ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.volume_pi_pi
theorem volume_pi_pi [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))]
    (s : ∀ i, Set (α i)) : volume (pi univ s) = ∏ i, volume (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] (s : (i : ι) → Set (α i))
Goal: MeasureTheory.volume (Set.univ.pi s) = ∏ i : ι, MeasureTheory.volume (s i)
-- MeasureTheory.volume_pi_ball
theorem volume_pi_ball [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))]
    [∀ i, MetricSpace (α i)] (x : ∀ i, α i) {r : ℝ} (hr : 0 < r) :
    volume (Metric.ball x r) = ∏ i, volume (Metric.ball (x i) r) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : ℝ} (hr : 0 < r)
Goal: MeasureTheory.volume (Metric.ball x r) = ∏ i : ι, MeasureTheory.volume (Metric.ball (x i) r)
-- MeasureTheory.volume_pi_closedBall
theorem volume_pi_closedBall [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))]
    [∀ i, MetricSpace (α i)] (x : ∀ i, α i) {r : ℝ} (hr : 0 ≤ r) :
    volume (Metric.closedBall x r) = ∏ i, volume (Metric.closedBall (x i) r) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : ℝ} (hr : 0 ≤ r)
Goal: MeasureTheory.volume (Metric.closedBall x r) = ∏ i : ι, MeasureTheory.volume (Metric.closedBall (x i) r)
-- MeasureTheory.measurePreserving_piEquivPiSubtypeProd
theorem measurePreserving_piEquivPiSubtypeProd (p : ι → Prop) [DecidablePred p] :
    MeasurePreserving (MeasurableEquiv.piEquivPiSubtypeProd α p) (Measure.pi μ)
      ((Measure.pi fun i : Subtype p => μ i).prod (Measure.pi fun i => μ i)) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (p : ι → Prop) [DecidablePred p]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piEquivPiSubtypeProd α p)) (MeasureTheory.Measure.pi μ)
  ((MeasureTheory.Measure.pi fun i => μ ↑i).prod (MeasureTheory.Measure.pi fun i => μ ↑i))
-- MeasureTheory.volume_preserving_piEquivPiSubtypeProd
theorem volume_preserving_piEquivPiSubtypeProd (α : ι → Type*)
    [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))] (p : ι → Prop)
    [DecidablePred p] : MeasurePreserving (MeasurableEquiv.piEquivPiSubtypeProd α p) 
Arguments: {ι : Type u_1} [Fintype ι] (α : ι → Type u_4) [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] (p : ι → Prop) [DecidablePred p]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piEquivPiSubtypeProd α p)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_piCongrLeft
theorem measurePreserving_piCongrLeft (f : ι' ≃ ι) :
    MeasurePreserving (MeasurableEquiv.piCongrLeft α f)
      (Measure.pi fun i' => μ (f i')) (Measure.pi μ) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [Fintype ι'] (f : ι' ≃ ι)
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piCongrLeft α f)) (MeasureTheory.Measure.pi fun i' => μ (f i'))
  (MeasureTheory.Measure.pi μ)
-- MeasureTheory.volume_measurePreserving_piCongrLeft
theorem volume_measurePreserving_piCongrLeft (α : ι → Type*) (f : ι' ≃ ι)
    [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))] :
    MeasurePreserving (MeasurableEquiv.piCongrLeft α f) volume volume 
Arguments: {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] (α : ι → Type u_4) (f : ι' ≃ ι) [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piCongrLeft α f)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_sumPiEquivProdPi_symm
theorem measurePreserving_sumPiEquivProdPi_symm {π : ι ⊕ ι' → Type*}
    {m : ∀ i, MeasurableSpace (π i)} (μ : ∀ i, Measure (π i)) [∀ i, SigmaFinite (μ i)] :
    MeasurePreserving (MeasurableEquiv.sumPiEquivProdPi π).symm
      ((Measure.pi fun i => μ (.inl i)).prod (Measure.pi fun i => μ (.inr i))) (Measure.pi μ) 
