ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.sdiff_addFundamentalFrontier
@[to_additive (attr := simp) MeasureTheory.sdiff_addFundamentalFrontier]
theorem sdiff_fundamentalFrontier : s \ fundamentalFrontier G s = fundamentalInterior G s 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α)
Goal: s \ MeasureTheory.addFundamentalFrontier G s = MeasureTheory.addFundamentalInterior G s
-- MeasureTheory.addFundamentalFrontier_vadd
@[to_additive (attr := simp) MeasureTheory.addFundamentalFrontier_vadd]
theorem fundamentalFrontier_smul [Group H] [MulAction H α] [SMulCommClass H G α] (g : H) :
    fundamentalFrontier G (g • s) = g • fundamentalFrontier G s 
Arguments: (G : Type u_1) {H : Type u_2} {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α) [AddGroup H] [AddAction H α] [VAddCommClass H G α] (g : H)
Goal: MeasureTheory.addFundamentalFrontier G (g +ᵥ s) = g +ᵥ MeasureTheory.addFundamentalFrontier G s
-- MeasureTheory.fundamentalFrontier_smul
@[to_additive (attr := simp) MeasureTheory.addFundamentalFrontier_vadd]
theorem fundamentalFrontier_smul [Group H] [MulAction H α] [SMulCommClass H G α] (g : H) :
    fundamentalFrontier G (g • s) = g • fundamentalFrontier G s 
Arguments: (G : Type u_1) {H : Type u_2} {α : Type u_3} [Group G] [MulAction G α] (s : Set α) [Group H] [MulAction H α] [SMulCommClass H G α] (g : H)
Goal: MeasureTheory.fundamentalFrontier G (g • s) = g • MeasureTheory.fundamentalFrontier G s
-- MeasureTheory.fundamentalInterior_smul
@[to_additive (attr := simp) MeasureTheory.addFundamentalInterior_vadd]
theorem fundamentalInterior_smul [Group H] [MulAction H α] [SMulCommClass H G α] (g : H) :
    fundamentalInterior G (g • s) = g • fundamentalInterior G s 
Arguments: (G : Type u_1) {H : Type u_2} {α : Type u_3} [Group G] [MulAction G α] (s : Set α) [Group H] [MulAction H α] [SMulCommClass H G α] (g : H)
Goal: MeasureTheory.fundamentalInterior G (g • s) = g • MeasureTheory.fundamentalInterior G s
-- MeasureTheory.addFundamentalInterior_vadd
@[to_additive (attr := simp) MeasureTheory.addFundamentalInterior_vadd]
theorem fundamentalInterior_smul [Group H] [MulAction H α] [SMulCommClass H G α] (g : H) :
    fundamentalInterior G (g • s) = g • fundamentalInterior G s 
Arguments: (G : Type u_1) {H : Type u_2} {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α) [AddGroup H] [AddAction H α] [VAddCommClass H G α] (g : H)
Goal: MeasureTheory.addFundamentalInterior G (g +ᵥ s) = g +ᵥ MeasureTheory.addFundamentalInterior G s
-- MeasureTheory.pairwise_disjoint_addFundamentalInterior
@[to_additive MeasureTheory.pairwise_disjoint_addFundamentalInterior]
theorem pairwise_disjoint_fundamentalInterior :
    Pairwise (Disjoint on fun g : G => g • fundamentalInterior G s) 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α)
Goal: Pairwise (Disjoint on fun g => g +ᵥ MeasureTheory.addFundamentalInterior G s)
-- MeasureTheory.pairwise_disjoint_fundamentalInterior
@[to_additive MeasureTheory.pairwise_disjoint_addFundamentalInterior]
theorem pairwise_disjoint_fundamentalInterior :
    Pairwise (Disjoint on fun g : G => g • fundamentalInterior G s) 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α)
Goal: Pairwise (Disjoint on fun g => g • MeasureTheory.fundamentalInterior G s)
-- MeasureTheory.NullMeasurableSet.addFundamentalFrontier
@[to_additive MeasureTheory.NullMeasurableSet.addFundamentalFrontier]
protected theorem NullMeasurableSet.fundamentalFrontier (hs : NullMeasurableSet s μ) :
    NullMeasurableSet (fundamentalFrontier G s) μ 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α) [Countable G] [MeasurableSpace G] [MeasurableSpace α] [MeasurableVAdd G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet (MeasureTheory.addFundamentalFrontier G s) μ
-- MeasureTheory.NullMeasurableSet.fundamentalFrontier
@[to_additive MeasureTheory.NullMeasurableSet.addFundamentalFrontier]
protected theorem NullMeasurableSet.fundamentalFrontier (hs : NullMeasurableSet s μ) :
    NullMeasurableSet (fundamentalFrontier G s) μ 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α) [Countable G] [MeasurableSpace G] [MeasurableSpace α] [MeasurableSMul G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet (MeasureTheory.fundamentalFrontier G s) μ
-- MeasureTheory.NullMeasurableSet.fundamentalInterior
@[to_additive MeasureTheory.NullMeasurableSet.addFundamentalInterior]
protected theorem NullMeasurableSet.fundamentalInterior (hs : NullMeasurableSet s μ) :
    NullMeasurableSet (fundamentalInterior G s) μ 
Arguments: (G : Type u_1) {α : Type u_3} [Group G] [MulAction G α] (s : Set α) [Countable G] [MeasurableSpace G] [MeasurableSpace α] [MeasurableSMul G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet (MeasureTheory.fundamentalInterior G s) μ
-- MeasureTheory.NullMeasurableSet.addFundamentalInterior
@[to_additive MeasureTheory.NullMeasurableSet.addFundamentalInterior]
protected theorem NullMeasurableSet.fundamentalInterior (hs : NullMeasurableSet s μ) :
    NullMeasurableSet (fundamentalInterior G s) μ 
Arguments: (G : Type u_1) {α : Type u_3} [AddGroup G] [AddAction G α] (s : Set α) [Countable G] [MeasurableSpace G] [MeasurableSpace α] [MeasurableVAdd G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet (MeasureTheory.addFundamentalInterior G s) μ
-- MeasureTheory.IsAddFundamentalDomain.measure_addFundamentalFrontier
@[to_additive MeasureTheory.IsAddFundamentalDomain.measure_addFundamentalFrontier]
theorem measure_fundamentalFrontier : μ (fundamentalFrontier G s) = 0 
Arguments: {G : Type u_1} {α : Type u_3} [Countable G] [AddGroup G] [AddAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: μ (MeasureTheory.addFundamentalFrontier G s) = 0
-- MeasureTheory.IsFundamentalDomain.measure_fundamentalFrontier
@[to_additive MeasureTheory.IsAddFundamentalDomain.measure_addFundamentalFrontier]
theorem measure_fundamentalFrontier : μ (fundamentalFrontier G s) = 0 
Arguments: {G : Type u_1} {α : Type u_3} [Countable G] [Group G] [MulAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.IsFundamentalDomain G s μ)
Goal: μ (MeasureTheory.fundamentalFrontier G s) = 0
-- MeasureTheory.IsFundamentalDomain.measure_fundamentalInterior
@[to_additive MeasureTheory.IsAddFundamentalDomain.measure_addFundamentalInterior]
theorem measure_fundamentalInterior : μ (fundamentalInterior G s) = μ s 
Arguments: {G : Type u_1} {α : Type u_3} [Countable G] [Group G] [MulAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.IsFundamentalDomain G s μ)
Goal: μ (MeasureTheory.fundamentalInterior G s) = μ s
-- MeasureTheory.IsAddFundamentalDomain.measure_addFundamentalInterior
@[to_additive MeasureTheory.IsAddFundamentalDomain.measure_addFundamentalInterior]
theorem measure_fundamentalInterior : μ (fundamentalInterior G s) = μ s 
Arguments: {G : Type u_1} {α : Type u_3} [Countable G] [AddGroup G] [AddAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: μ (MeasureTheory.addFundamentalInterior G s) = μ s
-- MeasureTheory.IsFundamentalDomain.fundamentalInterior
protected theorem fundamentalInterior : IsFundamentalDomain G (fundamentalInterior G s) μ 
Arguments: {G : Type u_1} {α : Type u_3} [Countable G] [Group G] [MulAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.IsFundamentalDomain G s μ) [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ]
Goal: MeasureTheory.IsFundamentalDomain G (MeasureTheory.fundamentalInterior G s) μ
-- MeasureTheory.measure_map_restrict_apply
@[to_additive addMeasure_map_restrict_apply]
lemma measure_map_restrict_apply (s : Set α) {U : Set (Quotient α_mod_G)}
    (meas_U : MeasurableSet U) :
    (μ.restrict s).map π U = μ ((π ⁻¹' U) ∩ s) 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α) {U : Set (Quotient (MulAction.orbitRel G α))} (meas_U : MeasurableSet U)
Goal: (MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (μ.restrict s)) U =
  μ (Quotient.mk (MulAction.orbitRel G α) ⁻¹' U ∩ s)
-- MeasureTheory.addMeasure_map_restrict_apply
@[to_additive addMeasure_map_restrict_apply]
lemma measure_map_restrict_apply (s : Set α) {U : Set (Quotient α_mod_G)}
    (meas_U : MeasurableSet U) :
    (μ.restrict s).map π U = μ ((π ⁻¹' U) ∩ s) 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α) {U : Set (Quotient (AddAction.orbitRel G α))} (meas_U : MeasurableSet U)
Goal: (MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (μ.restrict s)) U =
  μ (Quotient.mk (AddAction.orbitRel G α) ⁻¹' U ∩ s)
-- MeasureTheory.IsFundamentalDomain.quotientMeasure_eq
@[to_additive]
lemma IsFundamentalDomain.quotientMeasure_eq [Countable G] [MeasurableSpace G] {s t : Set α}
    [SMulInvariantMeasure G α μ] [MeasurableSMul G α] (fund_dom_s : IsFundamentalDomain G s μ)
    (fund_dom_t : IsFundamentalDomain G t μ) :
    (μ.restrict s).map π = (μ.restrict t).map π 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [Countable G] [MeasurableSpace G] {s t : Set α} [MeasureTheory.SMulInvariantMeasure G α μ] [MeasurableSMul G α] (fund_dom_s : MeasureTheory.IsFundamentalDomain G s μ) (fund_dom_t : MeasureTheory.IsFundamentalDomain G t μ)
Goal: MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (μ.restrict s) =
  MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (μ.restrict t)
-- MeasureTheory.IsAddFundamentalDomain.addQuotientMeasure_eq
@[to_additive]
lemma IsFundamentalDomain.quotientMeasure_eq [Countable G] [MeasurableSpace G] {s t : Set α}
    [SMulInvariantMeasure G α μ] [MeasurableSMul G α] (fund_dom_s : IsFundamentalDomain G s μ)
    (fund_dom_t : IsFundamentalDomain G t μ) :
    (μ.restrict s).map π = (μ.restrict t).map π 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [Countable G] [MeasurableSpace G] {s t : Set α} [MeasureTheory.VAddInvariantMeasure G α μ] [MeasurableVAdd G α] (fund_dom_s : MeasureTheory.IsAddFundamentalDomain G s μ) (fund_dom_t : MeasureTheory.IsAddFundamentalDomain G t μ)
Goal: MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (μ.restrict s) =
  MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (μ.restrict t)
-- MeasureTheory.HasAddFundamentalDomain.ExistsIsAddFundamentalDomain
class HasAddFundamentalDomain (G α : Type*) [Zero G] [VAdd G α] [MeasurableSpace α]
    (ν : Measure α := by volume_tac) : Prop where
  ExistsIsAddFundamentalDomain : ∃ s : Set α, IsAddFundamentalDomain G s ν
Arguments: {G : Type u_6} {α : Type u_7} [Zero G] [VAdd G α] [MeasurableSpace α] {ν : autoParam (MeasureTheory.Measure α) _auto✝} [self : MeasureTheory.HasAddFundamentalDomain G α ν]
Goal: ∃ s, MeasureTheory.IsAddFundamentalDomain G s ν
-- MeasureTheory.HasFundamentalDomain.ExistsIsFundamentalDomain
class HasFundamentalDomain (G : Type*) (α : Type*) [One G] [SMul G α] [MeasurableSpace α]
    (ν : Measure α := by volume_tac) : Prop where
  ExistsIsFundamentalDomain : ∃ (s : Set α), IsFundamentalDomain G s ν
Arguments: {G : Type u_6} {α : Type u_7} [One G] [SMul G α] [MeasurableSpace α] {ν : autoParam (MeasureTheory.Measure α) _auto✝} [self : MeasureTheory.HasFundamentalDomain G α ν]
Goal: ∃ s, MeasureTheory.IsFundamentalDomain G s ν
-- MeasureTheory.IsFundamentalDomain.hasFundamentalDomain
@[to_additive]
lemma IsFundamentalDomain.hasFundamentalDomain (ν : Measure α) {s : Set α}
    (fund_dom_s : IsFundamentalDomain G s ν) :
    HasFundamentalDomain G α ν 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] (ν : MeasureTheory.Measure α) {s : Set α} (fund_dom_s : MeasureTheory.IsFundamentalDomain G s ν)
Goal: MeasureTheory.HasFundamentalDomain G α ν
-- MeasureTheory.IsAddFundamentalDomain.hasAddFundamentalDomain
@[to_additive]
lemma IsFundamentalDomain.hasFundamentalDomain (ν : Measure α) {s : Set α}
    (fund_dom_s : IsFundamentalDomain G s ν) :
    HasFundamentalDomain G α ν 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] (ν : MeasureTheory.Measure α) {s : Set α} (fund_dom_s : MeasureTheory.IsAddFundamentalDomain G s ν)
Goal: MeasureTheory.HasAddFundamentalDomain G α ν
-- MeasureTheory.IsFundamentalDomain.covolume_eq_volume
@[to_additive]
lemma IsFundamentalDomain.covolume_eq_volume (ν : Measure α) [Countable G]
    [MeasurableSMul G α] [SMulInvariantMeasure G α ν] {s : Set α}
    (fund_dom_s : IsFundamentalDomain G s ν) : covolume G α ν = ν s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace G] [MeasurableSpace α] (ν : MeasureTheory.Measure α) [Countable G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α ν] {s : Set α} (fund_dom_s : MeasureTheory.IsFundamentalDomain G s ν)
Goal: MeasureTheory.covolume G α ν = ν s
-- MeasureTheory.IsAddFundamentalDomain.covolume_eq_volume
@[to_additive]
lemma IsFundamentalDomain.covolume_eq_volume (ν : Measure α) [Countable G]
    [MeasurableSMul G α] [SMulInvariantMeasure G α ν] {s : Set α}
    (fund_dom_s : IsFundamentalDomain G s ν) : covolume G α ν = ν s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace G] [MeasurableSpace α] (ν : MeasureTheory.Measure α) [Countable G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α ν] {s : Set α} (fund_dom_s : MeasureTheory.IsAddFundamentalDomain G s ν)
Goal: MeasureTheory.addCovolume G α ν = ν s
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.addProjection_respects_measure'
class AddQuotientMeasureEqMeasurePreimage (ν : Measure α := by volume_tac)
    (μ : Measure (Quotient α_mod_G)) : Prop where
  addProjection_respects_measure' : ∀ (t : Set α) (_ : IsAddFundamentalDomain G t ν),
    μ = (ν.restrict t).map π
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : autoParam (MeasureTheory.Measure α) _auto✝} {μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))} [self : MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] (t : Set α)
Goal: MeasureTheory.IsAddFundamentalDomain G t ν →
  μ = MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (MeasureTheory.Measure.restrict ν t)
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.projection_respects_measure'
class QuotientMeasureEqMeasurePreimage (ν : Measure α := by volume_tac)
    (μ : Measure (Quotient α_mod_G)) : Prop where
  projection_respects_measure' (t : Set α) : IsFundamentalDomain G t ν → μ = (ν.restrict t).map π
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : autoParam (MeasureTheory.Measure α) _auto✝} {μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))} [self : MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] (t : Set α)
Goal: MeasureTheory.IsFundamentalDomain G t ν →
  μ = MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (MeasureTheory.Measure.restrict ν t)
-- MeasureTheory.IsFundamentalDomain.projection_respects_measure
@[to_additive addProjection_respects_measure]
lemma IsFundamentalDomain.projection_respects_measure {ν : Measure α}
    (μ : Measure (Quotient α_mod_G)) [i : QuotientMeasureEqMeasurePreimage ν μ] {t : Set α}
    (fund_dom_t : IsFundamentalDomain G t ν) : μ = (ν.restrict t).map π 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} (μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [i : MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] {t : Set α} (fund_dom_t : MeasureTheory.IsFundamentalDomain G t ν)
Goal: μ = MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (ν.restrict t)
-- MeasureTheory.IsAddFundamentalDomain.addProjection_respects_measure
@[to_additive addProjection_respects_measure]
lemma IsFundamentalDomain.projection_respects_measure {ν : Measure α}
    (μ : Measure (Quotient α_mod_G)) [i : QuotientMeasureEqMeasurePreimage ν μ] {t : Set α}
    (fund_dom_t : IsFundamentalDomain G t ν) : μ = (ν.restrict t).map π 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} (μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [i : MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] {t : Set α} (fund_dom_t : MeasureTheory.IsAddFundamentalDomain G t ν)
Goal: μ = MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (ν.restrict t)
-- MeasureTheory.IsFundamentalDomain.projection_respects_measure_apply
@[to_additive addProjection_respects_measure_apply]
lemma IsFundamentalDomain.projection_respects_measure_apply {ν : Measure α}
    (μ : Measure (Quotient α_mod_G)) [i : QuotientMeasureEqMeasurePreimage ν μ] {t : Set α}
    (fund_dom_t : IsFundamentalDomain G t ν) {U : Set (Quotient α_mod_G)}
    (meas_U : MeasurableSet U) : μ U = ν (π ⁻¹' U ∩ t) 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} (μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [i : MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] {t : Set α} (fund_dom_t : MeasureTheory.IsFundamentalDomain G t ν) {U : Set (Quotient (MulAction.orbitRel G α))} (meas_U : MeasurableSet U)
Goal: μ U = ν (Quotient.mk (MulAction.orbitRel G α) ⁻¹' U ∩ t)
-- MeasureTheory.IsAddFundamentalDomain.addProjection_respects_measure_apply
@[to_additive addProjection_respects_measure_apply]
lemma IsFundamentalDomain.projection_respects_measure_apply {ν : Measure α}
    (μ : Measure (Quotient α_mod_G)) [i : QuotientMeasureEqMeasurePreimage ν μ] {t : Set α}
    (fund_dom_t : IsFundamentalDomain G t ν) {U : Set (Quotient α_mod_G)}
    (meas_U : MeasurableSet U) : μ U = ν (π ⁻¹' U ∩ t) 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} (μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [i : MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] {t : Set α} (fund_dom_t : MeasureTheory.IsAddFundamentalDomain G t ν) {U : Set (Quotient (AddAction.orbitRel G α))} (meas_U : MeasurableSet U)
Goal: μ U = ν (Quotient.mk (AddAction.orbitRel G α) ⁻¹' U ∩ t)
-- MeasureTheory.IsFundamentalDomain.quotientMeasureEqMeasurePreimage_quotientMeasure
@[to_additive]
lemma IsFundamentalDomain.quotientMeasureEqMeasurePreimage_quotientMeasure
    {s : Set α} (fund_dom_s : IsFundamentalDomain G s ν) :
    QuotientMeasureEqMeasurePreimage ν ((ν.restrict s).map π) 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.SMulInvariantMeasure G α ν] [MeasurableSMul G α] {s : Set α} (fund_dom_s : MeasureTheory.IsFundamentalDomain G s ν)
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν
  (MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (ν.restrict s))
-- MeasureTheory.IsAddFundamentalDomain.addQuotientMeasureEqMeasurePreimage_addQuotientMeasure
@[to_additive]
lemma IsFundamentalDomain.quotientMeasureEqMeasurePreimage_quotientMeasure
    {s : Set α} (fund_dom_s : IsFundamentalDomain G s ν) :
    QuotientMeasureEqMeasurePreimage ν ((ν.restrict s).map π) 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.VAddInvariantMeasure G α ν] [MeasurableVAdd G α] {s : Set α} (fund_dom_s : MeasureTheory.IsAddFundamentalDomain G s ν)
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν
  (MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (ν.restrict s))
-- MeasureTheory.IsAddFundamentalDomain.addQuotientMeasureEqMeasurePreimage
@[to_additive]
lemma IsFundamentalDomain.quotientMeasureEqMeasurePreimage {μ : Measure (Quotient α_mod_G)}
    {s : Set α} (fund_dom_s : IsFundamentalDomain G s ν) (h : μ = (ν.restrict s).map π) :
    QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.VAddInvariantMeasure G α ν] [MeasurableVAdd G α] {μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))} {s : Set α} (fund_dom_s : MeasureTheory.IsAddFundamentalDomain G s ν) (h : μ = MeasureTheory.Measure.map (Quotient.mk (AddAction.orbitRel G α)) (ν.restrict s))
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ
-- MeasureTheory.IsFundamentalDomain.quotientMeasureEqMeasurePreimage
@[to_additive]
lemma IsFundamentalDomain.quotientMeasureEqMeasurePreimage {μ : Measure (Quotient α_mod_G)}
    {s : Set α} (fund_dom_s : IsFundamentalDomain G s ν) (h : μ = (ν.restrict s).map π) :
    QuotientMeasureEqMeasurePreimage ν μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.SMulInvariantMeasure G α ν] [MeasurableSMul G α] {μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))} {s : Set α} (fund_dom_s : MeasureTheory.IsFundamentalDomain G s ν) (h : μ = MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (ν.restrict s))
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.unique
@[to_additive]
lemma QuotientMeasureEqMeasurePreimage.unique
    [hasFun : HasFundamentalDomain G α ν] (μ μ' : Measure (Quotient α_mod_G))
    [QuotientMeasureEqMeasurePreimage ν μ] [QuotientMeasureEqMeasurePreimage ν μ'] :
    μ = μ' 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [hasFun : MeasureTheory.HasFundamentalDomain G α ν] (μ μ' : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ']
Goal: μ = μ'
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.unique
@[to_additive]
lemma QuotientMeasureEqMeasurePreimage.unique
    [hasFun : HasFundamentalDomain G α ν] (μ μ' : Measure (Quotient α_mod_G))
    [QuotientMeasureEqMeasurePreimage ν μ] [QuotientMeasureEqMeasurePreimage ν μ'] :
    μ = μ' 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [hasFun : MeasureTheory.HasAddFundamentalDomain G α ν] (μ μ' : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ']
Goal: μ = μ'
-- MeasureTheory.IsAddFundamentalDomain.measurePreserving_add_quotient_mk
@[to_additive IsAddFundamentalDomain.measurePreserving_add_quotient_mk]
theorem IsFundamentalDomain.measurePreserving_quotient_mk
    {𝓕 : Set α} (h𝓕 : IsFundamentalDomain G 𝓕 ν)
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] :
    MeasurePreserving π (ν.restrict 𝓕) μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} {𝓕 : Set α} (h𝓕 : MeasureTheory.IsAddFundamentalDomain G 𝓕 ν) (μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ]
Goal: MeasureTheory.MeasurePreserving (Quotient.mk (AddAction.orbitRel G α)) (ν.restrict 𝓕) μ