ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isPiSystem_Iio
theorem isPiSystem_Iio : IsPiSystem (range Iio : Set (Set α)) 
Arguments: {α : Type u_1} [LinearOrder α]
Goal: IsPiSystem (Set.range Set.Iio)
-- isPiSystem_image_Ioi
theorem isPiSystem_image_Ioi (s : Set α) : IsPiSystem (Ioi '' s) 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α)
Goal: IsPiSystem (Set.Ioi '' s)
-- isPiSystem_Ioi
theorem isPiSystem_Ioi : IsPiSystem (range Ioi : Set (Set α)) 
Arguments: {α : Type u_1} [LinearOrder α]
Goal: IsPiSystem (Set.range Set.Ioi)
-- isPiSystem_image_Iic
theorem isPiSystem_image_Iic (s : Set α) : IsPiSystem (Iic '' s) 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α)
Goal: IsPiSystem (Set.Iic '' s)
-- isPiSystem_Iic
theorem isPiSystem_Iic : IsPiSystem (range Iic : Set (Set α)) 
Arguments: {α : Type u_1} [LinearOrder α]
Goal: IsPiSystem (Set.range Set.Iic)
-- isPiSystem_image_Ici
theorem isPiSystem_image_Ici (s : Set α) : IsPiSystem (Ici '' s) 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α)
Goal: IsPiSystem (Set.Ici '' s)
-- isPiSystem_Ici
theorem isPiSystem_Ici : IsPiSystem (range Ici : Set (Set α)) 
Arguments: {α : Type u_1} [LinearOrder α]
Goal: IsPiSystem (Set.range Set.Ici)
-- isPiSystem_Ixx_mem
theorem isPiSystem_Ixx_mem {Ixx : α → α → Set α} {p : α → α → Prop}
    (Hne : ∀ {a b}, (Ixx a b).Nonempty → p a b)
    (Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (s t : Set α) :
    IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), p l u ∧ Ixx l u = S } 
Arguments: {α : Type u_1} [LinearOrder α] {Ixx : α → α → Set α} {p : α → α → Prop} (Hne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b) (Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (s t : Set α)
Goal: IsPiSystem {S | ∃ l ∈ s, ∃ u ∈ t, p l u ∧ Ixx l u = S}
-- isPiSystem_Ixx
theorem isPiSystem_Ixx {Ixx : α → α → Set α} {p : α → α → Prop}
    (Hne : ∀ {a b}, (Ixx a b).Nonempty → p a b)
    (Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (f : ι → α)
    (g : ι' → α) : @IsPiSystem α { S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S } 
Arguments: {α : Type u_1} {ι : Sort u_2} {ι' : Sort u_3} [LinearOrder α] {Ixx : α → α → Set α} {p : α → α → Prop} (Hne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b) (Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (f : ι → α) (g : ι' → α)
Goal: IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S}
-- isPiSystem_Ioo_mem
theorem isPiSystem_Ioo_mem (s t : Set α) :
    IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), l < u ∧ Ioo l u = S } 
Arguments: {α : Type u_1} [LinearOrder α] (s t : Set α)
Goal: IsPiSystem {S | ∃ l ∈ s, ∃ u ∈ t, l < u ∧ Set.Ioo l u = S}
-- isPiSystem_Ioo
theorem isPiSystem_Ioo (f : ι → α) (g : ι' → α) :
    @IsPiSystem α { S | ∃ l u, f l < g u ∧ Ioo (f l) (g u) = S } 
Arguments: {α : Type u_1} {ι : Sort u_2} {ι' : Sort u_3} [LinearOrder α] (f : ι → α) (g : ι' → α)
Goal: IsPiSystem {S | ∃ l u, f l < g u ∧ Set.Ioo (f l) (g u) = S}
-- isPiSystem_Ioc_mem
theorem isPiSystem_Ioc_mem (s t : Set α) :
    IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), l < u ∧ Ioc l u = S } 
Arguments: {α : Type u_1} [LinearOrder α] (s t : Set α)
Goal: IsPiSystem {S | ∃ l ∈ s, ∃ u ∈ t, l < u ∧ Set.Ioc l u = S}
-- isPiSystem_Ioc
theorem isPiSystem_Ioc (f : ι → α) (g : ι' → α) :
    @IsPiSystem α { S | ∃ i j, f i < g j ∧ Ioc (f i) (g j) = S } 
Arguments: {α : Type u_1} {ι : Sort u_2} {ι' : Sort u_3} [LinearOrder α] (f : ι → α) (g : ι' → α)
Goal: IsPiSystem {S | ∃ i j, f i < g j ∧ Set.Ioc (f i) (g j) = S}
-- isPiSystem_Ico_mem
theorem isPiSystem_Ico_mem (s t : Set α) :
    IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), l < u ∧ Ico l u = S } 
Arguments: {α : Type u_1} [LinearOrder α] (s t : Set α)
Goal: IsPiSystem {S | ∃ l ∈ s, ∃ u ∈ t, l < u ∧ Set.Ico l u = S}
-- isPiSystem_Ico
theorem isPiSystem_Ico (f : ι → α) (g : ι' → α) :
    @IsPiSystem α { S | ∃ i j, f i < g j ∧ Ico (f i) (g j) = S } 
Arguments: {α : Type u_1} {ι : Sort u_2} {ι' : Sort u_3} [LinearOrder α] (f : ι → α) (g : ι' → α)
Goal: IsPiSystem {S | ∃ i j, f i < g j ∧ Set.Ico (f i) (g j) = S}
-- isPiSystem_Icc_mem
theorem isPiSystem_Icc_mem (s t : Set α) :
    IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), l ≤ u ∧ Icc l u = S } 
Arguments: {α : Type u_1} [LinearOrder α] (s t : Set α)
Goal: IsPiSystem {S | ∃ l ∈ s, ∃ u ∈ t, l ≤ u ∧ Set.Icc l u = S}
-- isPiSystem_Icc
theorem isPiSystem_Icc (f : ι → α) (g : ι' → α) :
    @IsPiSystem α { S | ∃ i j, f i ≤ g j ∧ Icc (f i) (g j) = S } 
Arguments: {α : Type u_1} {ι : Sort u_2} {ι' : Sort u_3} [LinearOrder α] (f : ι → α) (g : ι' → α)
Goal: IsPiSystem {S | ∃ i j, f i ≤ g j ∧ Set.Icc (f i) (g j) = S}
-- isPiSystem_generatePiSystem
theorem isPiSystem_generatePiSystem {α} (S : Set (Set α)) : IsPiSystem (generatePiSystem S) 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: IsPiSystem (generatePiSystem S)
-- subset_generatePiSystem_self
theorem subset_generatePiSystem_self {α} (S : Set (Set α)) : S ⊆ generatePiSystem S 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: S ⊆ generatePiSystem S
-- generatePiSystem_subset_self
theorem generatePiSystem_subset_self {α} {S : Set (Set α)} (h_S : IsPiSystem S) :
    generatePiSystem S ⊆ S 
Arguments: {α : Type u_1} {S : Set (Set α)} (h_S : IsPiSystem S)
Goal: generatePiSystem S ⊆ S
-- generatePiSystem_eq
theorem generatePiSystem_eq {α} {S : Set (Set α)} (h_pi : IsPiSystem S) : generatePiSystem S = S 
Arguments: {α : Type u_1} {S : Set (Set α)} (h_pi : IsPiSystem S)
Goal: generatePiSystem S = S
-- generatePiSystem_mono
theorem generatePiSystem_mono {α} {S T : Set (Set α)} (hST : S ⊆ T) :
    generatePiSystem S ⊆ generatePiSystem T 
Arguments: {α : Type u_1} {S T : Set (Set α)} (hST : S ⊆ T)
Goal: generatePiSystem S ⊆ generatePiSystem T
-- generatePiSystem_measurableSet
theorem generatePiSystem_measurableSet {α} [M : MeasurableSpace α] {S : Set (Set α)}
    (h_meas_S : ∀ s ∈ S, MeasurableSet s) (t : Set α) (h_in_pi : t ∈ generatePiSystem S) :
    MeasurableSet t 
Arguments: {α : Type u_1} [M : MeasurableSpace α] {S : Set (Set α)} (h_meas_S : ∀ s ∈ S, MeasurableSet s) (t : Set α) (h_in_pi : t ∈ generatePiSystem S)
Goal: MeasurableSet t
-- generateFrom_measurableSet_of_generatePiSystem
theorem generateFrom_measurableSet_of_generatePiSystem {α} {g : Set (Set α)} (t : Set α)
    (ht : t ∈ generatePiSystem g) : MeasurableSet[generateFrom g] t 
Arguments: {α : Type u_1} {g : Set (Set α)} (t : Set α) (ht : t ∈ generatePiSystem g)
Goal: MeasurableSet t
-- generateFrom_generatePiSystem_eq
theorem generateFrom_generatePiSystem_eq {α} {g : Set (Set α)} :
    generateFrom (generatePiSystem g) = generateFrom g 
Arguments: {α : Type u_1} {g : Set (Set α)}
Goal: MeasurableSpace.generateFrom (generatePiSystem g) = MeasurableSpace.generateFrom g
-- mem_generatePiSystem_iUnion_elim
theorem mem_generatePiSystem_iUnion_elim {α β} {g : β → Set (Set α)} (h_pi : ∀ b, IsPiSystem (g b))
    (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b, g b)) :
    ∃ (T : Finset β) (f : β → Set α), (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b 
Arguments: {α : Type u_1} {β : Type u_2} {g : β → Set (Set α)} (h_pi : ∀ (b : β), IsPiSystem (g b)) (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b, g b))
Goal: ∃ T f, t = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
-- mem_generatePiSystem_iUnion_elim'
theorem mem_generatePiSystem_iUnion_elim' {α β} {g : β → Set (Set α)} {s : Set β}
    (h_pi : ∀ b ∈ s, IsPiSystem (g b)) (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)) :
    ∃ (T : Finset β) (f : β → Set α), ↑T ⊆ s ∧ (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b 
Arguments: {α : Type u_1} {β : Type u_2} {g : β → Set (Set α)} {s : Set β} (h_pi : ∀ b ∈ s, IsPiSystem (g b)) (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b))
Goal: ∃ T f, ↑T ⊆ s ∧ t = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b
-- piiUnionInter_singleton
theorem piiUnionInter_singleton (π : ι → Set (Set α)) (i : ι) :
    piiUnionInter π {i} = π i ∪ {univ} 
Arguments: {α : Type u_1} {ι : Type u_2} (π : ι → Set (Set α)) (i : ι)
Goal: piiUnionInter π {i} = π i ∪ {Set.univ}
-- piiUnionInter_singleton_left
theorem piiUnionInter_singleton_left (s : ι → Set α) (S : Set ι) :
    piiUnionInter (fun i => ({s i} : Set (Set α))) S =
      { s' : Set α | ∃ (t : Finset ι) (_ : ↑t ⊆ S), s' = ⋂ i ∈ t, s i } 
Arguments: {α : Type u_1} {ι : Type u_2} (s : ι → Set α) (S : Set ι)
Goal: piiUnionInter (fun i => {s i}) S = {s' | ∃ t, ∃ (_ : ↑t ⊆ S), s' = ⋂ i ∈ t, s i}
-- generateFrom_piiUnionInter_singleton_left
theorem generateFrom_piiUnionInter_singleton_left (s : ι → Set α) (S : Set ι) :
    generateFrom (piiUnionInter (fun k => {s k}) S) = generateFrom { t | ∃ k ∈ S, s k = t } 
Arguments: {α : Type u_1} {ι : Type u_2} (s : ι → Set α) (S : Set ι)
Goal: MeasurableSpace.generateFrom (piiUnionInter (fun k => {s k}) S) = MeasurableSpace.generateFrom {t | ∃ k ∈ S, s k = t}
-- isPiSystem_piiUnionInter
theorem isPiSystem_piiUnionInter (π : ι → Set (Set α)) (hpi : ∀ x, IsPiSystem (π x)) (S : Set ι) :
    IsPiSystem (piiUnionInter π S) 
Arguments: {α : Type u_1} {ι : Type u_2} (π : ι → Set (Set α)) (hpi : ∀ (x : ι), IsPiSystem (π x)) (S : Set ι)
Goal: IsPiSystem (piiUnionInter π S)
-- piiUnionInter_mono_left
theorem piiUnionInter_mono_left {π π' : ι → Set (Set α)} (h_le : ∀ i, π i ⊆ π' i) (S : Set ι) :
    piiUnionInter π S ⊆ piiUnionInter π' S 
Arguments: {α : Type u_1} {ι : Type u_2} {π π' : ι → Set (Set α)} (h_le : ∀ (i : ι), π i ⊆ π' i) (S : Set ι)
Goal: piiUnionInter π S ⊆ piiUnionInter π' S
-- piiUnionInter_mono_right
theorem piiUnionInter_mono_right {π : ι → Set (Set α)} {S T : Set ι} (hST : S ⊆ T) :
    piiUnionInter π S ⊆ piiUnionInter π T 
Arguments: {α : Type u_1} {ι : Type u_2} {π : ι → Set (Set α)} {S T : Set ι} (hST : S ⊆ T)
Goal: piiUnionInter π S ⊆ piiUnionInter π T
-- generateFrom_piiUnionInter_le
theorem generateFrom_piiUnionInter_le {m : MeasurableSpace α} (π : ι → Set (Set α))
    (h : ∀ n, generateFrom (π n) ≤ m) (S : Set ι) : generateFrom (piiUnionInter π S) ≤ m 
Arguments: {α : Type u_1} {ι : Type u_2} {m : MeasurableSpace α} (π : ι → Set (Set α)) (h : ∀ (n : ι), MeasurableSpace.generateFrom (π n) ≤ m) (S : Set ι)
Goal: MeasurableSpace.generateFrom (piiUnionInter π S) ≤ m
-- subset_piiUnionInter
theorem subset_piiUnionInter {π : ι → Set (Set α)} {S : Set ι} {i : ι} (his : i ∈ S) :
    π i ⊆ piiUnionInter π S 
Arguments: {α : Type u_1} {ι : Type u_2} {π : ι → Set (Set α)} {S : Set ι} {i : ι} (his : i ∈ S)
Goal: π i ⊆ piiUnionInter π S
-- mem_piiUnionInter_of_measurableSet
theorem mem_piiUnionInter_of_measurableSet (m : ι → MeasurableSpace α) {S : Set ι} {i : ι}
    (hiS : i ∈ S) (s : Set α) (hs : MeasurableSet[m i] s) :
    s ∈ piiUnionInter (fun n => { s | MeasurableSet[m n] s }) S 
Arguments: {α : Type u_1} {ι : Type u_2} (m : ι → MeasurableSpace α) {S : Set ι} {i : ι} (hiS : i ∈ S) (s : Set α) (hs : MeasurableSet s)
Goal: s ∈ piiUnionInter (fun n => {s | MeasurableSet s}) S
-- le_generateFrom_piiUnionInter
theorem le_generateFrom_piiUnionInter {π : ι → Set (Set α)} (S : Set ι) {x : ι} (hxS : x ∈ S) :
    generateFrom (π x) ≤ generateFrom (piiUnionInter π S) 
Arguments: {α : Type u_1} {ι : Type u_2} {π : ι → Set (Set α)} (S : Set ι) {x : ι} (hxS : x ∈ S)
Goal: MeasurableSpace.generateFrom (π x) ≤ MeasurableSpace.generateFrom (piiUnionInter π S)
-- measurableSet_iSup_of_mem_piiUnionInter
theorem measurableSet_iSup_of_mem_piiUnionInter (m : ι → MeasurableSpace α) (S : Set ι) (t : Set α)
    (ht : t ∈ piiUnionInter (fun n => { s | MeasurableSet[m n] s }) S) :
    MeasurableSet[⨆ i ∈ S, m i] t 
Arguments: {α : Type u_1} {ι : Type u_2} (m : ι → MeasurableSpace α) (S : Set ι) (t : Set α) (ht : t ∈ piiUnionInter (fun n => {s | MeasurableSet s}) S)
Goal: MeasurableSet t
-- generateFrom_piiUnionInter_measurableSet
theorem generateFrom_piiUnionInter_measurableSet (m : ι → MeasurableSpace α) (S : Set ι) :
    generateFrom (piiUnionInter (fun n => { s | MeasurableSet[m n] s }) S) = ⨆ i ∈ S, m i 
Arguments: {α : Type u_1} {ι : Type u_2} (m : ι → MeasurableSpace α) (S : Set ι)
Goal: MeasurableSpace.generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) = ⨆ i ∈ S, m i
-- MeasurableSpace.DynkinSystem.has_iUnion_nat
structure DynkinSystem (α : Type*) where
    Has : Set α → Prop
    has_empty : Has ∅
    has_compl : ∀ {a}, Has a → Has aᶜ
    has_iUnion_nat : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ i, Has (f i)) → Has (⋃ i, f i)
Arguments: {α : Type u_2} (self : MeasurableSpace.DynkinSystem α) {f : ℕ → Set α}
Goal: Pairwise (Disjoint on f) → (∀ (i : ℕ), self.Has (f i)) → self.Has (⋃ i, f i)
-- MeasurableSpace.DynkinSystem.has_empty
structure DynkinSystem (α : Type*) where
    Has : Set α → Prop