Arguments: {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] {π : ι ⊕ ι' → Type u_4} {m : (i : ι ⊕ ι') → MeasurableSpace (π i)} (μ : (i : ι ⊕ ι') → MeasureTheory.Measure (π i)) [∀ (i : ι ⊕ ι'), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.sumPiEquivProdPi π).symm)
  ((MeasureTheory.Measure.pi fun i => μ (Sum.inl i)).prod (MeasureTheory.Measure.pi fun i => μ (Sum.inr i)))
  (MeasureTheory.Measure.pi μ)
-- MeasureTheory.volume_measurePreserving_sumPiEquivProdPi_symm
theorem volume_measurePreserving_sumPiEquivProdPi_symm (π : ι ⊕ ι' → Type*)
    [∀ i, MeasureSpace (π i)] [∀ i, SigmaFinite (volume : Measure (π i))] :
    MeasurePreserving (MeasurableEquiv.sumPiEquivProdPi π).symm volume volume 
Arguments: {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] (π : ι ⊕ ι' → Type u_4) [(i : ι ⊕ ι') → MeasureTheory.MeasureSpace (π i)] [∀ (i : ι ⊕ ι'), MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.sumPiEquivProdPi π).symm) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_sumPiEquivProdPi
theorem measurePreserving_sumPiEquivProdPi {π : ι ⊕ ι' → Type*} {_m : ∀ i, MeasurableSpace (π i)}
    (μ : ∀ i, Measure (π i)) [∀ i, SigmaFinite (μ i)] :
    MeasurePreserving (MeasurableEquiv.sumPiEquivProdPi π)
      (Measure.pi μ) ((Measure.pi fun i => μ (.inl i)).prod (Measure.pi fun i => μ (.inr i))) 
Arguments: {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] {π : ι ⊕ ι' → Type u_4} {_m : (i : ι ⊕ ι') → MeasurableSpace (π i)} (μ : (i : ι ⊕ ι') → MeasureTheory.Measure (π i)) [∀ (i : ι ⊕ ι'), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.sumPiEquivProdPi π)) (MeasureTheory.Measure.pi μ)
  ((MeasureTheory.Measure.pi fun i => μ (Sum.inl i)).prod (MeasureTheory.Measure.pi fun i => μ (Sum.inr i)))
-- MeasureTheory.volume_measurePreserving_sumPiEquivProdPi
theorem volume_measurePreserving_sumPiEquivProdPi (π : ι ⊕ ι' → Type*)
    [∀ i, MeasureSpace (π i)] [∀ i, SigmaFinite (volume : Measure (π i))] :
    MeasurePreserving (MeasurableEquiv.sumPiEquivProdPi π) volume volume 
Arguments: {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] (π : ι ⊕ ι' → Type u_4) [(i : ι ⊕ ι') → MeasureTheory.MeasureSpace (π i)] [∀ (i : ι ⊕ ι'), MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.sumPiEquivProdPi π)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_piFinSuccAbove
theorem measurePreserving_piFinSuccAbove {n : ℕ} {α : Fin (n + 1) → Type u}
    {m : ∀ i, MeasurableSpace (α i)} (μ : ∀ i, Measure (α i)) [∀ i, SigmaFinite (μ i)]
    (i : Fin (n + 1)) :
    MeasurePreserving (MeasurableEquiv.piFinSuccAbove α i) (Measure.pi μ)
      ((μ i).prod <| Measure.pi fun j => μ (i.succAbove j)) 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u} {m : (i : Fin (n + 1)) → MeasurableSpace (α i)} (μ : (i : Fin (n + 1)) → MeasureTheory.Measure (α i)) [∀ (i : Fin (n + 1)), MeasureTheory.SigmaFinite (μ i)] (i : Fin (n + 1))
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove α i)) (MeasureTheory.Measure.pi μ)
  ((μ i).prod (MeasureTheory.Measure.pi fun j => μ (i.succAbove j)))
-- MeasureTheory.volume_preserving_piFinSuccAbove
theorem volume_preserving_piFinSuccAbove {n : ℕ} (α : Fin (n + 1) → Type u)
    [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))] (i : Fin (n + 1)) :
    MeasurePreserving (MeasurableEquiv.piFinSuccAbove α i) 
Arguments: {n : ℕ} (α : Fin (n + 1) → Type u) [(i : Fin (n + 1)) → MeasureTheory.MeasureSpace (α i)] [∀ (i : Fin (n + 1)), MeasureTheory.SigmaFinite MeasureTheory.volume] (i : Fin (n + 1))
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove α i)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_piUnique
