ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.analyticAt_prod
theorem Finset.analyticAt_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
    {f : α → E → A} {c : E} (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
    AnalyticAt 𝕜 (fun z ↦ ∏ n ∈ N, f n z) c 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_9} [NormedCommRing A] [NormedAlgebra 𝕜 A] {f : α → E → A} {c : E} (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c)
Goal: AnalyticAt 𝕜 (fun z => ∏ n ∈ N, f n z) c
-- Finset.analyticOn_prod
theorem Finset.analyticOn_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
    {f : α → E → A} {s : Set E} (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s) :
    AnalyticOn 𝕜 (fun z ↦ ∏ n ∈ N, f n z) s 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_9} [NormedCommRing A] [NormedAlgebra 𝕜 A] {f : α → E → A} {s : Set E} (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s)
Goal: AnalyticOn 𝕜 (fun z => ∏ n ∈ N, f n z) s
-- LinearMap.isSymmetric_iff_sesqForm
theorem isSymmetric_iff_sesqForm (T : E →ₗ[𝕜] E) :
    T.IsSymmetric ↔ LinearMap.IsSelfAdjoint (R := 𝕜) (M := E) sesqFormOfInner T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →ₗ[𝕜] E)
Goal: T.IsSymmetric ↔ sesqFormOfInner.IsSelfAdjoint T
-- LinearMap.IsSymmetric.conj_inner_sym
theorem IsSymmetric.conj_inner_sym {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) (x y : E) :
    conj ⟪T x, y⟫ = ⟪T y, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x y : E)
Goal: (starRingEnd 𝕜) ⟪T x, y⟫_𝕜 = ⟪T y, x⟫_𝕜
-- LinearMap.IsSymmetric.apply_clm
@[simp]
theorem IsSymmetric.apply_clm {T : E →L[𝕜] E} (hT : IsSymmetric (T : E →ₗ[𝕜] E)) (x y : E) :
    ⟪T x, y⟫ = ⟪x, T y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →L[𝕜] E} (hT : (↑T).IsSymmetric) (x y : E)
Goal: ⟪T x, y⟫_𝕜 = ⟪x, T y⟫_𝕜
-- LinearMap.isSymmetric_zero
theorem isSymmetric_zero : (0 : E →ₗ[𝕜] E).IsSymmetric 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: LinearMap.IsSymmetric 0
-- LinearMap.isSymmetric_id
theorem isSymmetric_id : (LinearMap.id : E →ₗ[𝕜] E).IsSymmetric 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: LinearMap.id.IsSymmetric
-- LinearMap.IsSymmetric.add
theorem IsSymmetric.add {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric) :
    (T + S).IsSymmetric 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric)
Goal: (T + S).IsSymmetric
-- LinearMap.IsSymmetric.continuous
theorem IsSymmetric.continuous [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) :
    Continuous T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: Continuous ⇑T
-- LinearMap.IsSymmetric.coe_reApplyInnerSelf_apply
@[simp]
theorem IsSymmetric.coe_reApplyInnerSelf_apply {T : E →L[𝕜] E} (hT : IsSymmetric (T : E →ₗ[𝕜] E))
    (x : E) : (T.reApplyInnerSelf x : 𝕜) = ⟪T x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →L[𝕜] E} (hT : (↑T).IsSymmetric) (x : E)
Goal: ↑(T.reApplyInnerSelf x) = ⟪T x, x⟫_𝕜
-- LinearMap.IsSymmetric.restrict_invariant
theorem IsSymmetric.restrict_invariant {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) {V : Submodule 𝕜 E}
    (hV : ∀ v ∈ V, T v ∈ V) : IsSymmetric (T.restrict hV) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) {V : Submodule 𝕜 E} (hV : ∀ v ∈ V, T v ∈ V)
Goal: (T.restrict hV).IsSymmetric
-- LinearMap.IsSymmetric.restrictScalars
theorem IsSymmetric.restrictScalars {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) :
    @LinearMap.IsSymmetric ℝ E _ _ (InnerProductSpace.rclikeToReal 𝕜 E)
      (@LinearMap.restrictScalars ℝ 𝕜 _ _ _ _ _ _ (InnerProductSpace.rclikeToReal 𝕜 E).toModule
        (InnerProductSpace.rclikeToReal 𝕜 E).toModule _ _ _ T) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: (↑ℝ T).IsSymmetric
-- LinearMap.isSymmetric_iff_inner_map_self_real
theorem isSymmetric_iff_inner_map_self_real (T : V →ₗ[ℂ] V) :
    IsSymmetric T ↔ ∀ v : V, conj ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ 
Arguments: {V : Type u_6} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (T : V →ₗ[ℂ] V)
Goal: T.IsSymmetric ↔ ∀ (v : V), (starRingEnd ℂ) ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ
-- LinearMap.IsSymmetric.inner_map_polarization
theorem IsSymmetric.inner_map_polarization {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x y : E) :
    ⟪T x, y⟫ =
      (⟪T (x + y), x + y⟫ - ⟪T (x - y), x - y⟫ - I * ⟪T (x + (I : 𝕜) • y), x + (I : 𝕜) • y⟫ +
          I * ⟪T (x - (I : 𝕜) • y), x - (I : 𝕜) • y⟫) /
        4 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x y : E)
Goal: ⟪T x, y⟫_𝕜 =
  (⟪T (x + y), x + y⟫_𝕜 - ⟪T (x - y), x - y⟫_𝕜 - RCLike.I * ⟪T (x + RCLike.I • y), x + RCLike.I • y⟫_𝕜 +
      RCLike.I * ⟪T (x - RCLike.I • y), x - RCLike.I • y⟫_𝕜) /
    4
-- LinearMap.IsSymmetric.inner_map_self_eq_zero
theorem IsSymmetric.inner_map_self_eq_zero {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) :
    (∀ x, ⟪T x, x⟫ = 0) ↔ T = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: (∀ (x : E), ⟪T x, x⟫_𝕜 = 0) ↔ T = 0
-- MeasureTheory.IsAddFundamentalDomain.ae_covers
structure IsAddFundamentalDomain (G : Type*) {α : Type*} [Zero G] [VAdd G α] [MeasurableSpace α]
    (s : Set α) (μ : Measure α := by volume_tac) : Prop where
  protected nullMeasurableSet : NullMeasurableSet s μ
  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g +ᵥ x ∈ s
  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g +ᵥ s)
Arguments: {G : Type u_1} {α : Type u_2} [Zero G] [VAdd G α] [MeasurableSpace α] {s : Set α} {μ : autoParam (MeasureTheory.Measure α) _auto✝} (self : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: ∀ᵐ (x : α) ∂μ, ∃ g, g +ᵥ x ∈ s
-- MeasureTheory.IsAddFundamentalDomain.aedisjoint
structure IsAddFundamentalDomain (G : Type*) {α : Type*} [Zero G] [VAdd G α] [MeasurableSpace α]
    (s : Set α) (μ : Measure α := by volume_tac) : Prop where
  protected nullMeasurableSet : NullMeasurableSet s μ
  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g +ᵥ x ∈ s
  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g +ᵥ s)
Arguments: {G : Type u_1} {α : Type u_2} [Zero G] [VAdd G α] [MeasurableSpace α] {s : Set α} {μ : autoParam (MeasureTheory.Measure α) _auto✝} (self : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: Pairwise (MeasureTheory.AEDisjoint μ on fun g => g +ᵥ s)
-- MeasureTheory.IsAddFundamentalDomain.nullMeasurableSet
structure IsAddFundamentalDomain (G : Type*) {α : Type*} [Zero G] [VAdd G α] [MeasurableSpace α]
    (s : Set α) (μ : Measure α := by volume_tac) : Prop where
  protected nullMeasurableSet : NullMeasurableSet s μ
  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g +ᵥ x ∈ s
  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g +ᵥ s)
Arguments: {G : Type u_1} {α : Type u_2} [Zero G] [VAdd G α] [MeasurableSpace α] {s : Set α} {μ : autoParam (MeasureTheory.Measure α) _auto✝} (self : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.IsFundamentalDomain.aedisjoint
@[to_additive IsAddFundamentalDomain]
structure IsFundamentalDomain (G : Type*) {α : Type*} [One G] [SMul G α] [MeasurableSpace α]
    (s : Set α) (μ : Measure α := by volume_tac) : Prop where
  protected nullMeasurableSet : NullMeasurableSet s μ
  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s
  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g • s)
Arguments: {G : Type u_1} {α : Type u_2} [One G] [SMul G α] [MeasurableSpace α] {s : Set α} {μ : autoParam (MeasureTheory.Measure α) _auto✝} (self : MeasureTheory.IsFundamentalDomain G s μ)
Goal: Pairwise (MeasureTheory.AEDisjoint μ on fun g => g • s)
-- MeasureTheory.IsFundamentalDomain.nullMeasurableSet
@[to_additive IsAddFundamentalDomain]
structure IsFundamentalDomain (G : Type*) {α : Type*} [One G] [SMul G α] [MeasurableSpace α]
    (s : Set α) (μ : Measure α := by volume_tac) : Prop where
  protected nullMeasurableSet : NullMeasurableSet s μ
  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s
  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g • s)
Arguments: {G : Type u_1} {α : Type u_2} [One G] [SMul G α] [MeasurableSpace α] {s : Set α} {μ : autoParam (MeasureTheory.Measure α) _auto✝} (self : MeasureTheory.IsFundamentalDomain G s μ)
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.IsFundamentalDomain.ae_covers
@[to_additive IsAddFundamentalDomain]
structure IsFundamentalDomain (G : Type*) {α : Type*} [One G] [SMul G α] [MeasurableSpace α]
    (s : Set α) (μ : Measure α := by volume_tac) : Prop where
  protected nullMeasurableSet : NullMeasurableSet s μ
  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s
  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g • s)
Arguments: {G : Type u_1} {α : Type u_2} [One G] [SMul G α] [MeasurableSpace α] {s : Set α} {μ : autoParam (MeasureTheory.Measure α) _auto✝} (self : MeasureTheory.IsFundamentalDomain G s μ)
Goal: ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s
-- MeasureTheory.IsFundamentalDomain.mk'
@[to_additive "If for each `x : α`, exactly one of `g +ᵥ x`, `g : G`, belongs to a measurable set
`s`, then `s` is a fundamental domain for the additive action of `G` on `α`."]
theorem mk' (h_meas : NullMeasurableSet s μ) (h_exists : ∀ x : α, ∃! g : G, g • x ∈ s) :
    IsFundamentalDomain G s μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h_meas : MeasureTheory.NullMeasurableSet s μ) (h_exists : ∀ (x : α), ∃! g, g • x ∈ s)
Goal: MeasureTheory.IsFundamentalDomain G s μ
-- MeasureTheory.IsAddFundamentalDomain.mk'
@[to_additive "If for each `x : α`, exactly one of `g +ᵥ x`, `g : G`, belongs to a measurable set
`s`, then `s` is a fundamental domain for the additive action of `G` on `α`."]
theorem mk' (h_meas : NullMeasurableSet s μ) (h_exists : ∀ x : α, ∃! g : G, g • x ∈ s) :
    IsFundamentalDomain G s μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h_meas : MeasureTheory.NullMeasurableSet s μ) (h_exists : ∀ (x : α), ∃! g, g +ᵥ x ∈ s)
Goal: MeasureTheory.IsAddFundamentalDomain G s μ
-- MeasureTheory.IsAddFundamentalDomain.mk''
@[to_additive "For `s` to be a fundamental domain, it's enough to check
  `MeasureTheory.AEDisjoint (g +ᵥ s) s` for `g ≠ 0`."]
theorem mk'' (h_meas : NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s)
    (h_ae_disjoint : ∀ g, g ≠ (1 : G) → AEDisjoint μ (g • s) s)
    (h_qmp : ∀ g : G, QuasiMeasurePreserving ((g • ·) : α → α) μ μ) :
    IsFundamentalDomain G s μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h_meas : MeasureTheory.NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ (x : α) ∂μ, ∃ g, g +ᵥ x ∈ s) (h_ae_disjoint : ∀ (g : G), g ≠ 0 → MeasureTheory.AEDisjoint μ (g +ᵥ s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g +ᵥ x) μ μ)
Goal: MeasureTheory.IsAddFundamentalDomain G s μ
-- MeasureTheory.IsFundamentalDomain.mk''
@[to_additive "For `s` to be a fundamental domain, it's enough to check
  `MeasureTheory.AEDisjoint (g +ᵥ s) s` for `g ≠ 0`."]
theorem mk'' (h_meas : NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s)
    (h_ae_disjoint : ∀ g, g ≠ (1 : G) → AEDisjoint μ (g • s) s)
    (h_qmp : ∀ g : G, QuasiMeasurePreserving ((g • ·) : α → α) μ μ) :
    IsFundamentalDomain G s μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h_meas : MeasureTheory.NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s) (h_ae_disjoint : ∀ (g : G), g ≠ 1 → MeasureTheory.AEDisjoint μ (g • s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g • x) μ μ)
Goal: MeasureTheory.IsFundamentalDomain G s μ
-- MeasureTheory.IsAddFundamentalDomain.mk_of_measure_univ_le
@[to_additive
  "If a measurable space has a finite measure `μ` and a countable additive group `G` acts
  quasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient
  to check that its translates `g +ᵥ s` are (almost) disjoint and that the sum `∑' g, μ (g +ᵥ s)` is
  sufficiently large."]
theorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)
    (h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
    (h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)
    (h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [Countable G] (h_meas : MeasureTheory.NullMeasurableSet s μ) (h_ae_disjoint : ∀ (g : G), g ≠ 0 → MeasureTheory.AEDisjoint μ (g +ᵥ s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g +ᵥ x) μ μ) (h_measure_univ_le : μ Set.univ ≤ ∑' (g : G), μ (g +ᵥ s))
Goal: MeasureTheory.IsAddFundamentalDomain G s μ
-- MeasureTheory.IsFundamentalDomain.mk_of_measure_univ_le
@[to_additive
  "If a measurable space has a finite measure `μ` and a countable additive group `G` acts
  quasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient
  to check that its translates `g +ᵥ s` are (almost) disjoint and that the sum `∑' g, μ (g +ᵥ s)` is
  sufficiently large."]
theorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)
    (h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
    (h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)
    (h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [Countable G] (h_meas : MeasureTheory.NullMeasurableSet s μ) (h_ae_disjoint : ∀ (g : G), g ≠ 1 → MeasureTheory.AEDisjoint μ (g • s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g • x) μ μ) (h_measure_univ_le : μ Set.univ ≤ ∑' (g : G), μ (g • s))
Goal: MeasureTheory.IsFundamentalDomain G s μ
-- MeasureTheory.IsFundamentalDomain.iUnion_smul_ae_eq
@[to_additive]
theorem iUnion_smul_ae_eq (h : IsFundamentalDomain G s μ) : ⋃ g : G, g • s =ᵐ[μ] univ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IsFundamentalDomain G s μ)
Goal: ⋃ g, g • s =ᵐ[μ] Set.univ
-- MeasureTheory.IsAddFundamentalDomain.iUnion_vadd_ae_eq
@[to_additive]
theorem iUnion_smul_ae_eq (h : IsFundamentalDomain G s μ) : ⋃ g : G, g • s =ᵐ[μ] univ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: ⋃ g, g +ᵥ s =ᵐ[μ] Set.univ
-- MeasureTheory.IsAddFundamentalDomain.measure_ne_zero
@[to_additive]
theorem measure_ne_zero [Countable G] [SMulInvariantMeasure G α μ]
    (hμ : μ ≠ 0) (h : IsFundamentalDomain G s μ) : μ s ≠ 0 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [Countable G] [MeasureTheory.VAddInvariantMeasure G α μ] (hμ : μ ≠ 0) (h : MeasureTheory.IsAddFundamentalDomain G s μ)
Goal: μ s ≠ 0
-- MeasureTheory.IsFundamentalDomain.measure_ne_zero
@[to_additive]
theorem measure_ne_zero [Countable G] [SMulInvariantMeasure G α μ]
    (hμ : μ ≠ 0) (h : IsFundamentalDomain G s μ) : μ s ≠ 0 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [Countable G] [MeasureTheory.SMulInvariantMeasure G α μ] (hμ : μ ≠ 0) (h : MeasureTheory.IsFundamentalDomain G s μ)
Goal: μ s ≠ 0
-- MeasureTheory.IsAddFundamentalDomain.mono
@[to_additive]
theorem mono (h : IsFundamentalDomain G s μ) {ν : Measure α} (hle : ν ≪ μ) :
    IsFundamentalDomain G s ν 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IsAddFundamentalDomain G s μ) {ν : MeasureTheory.Measure α} (hle : ν ≪ μ)
Goal: MeasureTheory.IsAddFundamentalDomain G s ν
-- MeasureTheory.IsFundamentalDomain.mono
@[to_additive]
theorem mono (h : IsFundamentalDomain G s μ) {ν : Measure α} (hle : ν ≪ μ) :
    IsFundamentalDomain G s ν 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IsFundamentalDomain G s μ) {ν : MeasureTheory.Measure α} (hle : ν ≪ μ)
Goal: MeasureTheory.IsFundamentalDomain G s ν
-- MeasureTheory.IsAddFundamentalDomain.preimage_of_equiv
@[to_additive]
theorem preimage_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) {f : β → α}
    (hf : QuasiMeasurePreserving f ν μ) {e : G → H} (he : Bijective e)
    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f ⁻¹' s) ν 
Arguments: {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddGroup H] [AddAction G α] [MeasurableSpace α] [AddAction H β] [MeasurableSpace β] {s : Set α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} (h : MeasureTheory.IsAddFundamentalDomain G s μ) {f : β → α} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f ν μ) {e : G → H} (he : Function.Bijective e) (hef : ∀ (g : G), Function.Semiconj f (fun x => e g +ᵥ x) fun x => g +ᵥ x)
Goal: MeasureTheory.IsAddFundamentalDomain H (f ⁻¹' s) ν
-- MeasureTheory.IsFundamentalDomain.preimage_of_equiv
@[to_additive]
theorem preimage_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) {f : β → α}
    (hf : QuasiMeasurePreserving f ν μ) {e : G → H} (he : Bijective e)
    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f ⁻¹' s) ν 
Arguments: {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [Group G] [Group H] [MulAction G α] [MeasurableSpace α] [MulAction H β] [MeasurableSpace β] {s : Set α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} (h : MeasureTheory.IsFundamentalDomain G s μ) {f : β → α} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f ν μ) {e : G → H} (he : Function.Bijective e) (hef : ∀ (g : G), Function.Semiconj f (fun x => e g • x) fun x => g • x)
Goal: MeasureTheory.IsFundamentalDomain H (f ⁻¹' s) ν
-- MeasureTheory.IsFundamentalDomain.image_of_equiv
@[to_additive]
theorem image_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) (f : α ≃ β)
    (hf : QuasiMeasurePreserving f.symm ν μ) (e : H ≃ G)
    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f '' s) ν 
Arguments: {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [Group G] [Group H] [MulAction G α] [MeasurableSpace α] [MulAction H β] [MeasurableSpace β] {s : Set α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} (h : MeasureTheory.IsFundamentalDomain G s μ) (f : α ≃ β) (hf : MeasureTheory.Measure.QuasiMeasurePreserving (⇑f.symm) ν μ) (e : H ≃ G) (hef : ∀ (g : H), Function.Semiconj (⇑f) (fun x => e g • x) fun x => g • x)
Goal: MeasureTheory.IsFundamentalDomain H (⇑f '' s) ν
-- MeasureTheory.IsAddFundamentalDomain.image_of_equiv
@[to_additive]
theorem image_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) (f : α ≃ β)
    (hf : QuasiMeasurePreserving f.symm ν μ) (e : H ≃ G)
    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f '' s) ν 
Arguments: {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddGroup H] [AddAction G α] [MeasurableSpace α] [AddAction H β] [MeasurableSpace β] {s : Set α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} (h : MeasureTheory.IsAddFundamentalDomain G s μ) (f : α ≃ β) (hf : MeasureTheory.Measure.QuasiMeasurePreserving (⇑f.symm) ν μ) (e : H ≃ G) (hef : ∀ (g : H), Function.Semiconj (⇑f) (fun x => e g +ᵥ x) fun x => g +ᵥ x)
Goal: MeasureTheory.IsAddFundamentalDomain H (⇑f '' s) ν
-- MeasureTheory.IsAddFundamentalDomain.pairwise_aedisjoint_of_ac
@[to_additive]
theorem pairwise_aedisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
    Pairwise fun g₁ g₂ : G => AEDisjoint ν (g₁ • s) (g₂ • s) 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ ν : MeasureTheory.Measure α} (h : MeasureTheory.IsAddFundamentalDomain G s μ) (hν : ν ≪ μ)
Goal: Pairwise fun g₁ g₂ => MeasureTheory.AEDisjoint ν (g₁ +ᵥ s) (g₂ +ᵥ s)
-- MeasureTheory.IsFundamentalDomain.pairwise_aedisjoint_of_ac
@[to_additive]
theorem pairwise_aedisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
    Pairwise fun g₁ g₂ : G => AEDisjoint ν (g₁ • s) (g₂ • s) 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ ν : MeasureTheory.Measure α} (h : MeasureTheory.IsFundamentalDomain G s μ) (hν : ν ≪ μ)
Goal: Pairwise fun g₁ g₂ => MeasureTheory.AEDisjoint ν (g₁ • s) (g₂ • s)
-- MeasureTheory.IsFundamentalDomain.smul_of_comm
@[to_additive]
theorem smul_of_comm {G' : Type*} [Group G'] [MulAction G' α] [MeasurableSpace G']
    [MeasurableSMul G' α] [SMulInvariantMeasure G' α μ] [SMulCommClass G' G α]
    (h : IsFundamentalDomain G s μ) (g : G') : IsFundamentalDomain G (g • s) μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} {G' : Type u_6} [Group G'] [MulAction G' α] [MeasurableSpace G'] [MeasurableSMul G' α] [MeasureTheory.SMulInvariantMeasure G' α μ] [SMulCommClass G' G α] (h : MeasureTheory.IsFundamentalDomain G s μ) (g : G')
Goal: MeasureTheory.IsFundamentalDomain G (g • s) μ
-- MeasureTheory.IsAddFundamentalDomain.vadd_of_comm
@[to_additive]
theorem smul_of_comm {G' : Type*} [Group G'] [MulAction G' α] [MeasurableSpace G']
    [MeasurableSMul G' α] [SMulInvariantMeasure G' α μ] [SMulCommClass G' G α]
    (h : IsFundamentalDomain G s μ) (g : G') : IsFundamentalDomain G (g • s) μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} {G' : Type u_6} [AddGroup G'] [AddAction G' α] [MeasurableSpace G'] [MeasurableVAdd G' α] [MeasureTheory.VAddInvariantMeasure G' α μ] [VAddCommClass G' G α] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (g : G')
Goal: MeasureTheory.IsAddFundamentalDomain G (g +ᵥ s) μ
-- MeasureTheory.IsFundamentalDomain.nullMeasurableSet_smul
@[to_additive]
theorem nullMeasurableSet_smul (h : IsFundamentalDomain G s μ) (g : G) :
    NullMeasurableSet (g • s) μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] (h : MeasureTheory.IsFundamentalDomain G s μ) (g : G)
Goal: MeasureTheory.NullMeasurableSet (g • s) μ
-- MeasureTheory.IsAddFundamentalDomain.nullMeasurableSet_vadd
@[to_additive]
theorem nullMeasurableSet_smul (h : IsFundamentalDomain G s μ) (g : G) :
    NullMeasurableSet (g • s) μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} [MeasurableSpace G] [MeasurableVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] (h : MeasureTheory.IsAddFundamentalDomain G s μ) (g : G)
Goal: MeasureTheory.NullMeasurableSet (g +ᵥ s) μ