mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case refine_1
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
i' : HasFundamentalDomain G α 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
⊢ μ (Quotient.mk α_mod_G '' A n) < ∞

pp: case refine_2
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
i' : HasFundamentalDomain G α 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
⊢ ⋃ i, Quotient.mk α_mod_G '' A i = univ

Tactic:
·

Goal state after:
case refine_1
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
i' : HasFundamentalDomain G α 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
⊢ μ (Quotient.mk α_mod_G '' A n) < ∞
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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) atTop (𝓝 l)
x : R
this : Nonempty R
val✝ : OrderTop R
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)

pp: case inr
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) atTop (𝓝 l)
x : R
this : Nonempty R
val✝ : NoTopOrder R
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)

Tactic:
·

Goal state after:
case inl
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) atTop (𝓝 l)
x : R
this : Nonempty R
val✝ : OrderTop R
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
h : f ≤ᵐ[μ.restrict s] g
hE : CompleteSpace E
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
h : f ≤ᵐ[μ.restrict s] g
hE : ¬CompleteSpace E
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

Tactic:
·

Goal state after:
case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
h : f ≤ᵐ[μ.restrict s] g
hE : CompleteSpace E
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
h : f ≤ᵐ[μ.restrict s] g
hE : ¬CompleteSpace E
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

Tactic:
·

Goal state after:
case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
h : f ≤ᵐ[μ.restrict s] g
hE : ¬CompleteSpace E
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case refine_2
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
i' : HasFundamentalDomain G α 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
⊢ ⋃ i, Quotient.mk α_mod_G '' A i = univ

Tactic:
·

Goal state after:
case refine_2
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
i' : HasFundamentalDomain G α 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
⊢ ⋃ i, Quotient.mk α_mod_G '' A i = univ
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
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
⊢ 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
⊢ 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
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
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 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
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_1
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn f (toMeasurable μ s) μ

pp: case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn g (toMeasurable μ s) μ

pp: case refine_3
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ MeasurableSet (toMeasurable μ s)

pp: case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g x

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn f (toMeasurable μ s) 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
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) ≤ ↑(index (↑K₂) U)

pp: 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
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 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
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) ≤ ↑(index (↑K₂) U)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn g (toMeasurable μ s) μ

pp: case refine_3
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ MeasurableSet (toMeasurable μ s)

pp: case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g x

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn g (toMeasurable μ s) 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
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 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
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 0 < ↑(index (↑K₀) U)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_3
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ MeasurableSet (toMeasurable μ s)

pp: case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g x

Tactic:
·

Goal state after:
case refine_3
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ MeasurableSet (toMeasurable μ s)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g x

Tactic:
·

Goal state after:
case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g 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
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) ≤ ↑(index (↑K₁) U) + ↑(index (↑K₂) U)

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : 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
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) ≤ ↑(index (↑K₁) U) + ↑(index (↑K₂) 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
U : Set G
K₁ K₂ : 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
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ 0 < ↑(index (↑K₀) U)
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case pos
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
⊢ covolume G α ν < ∞

pp: case neg
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : ¬HasFundamentalDomain G α ν
⊢ covolume G α ν < ∞

Tactic:
·

Goal state after:
case pos
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
⊢ covolume G α ν < ∞
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case neg
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : ¬HasFundamentalDomain G α ν
⊢ covolume G α ν < ∞

Tactic:
·

Goal state after:
case neg
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : ¬HasFundamentalDomain G α ν
⊢ covolume G α ν < ∞
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : OrderBot R
⊢ f.measure (Iio x) = ∞

pp: case inr
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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : NoBotOrder R
⊢ f.measure (Iio x) = ∞

Tactic:
·

Goal state after:
case inl
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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : OrderBot R
⊢ f.measure (Iio x) = ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (f a)) μ

pp: case hs
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ ∀ x ∈ s, 1 ≤ ENNReal.ofReal (f x)

Tactic:
·

Goal state after:
case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (f a)) μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hs
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ ∀ x ∈ s, 1 ≤ ENNReal.ofReal (f x)

Tactic:
·

Goal state after:
case hs
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ ∀ x ∈ s, 1 ≤ ENNReal.ofReal (f x)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : ¬Integrable f μ
⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s

pp: case pos
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s

Tactic:
·

Goal state after:
case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : ¬Integrable f μ
⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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 u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
h✝ : IsEmpty R
⊢ f.measure univ < ∞

pp: case inr
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 u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
h✝ : Nonempty R
⊢ f.measure univ < ∞

Tactic:
·

Goal state after:
case inl
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 u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
h✝ : IsEmpty R
⊢ f.measure univ < ∞