theorem measurePreserving_piUnique {π : ι → Type*} [Unique ι] {m : ∀ i, MeasurableSpace (π i)}
    (μ : ∀ i, Measure (π i)) :
    MeasurePreserving (MeasurableEquiv.piUnique π) (Measure.pi μ) (μ default) 
Arguments: {ι : Type u_1} [Fintype ι] {π : ι → Type u_4} [Unique ι] {m : (i : ι) → MeasurableSpace (π i)} (μ : (i : ι) → MeasureTheory.Measure (π i))
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piUnique π)) (MeasureTheory.Measure.pi μ) (μ default)
-- MeasureTheory.volume_preserving_piUnique
theorem volume_preserving_piUnique (π : ι → Type*) [Unique ι] [∀ i, MeasureSpace (π i)] :
    MeasurePreserving (MeasurableEquiv.piUnique π) volume volume 
Arguments: {ι : Type u_1} [Fintype ι] (π : ι → Type u_4) [Unique ι] [(i : ι) → MeasureTheory.MeasureSpace (π i)]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piUnique π)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_funUnique
theorem measurePreserving_funUnique {β : Type u} {_m : MeasurableSpace β} (μ : Measure β)
    (α : Type v) [Unique α] :
    MeasurePreserving (MeasurableEquiv.funUnique α β) (Measure.pi fun _ : α => μ) μ 
Arguments: {β : Type u} {_m : MeasurableSpace β} (μ : MeasureTheory.Measure β) (α : Type v) [Unique α]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.funUnique α β)) (MeasureTheory.Measure.pi fun x => μ) μ
-- MeasureTheory.volume_preserving_funUnique
theorem volume_preserving_funUnique (α : Type u) (β : Type v) [Unique α] [MeasureSpace β] :
    MeasurePreserving (MeasurableEquiv.funUnique α β) volume volume 
Arguments: (α : Type u) (β : Type v) [Unique α] [MeasureTheory.MeasureSpace β]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.funUnique α β)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_piFinTwo
theorem measurePreserving_piFinTwo {α : Fin 2 → Type u} {m : ∀ i, MeasurableSpace (α i)}
    (μ : ∀ i, Measure (α i)) [∀ i, SigmaFinite (μ i)] :
    MeasurePreserving (MeasurableEquiv.piFinTwo α) (Measure.pi μ) ((μ 0).prod (μ 1)) 
Arguments: {α : Fin 2 → Type u} {m : (i : Fin 2) → MeasurableSpace (α i)} (μ : (i : Fin 2) → MeasureTheory.Measure (α i)) [∀ (i : Fin 2), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piFinTwo α)) (MeasureTheory.Measure.pi μ) ((μ 0).prod (μ 1))
-- MeasureTheory.volume_preserving_piFinTwo
theorem volume_preserving_piFinTwo (α : Fin 2 → Type u) [∀ i, MeasureSpace (α i)]
    [∀ i, SigmaFinite (volume : Measure (α i))] :
    MeasurePreserving (MeasurableEquiv.piFinTwo α) volume volume 
Arguments: (α : Fin 2 → Type u) [(i : Fin 2) → MeasureTheory.MeasureSpace (α i)] [∀ (i : Fin 2), MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piFinTwo α)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_finTwoArrow_vec
theorem measurePreserving_finTwoArrow_vec {α : Type u} {m : MeasurableSpace α} (μ ν : Measure α)
    [SigmaFinite μ] [SigmaFinite ν] :
    MeasurePreserving MeasurableEquiv.finTwoArrow (Measure.pi ![μ, ν]) (μ.prod ν) 
Arguments: {α : Type u} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
Goal: MeasureTheory.MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (MeasureTheory.Measure.pi ![μ, ν]) (μ.prod ν)
-- MeasureTheory.measurePreserving_finTwoArrow
theorem measurePreserving_finTwoArrow {α : Type u} {m : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] :
    MeasurePreserving MeasurableEquiv.finTwoArrow (Measure.pi fun _ => μ) (μ.prod μ) 
Arguments: {α : Type u} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: MeasureTheory.MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (MeasureTheory.Measure.pi fun x => μ) (μ.prod μ)
-- MeasureTheory.volume_preserving_finTwoArrow
theorem volume_preserving_finTwoArrow (α : Type u) [MeasureSpace α]