    has_empty : Has ∅
    has_compl : ∀ {a}, Has a → Has aᶜ
    has_iUnion_nat : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ i, Has (f i)) → Has (⋃ i, f i)
Arguments: {α : Type u_2} (self : MeasurableSpace.DynkinSystem α)
Goal: self.Has ∅
-- MeasurableSpace.DynkinSystem.has_compl
structure DynkinSystem (α : Type*) where
    Has : Set α → Prop
    has_empty : Has ∅
    has_compl : ∀ {a}, Has a → Has aᶜ
    has_iUnion_nat : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ i, Has (f i)) → Has (⋃ i, f i)
Arguments: {α : Type u_2} (self : MeasurableSpace.DynkinSystem α) {a : Set α}
Goal: self.Has a → self.Has aᶜ
-- MeasurableSpace.DynkinSystem.ext
@[ext]
theorem ext : ∀ {d₁ d₂ : DynkinSystem α}, (∀ s : Set α, d₁.Has s ↔ d₂.Has s) → d₁ = d₂
  
Arguments: {α : Type u_1} {d₁ d₂ : MeasurableSpace.DynkinSystem α}
Goal: (∀ (s : Set α), d₁.Has s ↔ d₂.Has s) → d₁ = d₂
-- MeasurableSpace.DynkinSystem.has_compl_iff
theorem has_compl_iff {a} : d.Has aᶜ ↔ d.Has a 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α) {a : Set α}
Goal: d.Has aᶜ ↔ d.Has a
-- MeasurableSpace.DynkinSystem.has_univ
theorem has_univ : d.Has univ 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α)
Goal: d.Has Set.univ
-- MeasurableSpace.DynkinSystem.has_iUnion
theorem has_iUnion {β} [Countable β] {f : β → Set α} (hd : Pairwise (Disjoint on f))
    (h : ∀ i, d.Has (f i)) : d.Has (⋃ i, f i) 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α) {β : Type u_2} [Countable β] {f : β → Set α} (hd : Pairwise (Disjoint on f)) (h : ∀ (i : β), d.Has (f i))
Goal: d.Has (⋃ i, f i)
-- MeasurableSpace.DynkinSystem.has_union
theorem has_union {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : Disjoint s₁ s₂) :
    d.Has (s₁ ∪ s₂) 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α) {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : Disjoint s₁ s₂)
Goal: d.Has (s₁ ∪ s₂)
-- MeasurableSpace.DynkinSystem.has_diff
theorem has_diff {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : s₂ ⊆ s₁) :
    d.Has (s₁ \ s₂) 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α) {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : s₂ ⊆ s₁)
Goal: d.Has (s₁ \ s₂)
-- MeasurableSpace.DynkinSystem.le_def
theorem le_def {α} {a b : DynkinSystem α} : a ≤ b ↔ a.Has ≤ b.Has 
Arguments: {α : Type u_2} {a b : MeasurableSpace.DynkinSystem α}
Goal: a ≤ b ↔ a.Has ≤ b.Has
-- MeasurableSpace.DynkinSystem.ofMeasurableSpace_le_ofMeasurableSpace_iff
theorem ofMeasurableSpace_le_ofMeasurableSpace_iff {m₁ m₂ : MeasurableSpace α} :
    ofMeasurableSpace m₁ ≤ ofMeasurableSpace m₂ ↔ m₁ ≤ m₂ 
Arguments: {α : Type u_1} {m₁ m₂ : MeasurableSpace α}
Goal: MeasurableSpace.DynkinSystem.ofMeasurableSpace m₁ ≤ MeasurableSpace.DynkinSystem.ofMeasurableSpace m₂ ↔ m₁ ≤ m₂
-- MeasurableSpace.DynkinSystem.generateHas_compl
theorem generateHas_compl {C : Set (Set α)} {s : Set α} : GenerateHas C sᶜ ↔ GenerateHas C s 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α}
Goal: MeasurableSpace.DynkinSystem.GenerateHas C sᶜ ↔ MeasurableSpace.DynkinSystem.GenerateHas C s
-- MeasurableSpace.DynkinSystem.generateHas_def
theorem generateHas_def {C : Set (Set α)} : (generate C).Has = GenerateHas C 
Arguments: {α : Type u_1} {C : Set (Set α)}
Goal: (MeasurableSpace.DynkinSystem.generate C).Has = MeasurableSpace.DynkinSystem.GenerateHas C
-- MeasurableSpace.DynkinSystem.ofMeasurableSpace_toMeasurableSpace
theorem ofMeasurableSpace_toMeasurableSpace
    (h_inter : ∀ s₁ s₂, d.Has s₁ → d.Has s₂ → d.Has (s₁ ∩ s₂)) :
    ofMeasurableSpace (d.toMeasurableSpace h_inter) = d 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α) (h_inter : ∀ (s₁ s₂ : Set α), d.Has s₁ → d.Has s₂ → d.Has (s₁ ∩ s₂))
Goal: MeasurableSpace.DynkinSystem.ofMeasurableSpace (d.toMeasurableSpace h_inter) = d
-- MeasurableSpace.DynkinSystem.generate_le
theorem generate_le {s : Set (Set α)} (h : ∀ t ∈ s, d.Has t) : generate s ≤ d 
Arguments: {α : Type u_1} (d : MeasurableSpace.DynkinSystem α) {s : Set (Set α)} (h : ∀ t ∈ s, d.Has t)
Goal: MeasurableSpace.DynkinSystem.generate s ≤ d
-- MeasurableSpace.DynkinSystem.generate_has_subset_generate_measurable
theorem generate_has_subset_generate_measurable {C : Set (Set α)} {s : Set α}
    (hs : (generate C).Has s) : MeasurableSet[generateFrom C] s 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} (hs : (MeasurableSpace.DynkinSystem.generate C).Has s)
Goal: MeasurableSet s
-- MeasurableSpace.DynkinSystem.generate_inter
theorem generate_inter {s : Set (Set α)} (hs : IsPiSystem s) {t₁ t₂ : Set α}
    (ht₁ : (generate s).Has t₁) (ht₂ : (generate s).Has t₂) : (generate s).Has (t₁ ∩ t₂) 
Arguments: {α : Type u_1} {s : Set (Set α)} (hs : IsPiSystem s) {t₁ t₂ : Set α} (ht₁ : (MeasurableSpace.DynkinSystem.generate s).Has t₁) (ht₂ : (MeasurableSpace.DynkinSystem.generate s).Has t₂)
Goal: (MeasurableSpace.DynkinSystem.generate s).Has (t₁ ∩ t₂)
-- MeasurableSpace.DynkinSystem.generateFrom_eq
theorem generateFrom_eq {s : Set (Set α)} (hs : IsPiSystem s) :
    generateFrom s = (generate s).toMeasurableSpace fun t₁ t₂ => generate_inter hs 
Arguments: {α : Type u_1} {s : Set (Set α)} (hs : IsPiSystem s)
Goal: MeasurableSpace.generateFrom s = (MeasurableSpace.DynkinSystem.generate s).toMeasurableSpace ⋯
-- MeasurableSpace.induction_on_inter
theorem induction_on_inter {C : Set α → Prop} {s : Set (Set α)} [m : MeasurableSpace α]
    (h_eq : m = generateFrom s) (h_inter : IsPiSystem s) (h_empty : C ∅) (h_basic : ∀ t ∈ s, C t)
    (h_compl : ∀ t, MeasurableSet t → C t → C tᶜ)
    (h_union :
      ∀ f : ℕ → Set α,
        Pairwise (Disjoint on f) → (∀ i, MeasurableSet (f i)) → (∀ i, C (f i)) → C (⋃ i, f i)) :
    ∀ ⦃t⦄, MeasurableSet t → C t 
Arguments: {α : Type u_1} {C : Set α → Prop} {s : Set (Set α)} [m : MeasurableSpace α] (h_eq : m = MeasurableSpace.generateFrom s) (h_inter : IsPiSystem s) (h_empty : C ∅) (h_basic : ∀ t ∈ s, C t) (h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ) (h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)) ⦃t : Set α⦄
Goal: MeasurableSet t → C t
-- MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge
theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ≥0) {ε : ℝ≥0∞}
    (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧
      (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [μ.WeaklyRegular] (f : MeasureTheory.SimpleFunc α ℝ≥0) {ε : ℝ≥0∞} (ε0 : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), ↑f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f x) ∂μ + ε
-- MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf : Measurable f) {ε : ℝ≥0∞}
    (εpos : ε ≠ 0) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [μ.WeaklyRegular] (f : α → ℝ≥0∞) (hf : Measurable f) {ε : ℝ≥0∞} (εpos : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ + ε
-- MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α → ℝ≥0)
    (fmeas : Measurable f) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [μ.WeaklyRegular] [MeasureTheory.SigmaFinite μ] (f : α → ℝ≥0) (fmeas : Measurable f) {ε : ℝ≥0∞} (ε0 : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
-- MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite μ] (f : α → ℝ≥0)
    (fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [μ.WeaklyRegular] [MeasureTheory.SigmaFinite μ] (f : α → ℝ≥0) (fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
-- MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal
theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : α → ℝ≥0)
    (fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, (f x : ℝ≥0∞) < g x) ∧
      LowerSemicontinuous g ∧
      (∀ᵐ x ∂μ, g x < ⊤) ∧
      Integrable (fun x => (g x).toReal) μ ∧ (∫ x, (g x).toReal ∂μ) < (∫ x, ↑(f x) ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] [MeasureTheory.SigmaFinite μ] (f : α → ℝ≥0) (fint : MeasureTheory.Integrable (fun x => ↑(f x)) μ) {ε : ℝ} (εpos : 0 < ε)
Goal: ∃ g,
  (∀ (x : α), ↑(f x) < g x) ∧
    LowerSemicontinuous g ∧
      (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧
        MeasureTheory.Integrable (fun x => (g x).toReal) μ ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), ↑(f x) ∂μ + ε
-- MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le
theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ≥0)
    (int_f : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
      (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] (f : MeasureTheory.SimpleFunc α ℝ≥0) (int_f : ∫⁻ (x : α), ↑(↑f x) ∂μ ≠ ⊤) {ε : ℝ≥0∞} (ε0 : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), g x ≤ ↑f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(↑f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
-- MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f : (∫⁻ x, f x ∂μ) ≠ ∞)
    {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
      (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] (f : α → ℝ≥0) (int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ⊤) {ε : ℝ≥0∞} (ε0 : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
-- MeasureTheory.exists_upperSemicontinuous_le_integral_le
theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
    (fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → ℝ≥0,
      (∀ x, g x ≤ f x) ∧
      UpperSemicontinuous g ∧
      Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, ↑(g x) ∂μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] (f : α → ℝ≥0) (fint : MeasureTheory.Integrable (fun x => ↑(f x)) μ) {ε : ℝ} (εpos : 0 < ε)
Goal: ∃ g,
  (∀ (x : α), g x ≤ f x) ∧
    UpperSemicontinuous g ∧
      MeasureTheory.Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ε ≤ ∫ (x : α), ↑(g x) ∂μ