mathlib-initiative/mathlib-tactics huggingface dataset:
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
a b : R
⊢ f.measure (Ioo a b) = ofReal (leftLim (↑f) b - ↑f a)

Tactic:
by

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
ν : Measure α
inst✝² : SMulInvariantMeasure G α ν
inst✝¹ : Countable G
inst✝ : MeasurableConstSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
t : Set α
fund_dom_t : IsFundamentalDomain G t ν
⊢ Measure.map (Quotient.mk α_mod_G) (ν.restrict s) = Measure.map (Quotient.mk α_mod_G) (ν.restrict t)

Tactic:
by

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
ν : Measure α
inst✝² : SMulInvariantMeasure G α ν
inst✝¹ : Countable G
inst✝ : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
h : μ = Measure.map (Quotient.mk α_mod_G) (ν.restrict s)
⊢ QuotientMeasureEqMeasurePreimage ν μ

Tactic:
by

Goal state after:
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
a b : R
hab : a < b
⊢ Disjoint {a} (Ioo a b)

Tactic:
by

Goal state after:
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
a b : R
⊢ f.measure (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a)

Tactic:
by

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
ν : Measure α
inst✝² : SMulInvariantMeasure G α ν
inst✝¹ : Countable G
inst✝ : MeasurableConstSMul G α
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
vol_s : ν s = 0
⊢ QuotientMeasureEqMeasurePreimage ν 0

Tactic:
by

Goal state after:
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
⊢ f.measure botSet = 0

Tactic:
by

Goal state after:
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
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 : ℕ), Quotient.mk α_mod_G '' A i ∈ univ

Tactic:
by

Goal state after:
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
val✝ : OrderBot R
⊢ Iic x = Icc ⊥ x

Tactic:
by

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
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
⊢ index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V

Tactic:
apply le_antisymm (index_union_le K₁ K₂ hV)
  rcases index_elim (K₁.2.union K₂.2) hV with ⟨s, h1s, h2s⟩; rw [← h2s]
  have (K : Set G) (hK : K ⊆ ⋃ g ∈ s, (g * ·) ⁻¹' V) :
      index K V ≤ {g ∈ s | ((g * ·) ⁻¹' V ∩ K).Nonempty}.card := by
    apply Nat.sInf_le; refine ⟨_, ?_, rfl⟩; rw [mem_ofPred_eq]
    intro g hg; rcases hK hg with ⟨_, ⟨g₀, rfl⟩, _, ⟨h1g₀, rfl⟩, h2g₀⟩
    simp only [mem_preimage] at h2g₀
    simp only [mem_iUnion]; use g₀; constructor; swap
    · simp only [Finset.mem_filter, h1g₀, true_and]; use g
      simp [hg, h2g₀]
    exact h2g₀
  refine
    le_trans
      (add_le_add (this K₁.1 <| Subset.trans subset_union_left h1s)
        (this K₂.1 <| Subset.trans subset_union_right h1s)) ?_
  rw [← Finset.card_union_of_disjoint, Finset.filter_union_right]
  · exact s.card_filter_le _
  apply Finset.disjoint_filter.mpr
  rintro g₁ _ ⟨g₂, h1g₂, h2g₂⟩ ⟨g₃, h1g₃, h2g₃⟩
  simp only [mem_preimage] at h1g₃ h1g₂
  refine h.le_bot (?_ : g₁⁻¹ ∈ _)
  constructor <;> simp only [Set.mem_inv, Set.mem_mul]
  · refine ⟨_, h2g₂, (g₁ * g₂)⁻¹, ?_, ?_⟩
    · simp only [inv_inv, h1g₂]
    · simp only [mul_inv_rev, mul_inv_cancel_left]
  · refine ⟨_, h2g₃, (g₁ * g₃)⁻¹, ?_, ?_⟩
    · simp only [inv_inv, h1g₃]
    · simp only [mul_inv_rev, mul_inv_cancel_left]

Goal state after:
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
⊢ μ.real {x | R < f x} * R < ∫ (x : X) in {x | R < f x}, f x ∂μ

Tactic:
by

Goal state after:
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
⊢ f.measure (Iic x) = ofReal (↑f x - l)

Tactic:
by

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
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
⊢ index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
X : Type u_5
m m0 : MeasurableSpace X
μ : Measure X
hm : m ≤ m0
f : X → E
hf_meas : StronglyMeasurable f
s : Set X
hs : MeasurableSet s
⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, f x ∂μ.trim hm

Tactic:
by

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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝¹ : PartialOrder X
x y : X
inst✝ : NullSingletonClass μ
⊢ ∫ (t : X) in Ico x y, f t ∂μ = ∫ (t : X) in Ioc x y, f t ∂μ

Tactic:
by

Goal state after:
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
⊢ f.measure (Iio x) = ofReal (leftLim (↑f) x - l)

Tactic:
by

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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝¹ : PartialOrder X
x y : X
inst✝ : NullSingletonClass μ
⊢ ∫ (t : X) in Icc x y, f t ∂μ = ∫ (t : X) in Ioo x y, f t ∂μ

Tactic:
by

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 : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
⊢ index ((fun h ↦ g * h) '' K) V ≤ index K V

Tactic:
by

Goal state after:
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) atTop (𝓝 l)
x : R
this : Nonempty R
val✝ : OrderTop R
⊢ Ici x = Icc x ⊤

Tactic:
by

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 : Set G
hK : IsCompact K
g : G
V : Set G
hV : (interior V).Nonempty
⊢ index ((fun h ↦ g * h) '' K) V = index K V

Tactic:
by

Goal state after:
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:
simp [integral, hE]

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✝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
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

Tactic:
by

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✝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 : MeasurableSet s
h : ∀ x ∈ s, f x ≤ g x
⊢ f ≤ᵐ[μ.restrict s] g

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ prehaar (↑K₀) U K ≤ ↑(index ↑K ↑K₀)

Tactic:
by

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✝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 : MeasurableSet s
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

Tactic:
by

Goal state after:
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
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
⊢ Surjective (Quotient.mk α_mod_G)

Tactic:
by

Goal state after:
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) atTop (𝓝 l)
x : R
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)

Tactic:
by

Goal state after:
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
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
⊢ SigmaFinite μ

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
h1K : IsCompact K
h2K : (interior K).Nonempty
⊢ 0 < prehaar (↑K₀) U { carrier := K, isCompact' := h1K }

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ prehaar (↑K₀) U K₁ ≤ prehaar (↑K₀) U K₂

Tactic:
by

Goal state after:
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
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
hasFun : HasFundamentalDomain G α ν
h : covolume G α ν ≠ ∞
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
⊢ Fact (ν 𝓕 < ∞)

Tactic:
by

Goal state after:
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
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
hasFun : HasFundamentalDomain G α ν
h : covolume G α ν ≠ ∞
⊢ IsFiniteMeasure μ

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) ≤ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Tactic:
by

Goal state after:
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) atTop (𝓝 l)
x : R
⊢ f.measure (Ioi x) = ofReal (l - ↑f x)

Tactic:
by

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✝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
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

Tactic:
by

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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
μ : Measure X
f : X → E
s : Set X
inst✝¹ : ClosedIciTopology E
inst✝ : CompleteSpace E
c : E
hs : MeasurableSet s
hμs : μ s ≠ ∞
hf : ∀ x ∈ s, c ≤ f x
hfint : IntegrableOn (fun x ↦ f x) s μ
⊢ μ.real s • c ≤ ∫ (x : X) in s, f x ∂μ

Tactic:
by

Goal state after:
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
hf : Tendsto (↑f) atTop atTop
x : R
⊢ f.measure (Ioi x) = ∞

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
s : Set X
f : X → ℝ
c : ℝ
hs : MeasurableSet s
hμs : μ s ≠ ∞
hf : ∀ x ∈ s, c ≤ f x
hfint : IntegrableOn (fun x ↦ f x) s μ
⊢ c * μ.real s ≤ ∫ (x : X) in s, f x ∂μ

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : ∀ᵐ (x : X) ∂μ, x ∈ s → 0 ≤ f x
⊢ 0 ≤ᵐ[μ.restrict s] f

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Tactic:
by

Goal state after:
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
hf : Tendsto (↑f) atTop atTop
x : R
⊢ f.measure (Ici x) = ∞

Tactic:
by

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

Goal state before:
pp: 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
⊢ covolume G α ν < ∞

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
g : G
K : Compacts G
⊢ prehaar (↑K₀) U (Compacts.map (fun x ↦ g * x) ⋯ K) = prehaar (↑K₀) U K

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in {y | 0 ≤ f y}, f x ∂μ

Tactic:
by

Goal state after:
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
hf : Tendsto (↑f) atBot atBot
x : R
⊢ f.measure (Iic x) = ∞

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ 0
⊢ f ≤ᵐ[μ.restrict s] 0

Tactic:
by

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
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
⊢ prehaar (↑K₀) U ∈ haarProduct ↑K₀

Tactic:
by

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ ∫ (x : X) in {y | f y ≤ 0}, f x ∂μ ≤ ∫ (x : X) in s, f x ∂μ

Tactic:
by

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
K₀ : PositiveCompacts G
⊢ IsCompact (haarProduct ↑K₀)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : SeminormedAddCommGroup E
c : InnerProductSpace 𝕜 E
x : E
⊢ 0 ≤ re (inner 𝕜 x x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : NormedAddCommGroup E
c : InnerProductSpace 𝕜 E
x : E
⊢ 0 ≤ re ⟪x, x⟫_𝕜

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : NormedAddCommGroup E
c : InnerProductSpace 𝕜 E
x : E
hx : ⟪x, x⟫_𝕜 = 0
⊢ ‖x‖ ^ 2 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x : F
⊢ im ⟪x, x⟫ = 0

Tactic:
by

Goal state after:
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
hf : Tendsto (↑f) atBot atBot
x : R
⊢ f.measure (Iio x) = ∞

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
l u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
⊢ f.measure univ = ofReal (u - l)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y z : F
⊢ ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫

Tactic:
by

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

Goal state before:
pp: 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
⊢ μ s ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)

Tactic:
by

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
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
⊢ ∀ i ∈ t, IsOpen[inst✝¹] i.carrier

Tactic:
by

Goal state after: