ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasurableSpace.generateFrom_empty
@[simp]
lemma generateFrom_empty : generateFrom (∅ : Set (Set α)) = ⊥ 
Arguments: {α : Type u_1}
Goal: MeasurableSpace.generateFrom ∅ = ⊥
-- MeasurableSpace.generateFrom_singleton_empty
theorem generateFrom_singleton_empty : generateFrom {∅} = (⊥ : MeasurableSpace α) 
Arguments: {α : Type u_1}
Goal: MeasurableSpace.generateFrom {∅} = ⊥
-- MeasurableSpace.generateFrom_singleton_univ
theorem generateFrom_singleton_univ : generateFrom {Set.univ} = (⊥ : MeasurableSpace α) 
Arguments: {α : Type u_1}
Goal: MeasurableSpace.generateFrom {Set.univ} = ⊥
-- MeasurableSpace.generateFrom_insert_univ
@[simp]
theorem generateFrom_insert_univ (S : Set (Set α)) :
    generateFrom (insert Set.univ S) = generateFrom S 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: MeasurableSpace.generateFrom (insert Set.univ S) = MeasurableSpace.generateFrom S
-- MeasurableSpace.generateFrom_insert_empty
@[simp]
theorem generateFrom_insert_empty (S : Set (Set α)) :
    generateFrom (insert ∅ S) = generateFrom S 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: MeasurableSpace.generateFrom (insert ∅ S) = MeasurableSpace.generateFrom S
-- MeasurableSpace.measurableSet_bot_iff
theorem measurableSet_bot_iff {s : Set α} : MeasurableSet[⊥] s ↔ s = ∅ ∨ s = univ 
Arguments: {α : Type u_1} {s : Set α}
Goal: MeasurableSet s ↔ s = ∅ ∨ s = Set.univ
-- MeasurableSpace.measurableSet_top
@[simp, measurability] theorem measurableSet_top {s : Set α} : MeasurableSet[⊤] s 
Arguments: {α : Type u_1} {s : Set α}
Goal: MeasurableSet s
-- MeasurableSpace.measurableSet_inf
@[simp, nolint simpNF] -- Porting note (#11215): TODO: `simpNF` claims that
-- this lemma doesn't simplify LHS
theorem measurableSet_inf {m₁ m₂ : MeasurableSpace α} {s : Set α} :
    MeasurableSet[m₁ ⊓ m₂] s ↔ MeasurableSet[m₁] s ∧ MeasurableSet[m₂] s 
Arguments: {α : Type u_1} {m₁ m₂ : MeasurableSpace α} {s : Set α}
Goal: MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s
-- MeasurableSpace.measurableSet_sInf
@[simp]
theorem measurableSet_sInf {ms : Set (MeasurableSpace α)} {s : Set α} :
    MeasurableSet[sInf ms] s ↔ ∀ m ∈ ms, MeasurableSet[m] s 
Arguments: {α : Type u_1} {ms : Set (MeasurableSpace α)} {s : Set α}
Goal: MeasurableSet s ↔ ∀ m ∈ ms, MeasurableSet s
-- MeasurableSpace.measurableSet_iInf
theorem measurableSet_iInf {ι} {m : ι → MeasurableSpace α} {s : Set α} :
    MeasurableSet[iInf m] s ↔ ∀ i, MeasurableSet[m i] s 
Arguments: {α : Type u_1} {ι : Sort u_7} {m : ι → MeasurableSpace α} {s : Set α}
Goal: MeasurableSet s ↔ ∀ (i : ι), MeasurableSet s
-- MeasurableSpace.measurableSet_sup
theorem measurableSet_sup {m₁ m₂ : MeasurableSpace α} {s : Set α} :
    MeasurableSet[m₁ ⊔ m₂] s ↔ GenerateMeasurable (MeasurableSet[m₁] ∪ MeasurableSet[m₂]) s 
Arguments: {α : Type u_1} {m₁ m₂ : MeasurableSpace α} {s : Set α}
Goal: MeasurableSet s ↔ MeasurableSpace.GenerateMeasurable (MeasurableSet ∪ MeasurableSet) s
-- MeasurableSpace.measurableSet_sSup
theorem measurableSet_sSup {ms : Set (MeasurableSpace α)} {s : Set α} :
    MeasurableSet[sSup ms] s ↔
      GenerateMeasurable { s : Set α | ∃ m ∈ ms, MeasurableSet[m] s } s 
Arguments: {α : Type u_1} {ms : Set (MeasurableSpace α)} {s : Set α}
Goal: MeasurableSet s ↔ MeasurableSpace.GenerateMeasurable {s | ∃ m ∈ ms, MeasurableSet s} s
-- MeasurableSpace.measurableSet_iSup
theorem measurableSet_iSup {ι} {m : ι → MeasurableSpace α} {s : Set α} :
    MeasurableSet[iSup m] s ↔ GenerateMeasurable { s : Set α | ∃ i, MeasurableSet[m i] s } s 
Arguments: {α : Type u_1} {ι : Sort u_7} {m : ι → MeasurableSpace α} {s : Set α}
Goal: MeasurableSet s ↔ MeasurableSpace.GenerateMeasurable {s | ∃ i, MeasurableSet s} s
-- MeasurableSpace.measurableSpace_iSup_eq
theorem measurableSpace_iSup_eq (m : ι → MeasurableSpace α) :
    ⨆ n, m n = generateFrom { s | ∃ n, MeasurableSet[m n] s } 
Arguments: {α : Type u_1} {ι : Sort u_6} (m : ι → MeasurableSpace α)
Goal: ⨆ n, m n = MeasurableSpace.generateFrom {s | ∃ n, MeasurableSet s}
-- MeasurableSpace.generateFrom_iUnion_measurableSet
theorem generateFrom_iUnion_measurableSet (m : ι → MeasurableSpace α) :
    generateFrom (⋃ n, { t | MeasurableSet[m n] t }) = ⨆ n, m n 
Arguments: {α : Type u_1} {ι : Sort u_6} (m : ι → MeasurableSpace α)
Goal: MeasurableSpace.generateFrom (⋃ n, {t | MeasurableSet t}) = ⨆ n, m n
-- measurable_id
@[measurability]
theorem measurable_id {_ : MeasurableSpace α} : Measurable (@id α) 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α}, Measurable id
-- measurable_id'
@[fun_prop, measurability]
theorem measurable_id' {_ : MeasurableSpace α} : Measurable fun a : α => a 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α}, Measurable fun a => a
-- Measurable.comp
protected theorem Measurable.comp {_ : MeasurableSpace α} {_ : MeasurableSpace β}
    {_ : MeasurableSpace γ} {g : β → γ} {f : α → β} (hg : Measurable g) (hf : Measurable f) :
    Measurable (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3}
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β} {x_2 : MeasurableSpace γ} {g : β → γ} {f : α → β},
  Measurable g → Measurable f → Measurable (g ∘ f)
-- Measurable.comp'
@[fun_prop, aesop safe 50 (rule_sets := [Measurable])]
protected theorem Measurable.comp' {_ : MeasurableSpace α} {_ : MeasurableSpace β}
    {_ : MeasurableSpace γ} {g : β → γ} {f : α → β} (hg : Measurable g) (hf : Measurable f) :
    Measurable (fun x => g (f x)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3}
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β} {x_2 : MeasurableSpace γ} {g : β → γ} {f : α → β},
  Measurable g → Measurable f → Measurable fun x => g (f x)
-- measurable_const
@[simp, fun_prop, measurability]
theorem measurable_const {_ : MeasurableSpace α} {_ : MeasurableSpace β} {a : α} :
    Measurable fun _ : β => a 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β} {a : α}, Measurable fun x => a
-- Measurable.le
theorem Measurable.le {α} {m m0 : MeasurableSpace α} {_ : MeasurableSpace β} (hm : m ≤ m0)
    {f : α → β} (hf : Measurable[m] f) : Measurable[m0] f 
Arguments: {β : Type u_2} {α : Type u_7} {m m0 : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace β}, m ≤ m0 → ∀ {f : α → β}, Measurable f → Measurable f
-- DiscreteMeasurableSpace.forall_measurableSet
class DiscreteMeasurableSpace (α : Type*) [MeasurableSpace α] : Prop where
    forall_measurableSet : ∀ s : Set α, MeasurableSet s
