, -f a ∂μ

Tactic:
]

Goal state after:
case e_a
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
⊢ ∫ (x : X) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ = ∫ (a : X) in {x | 0 ≤ f x}ᶜ, -f a ∂μ
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
⊢ ↑K ⊆ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V

Tactic:
]

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
⊢ ↑K ⊆ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case e_a
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}ᶜ
⊢ -f x = -f x

pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}ᶜ
⊢ f x ≤ 0

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}ᶜ
⊢ f x ≤ 0
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case neg
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : ¬IntegrableOn f s μ
⊢ 0 = 0

pp: case neg
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : ¬IntegrableOn f s μ
⊢ ¬Integrable f (μ.restrict t)

Tactic:
]

Goal state after:
case neg
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : ¬IntegrableOn f s μ
⊢ ¬Integrable f (μ.restrict t)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : ¬0 ≤ f x
⊢ f x ≤ 0

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : ¬0 ≤ f x
⊢ f x ≤ 0
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case neg
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : ¬IntegrableOn f t μ
⊢ ¬Integrable f (μ.restrict t)

Tactic:
]

Goal state after:
case neg
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
hfs : ¬IntegrableOn f t μ
⊢ ¬Integrable f (μ.restrict t)
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:
]

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
⊢ ∀ (a : G), g₁ * a ∈ ↑K₀ → a ∈ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
⊢ ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {a | ¬0 ≤ f a}, f x ∂μ =
    ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {x | f x < 0}, f x ∂μ

Tactic:
]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
⊢ ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {a | ¬0 ≤ f a}, f x ∂μ =
    ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {x | f x < 0}, f x ∂μ
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
htm : NullMeasurableSet t μ
ht : ∀ x ∈ t, ∀ y ∈ t, ∀ (g : G), g ≠ 1 → g • x ≠ y
g₁ g₂ : G
hne : g₁ ≠ g₂
⊢ ∀ ⦃a : α⦄, a ∈ g₁ • t → a ∉ g₂ • t

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
htm : NullMeasurableSet t μ
ht : ∀ x ∈ t, ∀ y ∈ t, ∀ (g : G), g ≠ 1 → g • x ≠ y
g₁ g₂ : G
hne : g₁ ≠ g₂
⊢ ∀ ⦃a : α⦄, a ∈ g₁ • t → a ∉ g₂ • t
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
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
g₃ : G
hg₃ : g₃ ∈ t
h2V : g₃ * g₁ * g₂ ∈ V
⊢ g₂ ∈ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V

Tactic:
]

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
g₃ : G
hg₃ : g₃ ∈ t
h2V : g₃ * g₁ * g₂ ∈ V
⊢ g₂ ∈ ⋃ g ∈ t * s, (fun h ↦ g * h) ⁻¹' V
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
s : Set X
μ : Measure X
inst✝ : CompleteSpace E
c : E
⊢ μ.real s • c = μ.real s • c

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
htm : NullMeasurableSet t μ
ht : ∀ x ∈ t, ∀ y ∈ t, ∀ (g : G), g ≠ 1 → g • x ≠ y
g₁ g₂ : G
hne : g₁ ≠ g₂
x : α
hx : x ∈ t
y : α
hy : y ∈ t
hxy : g₂ • y = g₁ • x
⊢ y = y

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : SecondCountableTopology R
⊢ MeasurableSpace.generateFrom (range Ioi) ≤ f.outer.caratheodory

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : SecondCountableTopology R
⊢ MeasurableSpace.generateFrom (range Ioi) ≤ f.outer.caratheodory
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
μ : Measure X
inst✝ : CompleteSpace E
e : E
s : Set X
s_meas : MeasurableSet s
⊢ ∫ (x : X) in s, e ∂μ = ∫ (x : X) in s, e ∂μ

Tactic:
]

Goal state after: