Theorem: MeasureTheory.QuotientMeasureEqMeasurePreimage.covolume_ne_top
File: Mathlib/MeasureTheory/Group/FundamentalDomain.lean

tactic: by_cases hasFun : HasFundamentalDomain G α ν
state_before: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
⊢ covolume G α ν < ⊤
state_after: case pos
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul 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 < ⊤

case neg
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul 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: obtain ⟨𝓕, h𝓕⟩ := hasFun.ExistsIsFundamentalDomain
state_before: case pos
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul 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 < ⊤
state_after: case pos.intro
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
⊢ covolume G α ν < ⊤

tactic: have H : μ univ < ∞ := IsFiniteMeasure.measure_univ_lt_top
state_before: case pos.intro
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
⊢ covolume G α ν < ⊤
state_after: case pos.intro
G : Type u_1
H✝ : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
H : μ univ < ⊤
⊢ covolume G α ν < ⊤

tactic: rw [h𝓕.projection_respects_measure_apply (μ := μ) MeasurableSet.univ] at H
state_before: case pos.intro
G : Type u_1
H✝ : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
H : μ univ < ⊤
⊢ covolume G α ν < ⊤
state_after: case pos.intro
G : Type u_1
H✝ : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
H : ν (Quotient.mk α_mod_G ⁻¹' univ ∩ 𝓕) < ⊤
⊢ covolume G α ν < ⊤

tactic: simpa [h𝓕.covolume_eq_volume ν] using H
state_before: case pos.intro
G : Type u_1
H✝ : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
H : ν (Quotient.mk α_mod_G ⁻¹' univ ∩ 𝓕) < ⊤
⊢ covolume G α ν < ⊤
state_after: no goals

tactic: simp [covolume, hasFun]
state_before: case neg
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁸ : Group G
inst✝⁷ : MulAction G α
inst✝⁶ : MeasurableSpace α
ν : Measure α
inst✝⁵ : Countable G
inst✝⁴ : MeasurableSpace G
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : MeasurableSMul 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 < ⊤
state_after: no goals