ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PreErgodic.prob_eq_zero_or_one
theorem prob_eq_zero_or_one [IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs : MeasurableSet s)
    (hs' : f ⁻¹' s = s) : μ s = 0 ∨ μ s = 1 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s = s)
Goal: μ s = 0 ∨ μ s = 1
-- PreErgodic.of_iterate
theorem of_iterate (n : ℕ) (hf : PreErgodic f^[n] μ) : PreErgodic f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {μ : MeasureTheory.Measure α} (n : ℕ) (hf : PreErgodic f^[n] μ)
Goal: PreErgodic f μ
-- MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate
theorem preErgodic_of_preErgodic_conjugate (hg : MeasurePreserving g μ μ') (hf : PreErgodic f μ)
    {f' : β → β} (h_comm : g ∘ f = f' ∘ g) : PreErgodic f' μ' 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {μ : MeasureTheory.Measure α} {β : Type u_2} {m' : MeasurableSpace β} {μ' : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g μ μ') (hf : PreErgodic f μ) {f' : β → β} (h_comm : g ∘ f = f' ∘ g)
Goal: PreErgodic f' μ'
-- MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff
theorem preErgodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
    PreErgodic (e ∘ f ∘ e.symm) μ' ↔ PreErgodic f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {μ : MeasureTheory.Measure α} {β : Type u_2} {m' : MeasurableSpace β} {μ' : MeasureTheory.Measure β} {e : α ≃ᵐ β} (h : MeasureTheory.MeasurePreserving (⇑e) μ μ')
Goal: PreErgodic (⇑e ∘ f ∘ ⇑e.symm) μ' ↔ PreErgodic f μ
-- MeasureTheory.MeasurePreserving.ergodic_conjugate_iff
theorem ergodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
    Ergodic (e ∘ f ∘ e.symm) μ' ↔ Ergodic f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {μ : MeasureTheory.Measure α} {β : Type u_2} {m' : MeasurableSpace β} {μ' : MeasureTheory.Measure β} {e : α ≃ᵐ β} (h : MeasureTheory.MeasurePreserving (⇑e) μ μ')
Goal: Ergodic (⇑e ∘ f ∘ ⇑e.symm) μ' ↔ Ergodic f μ
-- QuasiErgodic.ae_empty_or_univ'
theorem ae_empty_or_univ' (hf : QuasiErgodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s =ᵐ[μ] s) :
    s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : QuasiErgodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s =ᵐ[μ] s)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- QuasiErgodic.ae_empty_or_univ₀
theorem ae_empty_or_univ₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ)
    (hs : f ⁻¹' s =ᵐ[μ] s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : QuasiErgodic f μ) (hsm : MeasureTheory.NullMeasurableSet s μ) (hs : f ⁻¹' s =ᵐ[μ] s)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- QuasiErgodic.ae_mem_or_ae_nmem₀
theorem ae_mem_or_ae_nmem₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ)
    (hs : f ⁻¹' s =ᵐ[μ] s) :
    (∀ᵐ x ∂μ, x ∈ s) ∨ ∀ᵐ x ∂μ, x ∉ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : QuasiErgodic f μ) (hsm : MeasureTheory.NullMeasurableSet s μ) (hs : f ⁻¹' s =ᵐ[μ] s)
Goal: (∀ᵐ (x : α) ∂μ, x ∈ s) ∨ ∀ᵐ (x : α) ∂μ, x ∉ s
-- Ergodic.quasiErgodic
theorem quasiErgodic (hf : Ergodic f μ) : QuasiErgodic f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {μ : MeasureTheory.Measure α} (hf : Ergodic f μ)
Goal: QuasiErgodic f μ
-- Ergodic.ae_empty_or_univ_of_preimage_ae_le'
theorem ae_empty_or_univ_of_preimage_ae_le' (hf : Ergodic f μ) (hs : MeasurableSet s)
    (hs' : f ⁻¹' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : Ergodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ⊤)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- Ergodic.ae_empty_or_univ_of_ae_le_preimage'
theorem ae_empty_or_univ_of_ae_le_preimage' (hf : Ergodic f μ) (hs : MeasurableSet s)
    (hs' : s ≤ᵐ[μ] f ⁻¹' s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : Ergodic f μ) (hs : MeasurableSet s) (hs' : s ≤ᵐ[μ] f ⁻¹' s) (h_fin : μ s ≠ ⊤)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- Ergodic.ae_empty_or_univ_of_image_ae_le'
theorem ae_empty_or_univ_of_image_ae_le' (hf : Ergodic f μ) (hs : MeasurableSet s)
    (hs' : f '' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : Ergodic f μ) (hs : MeasurableSet s) (hs' : f '' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ⊤)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- Ergodic.ae_empty_or_univ_of_preimage_ae_le
theorem ae_empty_or_univ_of_preimage_ae_le (hf : Ergodic f μ) (hs : MeasurableSet s)
    (hs' : f ⁻¹' s ≤ᵐ[μ] s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (hf : Ergodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s ≤ᵐ[μ] s)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- Ergodic.ae_empty_or_univ_of_ae_le_preimage
theorem ae_empty_or_univ_of_ae_le_preimage (hf : Ergodic f μ) (hs : MeasurableSet s)
    (hs' : s ≤ᵐ[μ] f ⁻¹' s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (hf : Ergodic f μ) (hs : MeasurableSet s) (hs' : s ≤ᵐ[μ] f ⁻¹' s)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- Ergodic.ae_empty_or_univ_of_image_ae_le
theorem ae_empty_or_univ_of_image_ae_le (hf : Ergodic f μ) (hs : MeasurableSet s)
    (hs' : f '' s ≤ᵐ[μ] s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (hf : Ergodic f μ) (hs : MeasurableSet s) (hs' : f '' s ≤ᵐ[μ] s)
Goal: s =ᵐ[μ] ∅ ∨ s =ᵐ[μ] Set.univ
-- CategoryTheory.SplitMono.id
/- Porting note(#5171): removed @[nolint has_nonempty_instance] -/
/- Porting note: `@[ext]` used to accept lemmas like this. Now we add an aesop rule -/
@[ext, aesop apply safe (rule_sets := [CategoryTheory])]
structure SplitMono {X Y : C} (f : X ⟶ Y) where
    retraction : Y ⟶ X
    id : f ≫ retraction = 𝟙 X := by aesop_cat
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (self : CategoryTheory.SplitMono f)
Goal: CategoryTheory.CategoryStruct.comp f self.retraction = CategoryTheory.CategoryStruct.id X
-- CategoryTheory.IsSplitMono.exists_splitMono
class IsSplitMono {X Y : C} (f : X ⟶ Y) : Prop where
    exists_splitMono : Nonempty (SplitMono f)
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} [self : CategoryTheory.IsSplitMono f]
Goal: Nonempty (CategoryTheory.SplitMono f)
-- CategoryTheory.IsSplitMono.mk'
theorem IsSplitMono.mk' {X Y : C} {f : X ⟶ Y} (sm : SplitMono f) : IsSplitMono f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (sm : CategoryTheory.SplitMono f)
Goal: CategoryTheory.IsSplitMono f
-- CategoryTheory.SplitEpi.id
/- Porting note(#5171): removed @[nolint has_nonempty_instance] -/
/- Porting note: `@[ext]` used to accept lemmas like this. Now we add an aesop rule -/
@[ext, aesop apply safe (rule_sets := [CategoryTheory])]
structure SplitEpi {X Y : C} (f : X ⟶ Y) where
    section_ : Y ⟶ X
    id : section_ ≫ f = 𝟙 Y := by aesop_cat
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (self : CategoryTheory.SplitEpi f)
Goal: CategoryTheory.CategoryStruct.comp self.section_ f = CategoryTheory.CategoryStruct.id Y
-- CategoryTheory.IsSplitEpi.exists_splitEpi
class IsSplitEpi {X Y : C} (f : X ⟶ Y) : Prop where
    exists_splitEpi : Nonempty (SplitEpi f)
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} [self : CategoryTheory.IsSplitEpi f]
Goal: Nonempty (CategoryTheory.SplitEpi f)
-- CategoryTheory.IsSplitEpi.mk'
theorem IsSplitEpi.mk' {X Y : C} {f : X ⟶ Y} (se : SplitEpi f) : IsSplitEpi f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (se : CategoryTheory.SplitEpi f)
Goal: CategoryTheory.IsSplitEpi f
-- CategoryTheory.IsSplitMono.id
@[reassoc (attr := simp)]
theorem IsSplitMono.id {X Y : C} (f : X ⟶ Y) [hf : IsSplitMono f] : f ≫ retraction f = 𝟙 X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [hf : CategoryTheory.IsSplitMono f]
Goal: CategoryTheory.CategoryStruct.comp f (CategoryTheory.retraction f) = CategoryTheory.CategoryStruct.id X
-- CategoryTheory.isIso_of_epi_of_isSplitMono
theorem isIso_of_epi_of_isSplitMono {X Y : C} (f : X ⟶ Y) [IsSplitMono f] [Epi f] : IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.IsSplitMono f] [CategoryTheory.Epi f]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.IsSplitEpi.id
@[reassoc (attr := simp)]
theorem IsSplitEpi.id {X Y : C} (f : X ⟶ Y) [hf : IsSplitEpi f] : section_ f ≫ f = 𝟙 Y 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [hf : CategoryTheory.IsSplitEpi f]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.section_ f) f = CategoryTheory.CategoryStruct.id Y
-- CategoryTheory.isIso_of_mono_of_isSplitEpi
theorem isIso_of_mono_of_isSplitEpi {X Y : C} (f : X ⟶ Y) [Mono f] [IsSplitEpi f] : IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] [CategoryTheory.IsSplitEpi f]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.SplitMono.mono
theorem SplitMono.mono {X Y : C} {f : X ⟶ Y} (sm : SplitMono f) : Mono f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (sm : CategoryTheory.SplitMono f)
Goal: CategoryTheory.Mono f
-- CategoryTheory.SplitEpi.epi
theorem SplitEpi.epi {X Y : C} {f : X ⟶ Y} (se : SplitEpi f) : Epi f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (se : CategoryTheory.SplitEpi f)
Goal: CategoryTheory.Epi f
-- CategoryTheory.IsIso.of_mono_retraction'
theorem IsIso.of_mono_retraction' {X Y : C} {f : X ⟶ Y} (hf : SplitMono f) [Mono <| hf.retraction] :
    IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (hf : CategoryTheory.SplitMono f) [CategoryTheory.Mono hf.retraction]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.IsIso.of_mono_retraction
theorem IsIso.of_mono_retraction {X Y : C} (f : X ⟶ Y) [hf : IsSplitMono f]
    [hf' : Mono <| retraction f] : IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [hf : CategoryTheory.IsSplitMono f] [hf' : CategoryTheory.Mono (CategoryTheory.retraction f)]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.IsIso.of_epi_section'
theorem IsIso.of_epi_section' {X Y : C} {f : X ⟶ Y} (hf : SplitEpi f) [Epi <| hf.section_] :
    IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {f : X ⟶ Y} (hf : CategoryTheory.SplitEpi f) [CategoryTheory.Epi hf.section_]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.IsIso.of_epi_section
theorem IsIso.of_epi_section {X Y : C} (f : X ⟶ Y) [hf : IsSplitEpi f] [hf' : Epi <| section_ f] :
    IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [hf : CategoryTheory.IsSplitEpi f] [hf' : CategoryTheory.Epi (CategoryTheory.section_ f)]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.SplitMonoCategory.isSplitMono_of_mono
class SplitMonoCategory : Prop where
    isSplitMono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [Mono f], IsSplitMono f
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [self : CategoryTheory.SplitMonoCategory C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f]
Goal: CategoryTheory.IsSplitMono f
-- CategoryTheory.SplitEpiCategory.isSplitEpi_of_epi
class SplitEpiCategory : Prop where
    isSplitEpi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [Epi f], IsSplitEpi f
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [self : CategoryTheory.SplitEpiCategory C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f]
Goal: CategoryTheory.IsSplitEpi f
-- CategoryTheory.isSplitMono_of_mono
theorem isSplitMono_of_mono [SplitMonoCategory C] {X Y : C} (f : X ⟶ Y) [Mono f] : IsSplitMono f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.SplitMonoCategory C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f]
Goal: CategoryTheory.IsSplitMono f
-- CategoryTheory.isSplitEpi_of_epi
theorem isSplitEpi_of_epi [SplitEpiCategory C] {X Y : C} (f : X ⟶ Y) [Epi f] : IsSplitEpi f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.SplitEpiCategory C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f]
Goal: CategoryTheory.IsSplitEpi f
-- CategoryTheory.SplitMono.map_retraction
@[simps]
def SplitMono.map {X Y : C} {f : X ⟶ Y} (sm : SplitMono f) (F : C ⥤ D) : SplitMono (F.map f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X Y : C} {f : X ⟶ Y} (sm : CategoryTheory.SplitMono f) (F : CategoryTheory.Functor C D)
Goal: (sm.map F).retraction = F.map sm.retraction
-- CategoryTheory.SplitEpi.map_section_
@[simps]
def SplitEpi.map {X Y : C} {f : X ⟶ Y} (se : SplitEpi f) (F : C ⥤ D) : SplitEpi (F.map f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X Y : C} {f : X ⟶ Y} (se : CategoryTheory.SplitEpi f) (F : CategoryTheory.Functor C D)
Goal: (se.map F).section_ = F.map se.section_
-- galois_poly_separable
theorem galois_poly_separable {K : Type*} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
    Separable (X ^ q - X : K[X]) 
Arguments: {K : Type u_1} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q)
Goal: (Polynomial.X ^ q - Polynomial.X).Separable
-- GaloisField.finrank
theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n 
Arguments: (p : ℕ) [h_prime : Fact (Nat.Prime p)] {n : ℕ} (h : n ≠ 0)
Goal: FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n
-- GaloisField.card
theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n 
Arguments: (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) (h : n ≠ 0)
Goal: Fintype.card (GaloisField p n) = p ^ n
-- GaloisField.splits_zmod_X_pow_sub_X
theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) 
Arguments: (p : ℕ) [h_prime : Fact (Nat.Prime p)]
Goal: Polynomial.Splits (RingHom.id (ZMod p)) (Polynomial.X ^ p - Polynomial.X)
-- GaloisField.splits_X_pow_card_sub_X
theorem splits_X_pow_card_sub_X : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) 
Arguments: (p : ℕ) [h_prime : Fact (Nat.Prime p)] {K : Type u_1} [Field K] [Fintype K] [Algebra (ZMod p) K]
Goal: Polynomial.Splits (algebraMap (ZMod p) K) (Polynomial.X ^ Fintype.card K - Polynomial.X)
-- GaloisField.isSplittingField_of_card_eq
theorem isSplittingField_of_card_eq (h : Fintype.card K = p ^ n) :
    IsSplittingField (ZMod p) K (X ^ p ^ n - X) 
Arguments: (p : ℕ) [h_prime : Fact (Nat.Prime p)] (n : ℕ) {K : Type u_1} [Field K] [Fintype K] [Algebra (ZMod p) K] (h : Fintype.card K = p ^ n)
Goal: Polynomial.IsSplittingField (ZMod p) K (Polynomial.X ^ p ^ n - Polynomial.X)
-- MeasureTheory.MeasurePreserving.map_eq
structure MeasurePreserving (f : α → β)
  (μa : Measure α := by volume_tac) (μb : Measure β := by volume_tac) : Prop where
  protected measurable : Measurable f
  protected map_eq : map f μa = μb
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {μa : autoParam (MeasureTheory.Measure α) _auto✝} {μb : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : MeasureTheory.MeasurePreserving f μa μb)
Goal: MeasureTheory.Measure.map f μa = μb
-- MeasureTheory.MeasurePreserving.measurable
structure MeasurePreserving (f : α → β)
  (μa : Measure α := by volume_tac) (μb : Measure β := by volume_tac) : Prop where
  protected measurable : Measurable f
  protected map_eq : map f μa = μb
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {μa : autoParam (MeasureTheory.Measure α) _auto✝} {μb : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : MeasureTheory.MeasurePreserving f μa μb)
Goal: Measurable f
-- Measurable.measurePreserving
protected theorem _root_.Measurable.measurePreserving
    {f : α → β} (h : Measurable f) (μa : Measure α) : MeasurePreserving f μa (map f μa) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (h : Measurable f) (μa : MeasureTheory.Measure α)
Goal: MeasureTheory.MeasurePreserving f μa (MeasureTheory.Measure.map f μa)
-- MeasureTheory.MeasurePreserving.id
protected theorem id (μ : Measure α) : MeasurePreserving id μ μ 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.MeasurePreserving id μ μ
-- MeasureTheory.MeasurePreserving.aemeasurable
protected theorem aemeasurable {f : α → β} (hf : MeasurePreserving f μa μb) : AEMeasurable f μa 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb)
Goal: AEMeasurable f μa
-- MeasureTheory.MeasurePreserving.of_isEmpty
@[nontriviality]
theorem of_isEmpty [IsEmpty β] (f : α → β) (μa : Measure α) (μb : Measure β) :
    MeasurePreserving f μa μb 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [IsEmpty β] (f : α → β) (μa : MeasureTheory.Measure α) (μb : MeasureTheory.Measure β)
Goal: MeasureTheory.MeasurePreserving f μa μb
-- MeasureTheory.MeasurePreserving.symm
theorem symm (e : α ≃ᵐ β) {μa : Measure α} {μb : Measure β} (h : MeasurePreserving e μa μb) :
    MeasurePreserving e.symm μb μa 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} (h : MeasureTheory.MeasurePreserving (⇑e) μa μb)
Goal: MeasureTheory.MeasurePreserving (⇑e.symm) μb μa
-- MeasureTheory.MeasurePreserving.restrict_preimage
theorem restrict_preimage {f : α → β} (hf : MeasurePreserving f μa μb) {s : Set β}
    (hs : MeasurableSet s) : MeasurePreserving f (μa.restrict (f ⁻¹' s)) (μb.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.MeasurePreserving f (μa.restrict (f ⁻¹' s)) (μb.restrict s)
-- MeasureTheory.MeasurePreserving.restrict_preimage_emb
theorem restrict_preimage_emb {f : α → β} (hf : MeasurePreserving f μa μb)
    (h₂ : MeasurableEmbedding f) (s : Set β) :
    MeasurePreserving f (μa.restrict (f ⁻¹' s)) (μb.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) (s : Set β)
Goal: MeasureTheory.MeasurePreserving f (μa.restrict (f ⁻¹' s)) (μb.restrict s)
-- MeasureTheory.MeasurePreserving.restrict_image_emb
theorem restrict_image_emb {f : α → β} (hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f)
    (s : Set α) : MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s)) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) (s : Set α)
Goal: MeasureTheory.MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s))
-- MeasureTheory.MeasurePreserving.aemeasurable_comp_iff
theorem aemeasurable_comp_iff {f : α → β} (hf : MeasurePreserving f μa μb)
    (h₂ : MeasurableEmbedding f) {g : β → γ} : AEMeasurable (g ∘ f) μa ↔ AEMeasurable g μb 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ}
Goal: AEMeasurable (g ∘ f) μa ↔ AEMeasurable g μb
-- MeasureTheory.MeasurePreserving.quasiMeasurePreserving
protected theorem quasiMeasurePreserving {f : α → β} (hf : MeasurePreserving f μa μb) :
    QuasiMeasurePreserving f μa μb 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f μa μb
-- MeasureTheory.MeasurePreserving.comp
protected theorem comp {g : β → γ} {f : α → β} (hg : MeasurePreserving g μb μc)
    (hf : MeasurePreserving f μa μb) : MeasurePreserving (g ∘ f) μa μc 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {g : β → γ} {f : α → β} (hg : MeasureTheory.MeasurePreserving g μb μc) (hf : MeasureTheory.MeasurePreserving f μa μb)
Goal: MeasureTheory.MeasurePreserving (g ∘ f) μa μc
-- MeasureTheory.MeasurePreserving.trans
protected theorem trans {e : α ≃ᵐ β} {e' : β ≃ᵐ γ}
    {μa : Measure α} {μb : Measure β} {μc : Measure γ}
    (h : MeasurePreserving e μa μb) (h' : MeasurePreserving e' μb μc) :
    MeasurePreserving (e.trans e') μa μc 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {e : α ≃ᵐ β} {e' : β ≃ᵐ γ} {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} (h : MeasureTheory.MeasurePreserving (⇑e) μa μb) (h' : MeasureTheory.MeasurePreserving (⇑e') μb μc)
Goal: MeasureTheory.MeasurePreserving (⇑(e.trans e')) μa μc
-- MeasureTheory.MeasurePreserving.comp_left_iff
protected theorem comp_left_iff {g : α → β} {e : β ≃ᵐ γ} (h : MeasurePreserving e μb μc) :
    MeasurePreserving (e ∘ g) μa μc ↔ MeasurePreserving g μa μb 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {g : α → β} {e : β ≃ᵐ γ} (h : MeasureTheory.MeasurePreserving (⇑e) μb μc)
Goal: MeasureTheory.MeasurePreserving (⇑e ∘ g) μa μc ↔ MeasureTheory.MeasurePreserving g μa μb
-- MeasureTheory.MeasurePreserving.comp_right_iff
protected theorem comp_right_iff {g : α → β} {e : γ ≃ᵐ α} (h : MeasurePreserving e μc μa) :
    MeasurePreserving (g ∘ e) μc μb ↔ MeasurePreserving g μa μb 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {g : α → β} {e : γ ≃ᵐ α} (h : MeasureTheory.MeasurePreserving (⇑e) μc μa)
Goal: MeasureTheory.MeasurePreserving (g ∘ ⇑e) μc μb ↔ MeasureTheory.MeasurePreserving g μa μb
-- MeasureTheory.MeasurePreserving.sigmaFinite
protected theorem sigmaFinite {f : α → β} (hf : MeasurePreserving f μa μb) [SigmaFinite μb] :
    SigmaFinite μa 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) [MeasureTheory.SigmaFinite μb]
Goal: MeasureTheory.SigmaFinite μa
-- MeasureTheory.MeasurePreserving.measure_preimage
theorem measure_preimage {f : α → β} (hf : MeasurePreserving f μa μb) {s : Set β}
    (hs : NullMeasurableSet s μb) : μa (f ⁻¹' s) = μb s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) {s : Set β} (hs : MeasureTheory.NullMeasurableSet s μb)
Goal: μa (f ⁻¹' s) = μb s