    [SigmaFinite (volume : Measure α)] :
    MeasurePreserving (@MeasurableEquiv.finTwoArrow α _) volume volume 
Arguments: (α : Type u) [MeasureTheory.MeasureSpace α] [MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: MeasureTheory.MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_pi_empty
theorem measurePreserving_pi_empty {ι : Type u} {α : ι → Type v} [Fintype ι] [IsEmpty ι]
    {m : ∀ i, MeasurableSpace (α i)} (μ : ∀ i, Measure (α i)) :
    MeasurePreserving (MeasurableEquiv.ofUniqueOfUnique (∀ i, α i) Unit) (Measure.pi μ)
      (Measure.dirac ()) 
Arguments: {ι : Type u} {α : ι → Type v} [Fintype ι] [IsEmpty ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i))
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit)) (MeasureTheory.Measure.pi μ)
  (MeasureTheory.Measure.dirac ())
-- MeasureTheory.volume_preserving_pi_empty
theorem volume_preserving_pi_empty {ι : Type u} (α : ι → Type v) [Fintype ι] [IsEmpty ι]
    [∀ i, MeasureSpace (α i)] :
    MeasurePreserving (MeasurableEquiv.ofUniqueOfUnique (∀ i, α i) Unit) volume volume 
Arguments: {ι : Type u} (α : ι → Type v) [Fintype ι] [IsEmpty ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit)) MeasureTheory.volume
  MeasureTheory.volume
-- MeasureTheory.measurePreserving_piFinsetUnion
theorem measurePreserving_piFinsetUnion [DecidableEq ι] {s t : Finset ι} (h : Disjoint s t)
    (μ : ∀ i, Measure (α i)) [∀ i, SigmaFinite (μ i)] :
    MeasurePreserving (MeasurableEquiv.piFinsetUnion α h)
      ((Measure.pi fun i : s ↦ μ i).prod (Measure.pi fun i : t ↦ μ i))
      (Measure.pi fun i : ↥(s ∪ t) ↦ μ i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} {m : (i : ι) → MeasurableSpace (α i)} [DecidableEq ι] {s t : Finset ι} (h : Disjoint s t) (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piFinsetUnion α h))
  ((MeasureTheory.Measure.pi fun i => μ ↑i).prod (MeasureTheory.Measure.pi fun i => μ ↑i))
  (MeasureTheory.Measure.pi fun i => μ ↑i)
-- MeasureTheory.volume_preserving_piFinsetUnion
theorem volume_preserving_piFinsetUnion (α : ι → Type*) [DecidableEq ι] {s t : Finset ι}
    (h : Disjoint s t) [∀ i, MeasureSpace (α i)] [∀ i, SigmaFinite (volume : Measure (α i))] :
    MeasurePreserving (MeasurableEquiv.piFinsetUnion α h) volume volume 
Arguments: {ι : Type u_1} (α : ι → Type u_4) [DecidableEq ι] {s t : Finset ι} (h : Disjoint s t) [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.piFinsetUnion α h)) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.measurePreserving_pi
theorem measurePreserving_pi {β : ι → Type*} [∀ i, MeasurableSpace (β i)]
    (ν : (i : ι) → Measure (β i)) {f : (i : ι) → (α i) → (β i)} [∀ i, SigmaFinite (ν i)]
    (hf : ∀ i, MeasurePreserving (f i) (μ i) (ν i)) :
    MeasurePreserving (fun a i ↦ f i (a i)) (Measure.pi μ) (Measure.pi ν) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)) {β : ι → Type u_4} [(i : ι) → MeasurableSpace (β i)] (ν : (i : ι) → MeasureTheory.Measure (β i)) {f : (i : ι) → α i → β i} [∀ (i : ι), MeasureTheory.SigmaFinite (ν i)] (hf : ∀ (i : ι), MeasureTheory.MeasurePreserving (f i) (μ i) (ν i))
Goal: MeasureTheory.MeasurePreserving (fun a i => f i (a i)) (MeasureTheory.Measure.pi μ) (MeasureTheory.Measure.pi ν)
-- MeasureTheory.volume_preserving_pi
theorem volume_preserving_pi {α' β' : ι → Type*} [∀ i, MeasureSpace (α' i)]
    [∀ i, MeasureSpace (β' i)] [∀ i, SigmaFinite (volume : Measure (β' i))]
    {f : (i : ι) → (α' i) → (β' i)} (hf : ∀ i, MeasurePreserving (f i)) :
    MeasurePreserving (fun (a : (i : ι) → α' i) (i : ι) ↦ (f i) (a i)) 