Arguments: {α : Type u_7} [MeasurableSpace α] [self : DiscreteMeasurableSpace α] (s : Set α)
Goal: MeasurableSet s
-- measurableSet_discrete
@[measurability] lemma measurableSet_discrete (s : Set α) : MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] [DiscreteMeasurableSpace α] (s : Set α)
Goal: MeasurableSet s
-- measurable_discrete
@[measurability]
lemma measurable_discrete (f : α → β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [DiscreteMeasurableSpace α] (f : α → β)
Goal: Measurable f
-- MeasurableSpace.CountablyGenerated.isCountablyGenerated
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
  isCountablyGenerated : ∃ b : Set (Set α), b.Countable ∧ m = generateFrom b
Arguments: {α : Type u_3} [m : MeasurableSpace α] [self : MeasurableSpace.CountablyGenerated α]
Goal: ∃ b, b.Countable ∧ m = MeasurableSpace.generateFrom b
-- MeasurableSpace.countable_countableGeneratingSet
lemma countable_countableGeneratingSet [MeasurableSpace α] [h : CountablyGenerated α] :
    Set.Countable (countableGeneratingSet α) 
Arguments: {α : Type u_1} [MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α]
Goal: (MeasurableSpace.countableGeneratingSet α).Countable
-- MeasurableSpace.generateFrom_countableGeneratingSet
lemma generateFrom_countableGeneratingSet [m : MeasurableSpace α] [h : CountablyGenerated α] :
    generateFrom (countableGeneratingSet α) = m 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α]
Goal: MeasurableSpace.generateFrom (MeasurableSpace.countableGeneratingSet α) = m
-- MeasurableSpace.empty_mem_countableGeneratingSet
lemma empty_mem_countableGeneratingSet [MeasurableSpace α] [CountablyGenerated α] :
    ∅ ∈ countableGeneratingSet α 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α]
Goal: ∅ ∈ MeasurableSpace.countableGeneratingSet α
-- MeasurableSpace.nonempty_countableGeneratingSet
lemma nonempty_countableGeneratingSet [MeasurableSpace α] [CountablyGenerated α] :
    Set.Nonempty (countableGeneratingSet α) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α]
Goal: (MeasurableSpace.countableGeneratingSet α).Nonempty
-- MeasurableSpace.measurableSet_countableGeneratingSet
lemma measurableSet_countableGeneratingSet [MeasurableSpace α] [CountablyGenerated α]
    {s : Set α} (hs : s ∈ countableGeneratingSet α) :
    MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] {s : Set α} (hs : s ∈ MeasurableSpace.countableGeneratingSet α)
Goal: MeasurableSet s
-- MeasurableSpace.generateFrom_natGeneratingSequence
lemma generateFrom_natGeneratingSequence (α : Type*) [m : MeasurableSpace α]
    [CountablyGenerated α] : generateFrom (range (natGeneratingSequence _)) = m 
Arguments: (α : Type u_3) [m : MeasurableSpace α] [MeasurableSpace.CountablyGenerated α]
Goal: MeasurableSpace.generateFrom (Set.range (MeasurableSpace.natGeneratingSequence α)) = m
-- MeasurableSpace.measurableSet_natGeneratingSequence
lemma measurableSet_natGeneratingSequence [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
    MeasurableSet (natGeneratingSequence α n) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: MeasurableSet (MeasurableSpace.natGeneratingSequence α n)
-- MeasurableSpace.CountablyGenerated.comap
theorem CountablyGenerated.comap [m : MeasurableSpace β] [h : CountablyGenerated β] (f : α → β) :
    @CountablyGenerated α (.comap f m) 
Arguments: {α : Type u_1} {β : Type u_2} [m : MeasurableSpace β] [h : MeasurableSpace.CountablyGenerated β] (f : α → β)
Goal: MeasurableSpace.CountablyGenerated α
-- MeasurableSpace.CountablyGenerated.sup
theorem CountablyGenerated.sup {m₁ m₂ : MeasurableSpace β} (h₁ : @CountablyGenerated β m₁)
    (h₂ : @CountablyGenerated β m₂) : @CountablyGenerated β (m₁ ⊔ m₂) 
Arguments: {β : Type u_2} {m₁ m₂ : MeasurableSpace β} (h₁ : MeasurableSpace.CountablyGenerated β) (h₂ : MeasurableSpace.CountablyGenerated β)
Goal: MeasurableSpace.CountablyGenerated β
-- MeasurableSpace.SeparatesPoints.separates
class SeparatesPoints (α : Type*) [m : MeasurableSpace α] : Prop where
  separates : ∀ x y : α, (∀ s, MeasurableSet s → (x ∈ s → y ∈ s)) → x = y
Arguments: {α : Type u_3} [m : MeasurableSpace α] [self : MeasurableSpace.SeparatesPoints α] (x y : α)
Goal: (∀ (s : Set α), MeasurableSet s → x ∈ s → y ∈ s) → x = y
-- MeasurableSpace.separatesPoints_def
theorem separatesPoints_def [MeasurableSpace α] [hs : SeparatesPoints α] {x y : α}
    (h : ∀ s, MeasurableSet s → (x ∈ s → y ∈ s)) : x = y 
Arguments: {α : Type u_1} [MeasurableSpace α] [hs : MeasurableSpace.SeparatesPoints α] {x y : α} (h : ∀ (s : Set α), MeasurableSet s → x ∈ s → y ∈ s)
Goal: x = y
-- MeasurableSpace.exists_measurableSet_of_ne
theorem exists_measurableSet_of_ne [MeasurableSpace α] [SeparatesPoints α] {x y : α}
    (h : x ≠ y) : ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.SeparatesPoints α] {x y : α} (h : x ≠ y)
Goal: ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s
-- MeasurableSpace.separatesPoints_iff
theorem separatesPoints_iff [MeasurableSpace α] : SeparatesPoints α ↔
    ∀ x y : α, (∀ s, MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasurableSpace.SeparatesPoints α ↔ ∀ (x y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y
-- MeasurableSpace.separating_of_generateFrom
theorem separating_of_generateFrom (S : Set (Set α))
    [h : @SeparatesPoints α (generateFrom S)] :
    ∀ x y : α, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y 
Arguments: {α : Type u_1} (S : Set (Set α)) [h : MeasurableSpace.SeparatesPoints α] (x y : α)
Goal: (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
-- MeasurableSpace.SeparatesPoints.mono
theorem SeparatesPoints.mono {m m' : MeasurableSpace α} [hsep : @SeparatesPoints _ m] (h : m ≤ m') :
    @SeparatesPoints _ m' 
Arguments: {α : Type u_1} {m m' : MeasurableSpace α} [hsep : MeasurableSpace.SeparatesPoints α] (h : m ≤ m')
Goal: MeasurableSpace.SeparatesPoints α
-- MeasurableSpace.CountablySeparated.countably_separated
class CountablySeparated (α : Type*) [MeasurableSpace α] : Prop where
  countably_separated : HasCountableSeparatingOn α MeasurableSet univ
Arguments: {α : Type u_3} [MeasurableSpace α] [self : MeasurableSpace.CountablySeparated α]
Goal: HasCountableSeparatingOn α MeasurableSet Set.univ
-- MeasurableSpace.countablySeparated_def
theorem countablySeparated_def [MeasurableSpace α] :
    CountablySeparated α ↔ HasCountableSeparatingOn α MeasurableSet univ 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasurableSpace.CountablySeparated α ↔ HasCountableSeparatingOn α MeasurableSet Set.univ
-- MeasurableSpace.CountablySeparated.mono
theorem CountablySeparated.mono {m m' : MeasurableSpace α} [hsep : @CountablySeparated _ m]
    (h : m ≤ m') : @CountablySeparated _ m' 
Arguments: {α : Type u_1} {m m' : MeasurableSpace α} [hsep : MeasurableSpace.CountablySeparated α] (h : m ≤ m')
Goal: MeasurableSpace.CountablySeparated α
-- MeasurableSpace.CountablySeparated.subtype_iff
theorem CountablySeparated.subtype_iff [MeasurableSpace α] {s : Set α} :
    CountablySeparated s ↔ HasCountableSeparatingOn α MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α}
Goal: MeasurableSpace.CountablySeparated ↑s ↔ HasCountableSeparatingOn α MeasurableSet s
-- MeasurableSpace.exists_countablyGenerated_le_of_countablySeparated
theorem exists_countablyGenerated_le_of_countablySeparated [m : MeasurableSpace α]
    [h : CountablySeparated α] :
    ∃ m' : MeasurableSpace α, @CountablyGenerated _ m' ∧ @SeparatesPoints _ m' ∧ m' ≤ m 
Arguments: (α : Type u_1) [m : MeasurableSpace α] [h : MeasurableSpace.CountablySeparated α]
Goal: ∃ m', MeasurableSpace.CountablyGenerated α ∧ MeasurableSpace.SeparatesPoints α ∧ m' ≤ m
-- MeasurableSpace.measurable_mapNatBool
theorem measurable_mapNatBool [MeasurableSpace α] [CountablyGenerated α] :
    Measurable (mapNatBool α) 
Arguments: (α : Type u_1) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α]
Goal: Measurable (MeasurableSpace.mapNatBool α)
-- MeasurableSpace.injective_mapNatBool
theorem injective_mapNatBool [MeasurableSpace α] [CountablyGenerated α]
    [SeparatesPoints α] : Injective (mapNatBool α) 
Arguments: (α : Type u_1) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] [MeasurableSpace.SeparatesPoints α]
Goal: Function.Injective (MeasurableSpace.mapNatBool α)
-- MeasurableSpace.measurableEquiv_nat_bool_of_countablyGenerated
theorem measurableEquiv_nat_bool_of_countablyGenerated [MeasurableSpace α]
    [CountablyGenerated α] [SeparatesPoints α] :
    ∃ s : Set (ℕ → Bool), Nonempty (α ≃ᵐ s) 
Arguments: (α : Type u_1) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] [MeasurableSpace.SeparatesPoints α]
Goal: ∃ s, Nonempty (α ≃ᵐ ↑s)
-- MeasurableSpace.measurable_injection_nat_bool_of_countablySeparated
theorem measurable_injection_nat_bool_of_countablySeparated [MeasurableSpace α]
    [CountablySeparated α] : ∃ f : α → ℕ → Bool, Measurable f ∧ Injective f 
Arguments: (α : Type u_1) [MeasurableSpace α] [MeasurableSpace.CountablySeparated α]
Goal: ∃ f, Measurable f ∧ Function.Injective f
-- MeasurableSpace.measurableSingletonClass_of_countablySeparated
theorem measurableSingletonClass_of_countablySeparated
    [MeasurableSpace α] [CountablySeparated α] :
    MeasurableSingletonClass α 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablySeparated α]
Goal: MeasurableSingletonClass α
-- MeasurableSpace.measurableSet_succ_memPartition
lemma measurableSet_succ_memPartition (t : ℕ → Set α) (n : ℕ) {s : Set α}
    (hs : s ∈ memPartition t n) :
    MeasurableSet[generateFrom (memPartition t (n + 1))] s 
Arguments: {α : Type u_1} (t : ℕ → Set α) (n : ℕ) {s : Set α} (hs : s ∈ memPartition t n)
Goal: MeasurableSet s
-- MeasurableSpace.generateFrom_memPartition_le_succ
lemma generateFrom_memPartition_le_succ (t : ℕ → Set α) (n : ℕ) :