Arguments: {ι : Type u_1} [Fintype ι] {α' : ι → Type u_4} {β' : ι → Type u_5} [(i : ι) → MeasureTheory.MeasureSpace (α' i)] [(i : ι) → MeasureTheory.MeasureSpace (β' i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] {f : (i : ι) → α' i → β' i} (hf : ∀ (i : ι), MeasureTheory.MeasurePreserving (f i) MeasureTheory.volume MeasureTheory.volume)
Goal: MeasureTheory.MeasurePreserving (fun a i => f i (a i)) MeasureTheory.volume MeasureTheory.volume
-- StandardBorelSpace.polish
class StandardBorelSpace [MeasurableSpace α] : Prop where
    polish : ∃ _ : TopologicalSpace α, BorelSpace α ∧ PolishSpace α
Arguments: {α : Type u_1} [MeasurableSpace α] [self : StandardBorelSpace α]
Goal: ∃ x, BorelSpace α ∧ PolishSpace α
-- eq_borel_upgradeStandardBorel
theorem eq_borel_upgradeStandardBorel [MeasurableSpace α] [StandardBorelSpace α] :
    ‹MeasurableSpace α› = @borel _ (upgradeStandardBorel α).toTopologicalSpace 
Arguments: (α : Type u_1) [MeasurableSpace α] [StandardBorelSpace α]
Goal: inst✝ = borel α
-- MeasureTheory.AnalyticSet_def
irreducible_def AnalyticSet (s : Set α) : Prop 
Arguments: {α : Type u_3} [TopologicalSpace α] (s : Set α)
Goal: MeasureTheory.AnalyticSet s = (s = ∅ ∨ ∃ f, Continuous f ∧ Set.range f = s)
-- MeasureTheory.analyticSet_empty
theorem analyticSet_empty : AnalyticSet (∅ : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: MeasureTheory.AnalyticSet ∅
-- MeasureTheory.analyticSet_range_of_polishSpace
theorem analyticSet_range_of_polishSpace {β : Type*} [TopologicalSpace β] [PolishSpace β]
    {f : β → α} (f_cont : Continuous f) : AnalyticSet (range f) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] [PolishSpace β] {f : β → α} (f_cont : Continuous f)
Goal: MeasureTheory.AnalyticSet (Set.range f)
-- IsOpen.analyticSet_image
theorem _root_.IsOpen.analyticSet_image {β : Type*} [TopologicalSpace β] [PolishSpace β]
    {s : Set β} (hs : IsOpen s) {f : β → α} (f_cont : Continuous f) : AnalyticSet (f '' s) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] [PolishSpace β] {s : Set β} (hs : IsOpen s) {f : β → α} (f_cont : Continuous f)
Goal: MeasureTheory.AnalyticSet (f '' s)
-- MeasureTheory.analyticSet_iff_exists_polishSpace_range
theorem analyticSet_iff_exists_polishSpace_range {s : Set α} :
    AnalyticSet s ↔
      ∃ (β : Type) (h : TopologicalSpace β) (_ : @PolishSpace β h) (f : β → α),
        @Continuous _ _ h _ f ∧ range f = s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α}
Goal: MeasureTheory.AnalyticSet s ↔ ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous f ∧ Set.range f = s
-- MeasureTheory.AnalyticSet.image_of_continuousOn
theorem AnalyticSet.image_of_continuousOn {β : Type*} [TopologicalSpace β] {s : Set α}
    (hs : AnalyticSet s) {f : α → β} (hf : ContinuousOn f s) : AnalyticSet (f '' s) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] {s : Set α} (hs : MeasureTheory.AnalyticSet s) {f : α → β} (hf : ContinuousOn f s)
Goal: MeasureTheory.AnalyticSet (f '' s)
-- MeasureTheory.AnalyticSet.image_of_continuous
theorem AnalyticSet.image_of_continuous {β : Type*} [TopologicalSpace β] {s : Set α}
    (hs : AnalyticSet s) {f : α → β} (hf : Continuous f) : AnalyticSet (f '' s) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] {s : Set α} (hs : MeasureTheory.AnalyticSet s) {f : α → β} (hf : Continuous f)
Goal: MeasureTheory.AnalyticSet (f '' s)
-- MeasureTheory.AnalyticSet.iInter
theorem AnalyticSet.iInter [hι : Nonempty ι] [Countable ι] [T2Space α] {s : ι → Set α}
    (hs : ∀ n, AnalyticSet (s n)) : AnalyticSet (⋂ n, s n) 
Arguments: {α : Type u_1} {ι : Type u_2} [TopologicalSpace α] [hι : Nonempty ι] [Countable ι] [T2Space α] {s : ι → Set α} (hs : ∀ (n : ι), MeasureTheory.AnalyticSet (s n))
Goal: MeasureTheory.AnalyticSet (⋂ n, s n)
-- MeasureTheory.AnalyticSet.iUnion
theorem AnalyticSet.iUnion [Countable ι] {s : ι → Set α} (hs : ∀ n, AnalyticSet (s n)) :
    AnalyticSet (⋃ n, s n) 
Arguments: {α : Type u_1} {ι : Type u_2} [TopologicalSpace α] [Countable ι] {s : ι → Set α} (hs : ∀ (n : ι), MeasureTheory.AnalyticSet (s n))
Goal: MeasureTheory.AnalyticSet (⋃ n, s n)
-- IsClosed.analyticSet
theorem _root_.IsClosed.analyticSet [PolishSpace α] {s : Set α} (hs : IsClosed s) :
    AnalyticSet s 
Arguments: {α : Type u_1} [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsClosed s)
Goal: MeasureTheory.AnalyticSet s
-- MeasurableSet.isClopenable
theorem _root_.MeasurableSet.isClopenable [PolishSpace α] [MeasurableSpace α] [BorelSpace α]
    {s : Set α} (hs : MeasurableSet s) : IsClopenable s 
Arguments: {α : Type u_1} [TopologicalSpace α] [PolishSpace α] [MeasurableSpace α] [BorelSpace α] {s : Set α} (hs : MeasurableSet s)
Goal: PolishSpace.IsClopenable s
-- MeasurableSet.analyticSet
theorem _root_.MeasurableSet.analyticSet {α : Type*} [t : TopologicalSpace α] [PolishSpace α]
    [MeasurableSpace α] [BorelSpace α] {s : Set α} (hs : MeasurableSet s) : AnalyticSet s 
Arguments: {α : Type u_3} [t : TopologicalSpace α] [PolishSpace α] [MeasurableSpace α] [BorelSpace α] {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.AnalyticSet s
-- Measurable.exists_continuous
theorem _root_.Measurable.exists_continuous {α β : Type*} [t : TopologicalSpace α] [PolishSpace α]
    [MeasurableSpace α] [BorelSpace α] [tβ : TopologicalSpace β] [MeasurableSpace β]
    [OpensMeasurableSpace β] {f : α → β} [SecondCountableTopology (range f)] (hf : Measurable f) :
    ∃ t' : TopologicalSpace α, t' ≤ t ∧ @Continuous α β t' tβ f ∧ @PolishSpace α t' 
Arguments: {α : Type u_3} {β : Type u_4} [t : TopologicalSpace α] [PolishSpace α] [MeasurableSpace α] [BorelSpace α] [tβ : TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] {f : α → β} [SecondCountableTopology ↑(Set.range f)] (hf : Measurable f)
Goal: ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- MeasurableSet.analyticSet_image
theorem _root_.MeasurableSet.analyticSet_image {X Y : Type*} [MeasurableSpace X]
    [StandardBorelSpace X] [TopologicalSpace Y] [MeasurableSpace Y]
    [OpensMeasurableSpace Y] {f : X → Y} [SecondCountableTopology (range f)] {s : Set X}
    (hs : MeasurableSet s) (hf : Measurable f) : AnalyticSet (f '' s) 
Arguments: {X : Type u_3} {Y : Type u_4} [MeasurableSpace X] [StandardBorelSpace X] [TopologicalSpace Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] {f : X → Y} [SecondCountableTopology ↑(Set.range f)] {s : Set X} (hs : MeasurableSet s) (hf : Measurable f)
Goal: MeasureTheory.AnalyticSet (f '' s)
-- MeasureTheory.AnalyticSet.preimage
protected lemma AnalyticSet.preimage {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    [PolishSpace X] [T2Space Y] {s : Set Y} (hs : AnalyticSet s) {f : X → Y} (hf : Continuous f) :
    AnalyticSet (f ⁻¹' s) 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [PolishSpace X] [T2Space Y] {s : Set Y} (hs : MeasureTheory.AnalyticSet s) {f : X → Y} (hf : Continuous f)
Goal: MeasureTheory.AnalyticSet (f ⁻¹' s)
-- MeasureTheory.MeasurablySeparable.iUnion
theorem MeasurablySeparable.iUnion [Countable ι] {α : Type*} [MeasurableSpace α] {s t : ι → Set α}
    (h : ∀ m n, MeasurablySeparable (s m) (t n)) : MeasurablySeparable (⋃ n, s n) (⋃ m, t m) 
Arguments: {ι : Type u_2} [Countable ι] {α : Type u_3} [MeasurableSpace α] {s t : ι → Set α} (h : ∀ (m n : ι), MeasureTheory.MeasurablySeparable (s m) (t n))
Goal: MeasureTheory.MeasurablySeparable (⋃ n, s n) (⋃ m, t m)