    generateFrom (memPartition t n) ≤ generateFrom (memPartition t (n + 1)) 
Arguments: {α : Type u_1} (t : ℕ → Set α) (n : ℕ)
Goal: MeasurableSpace.generateFrom (memPartition t n) ≤ MeasurableSpace.generateFrom (memPartition t (n + 1))
-- MeasurableSpace.measurableSet_generateFrom_memPartition_iff
lemma measurableSet_generateFrom_memPartition_iff (t : ℕ → Set α) (n : ℕ) (s : Set α) :
    MeasurableSet[generateFrom (memPartition t n)] s
      ↔ ∃ S : Finset (Set α), ↑S ⊆ memPartition t n ∧ s = ⋃₀ S 
Arguments: {α : Type u_1} (t : ℕ → Set α) (n : ℕ) (s : Set α)
Goal: MeasurableSet s ↔ ∃ S, ↑S ⊆ memPartition t n ∧ s = ⋃₀ ↑S
-- MeasurableSpace.measurableSet_generateFrom_memPartition
lemma measurableSet_generateFrom_memPartition (t : ℕ → Set α) (n : ℕ) :
    MeasurableSet[generateFrom (memPartition t (n + 1))] (t n) 
Arguments: {α : Type u_1} (t : ℕ → Set α) (n : ℕ)
Goal: MeasurableSet (t n)
-- MeasurableSpace.generateFrom_iUnion_memPartition
lemma generateFrom_iUnion_memPartition (t : ℕ → Set α) :
    generateFrom (⋃ n, memPartition t n) = generateFrom (range t) 
Arguments: {α : Type u_1} (t : ℕ → Set α)
Goal: MeasurableSpace.generateFrom (⋃ n, memPartition t n) = MeasurableSpace.generateFrom (Set.range t)
-- MeasurableSpace.generateFrom_memPartition_le_range
lemma generateFrom_memPartition_le_range (t : ℕ → Set α) (n : ℕ) :
    generateFrom (memPartition t n) ≤ generateFrom (range t) 
Arguments: {α : Type u_1} (t : ℕ → Set α) (n : ℕ)
Goal: MeasurableSpace.generateFrom (memPartition t n) ≤ MeasurableSpace.generateFrom (Set.range t)
-- MeasurableSpace.generateFrom_iUnion_memPartition_le
lemma generateFrom_iUnion_memPartition_le [m : MeasurableSpace α] {t : ℕ → Set α}
    (ht : ∀ n, MeasurableSet (t n)) :
    generateFrom (⋃ n, memPartition t n) ≤ m 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n))
Goal: MeasurableSpace.generateFrom (⋃ n, memPartition t n) ≤ m
-- MeasurableSpace.generateFrom_memPartition_le
lemma generateFrom_memPartition_le [m : MeasurableSpace α] {t : ℕ → Set α}
    (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) :
    generateFrom (memPartition t n) ≤ m 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ)
Goal: MeasurableSpace.generateFrom (memPartition t n) ≤ m
-- MeasurableSpace.measurableSet_memPartition
lemma measurableSet_memPartition [MeasurableSpace α] {t : ℕ → Set α}
    (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) {s : Set α} (hs : s ∈ memPartition t n) :
    MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ) {s : Set α} (hs : s ∈ memPartition t n)
Goal: MeasurableSet s
-- MeasurableSpace.measurableSet_memPartitionSet
lemma measurableSet_memPartitionSet [MeasurableSpace α] {t : ℕ → Set α}
    (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) (a : α) :
    MeasurableSet (memPartitionSet t n a) 
Arguments: {α : Type u_1} [MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ) (a : α)
Goal: MeasurableSet (memPartitionSet t n a)
-- MeasurableSpace.measurableSet_enumerateCountable_countableGeneratingSet
lemma measurableSet_enumerateCountable_countableGeneratingSet
    (α : Type*) [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
    MeasurableSet (enumerateCountable (countable_countableGeneratingSet (α := α)) ∅ n) 
Arguments: (α : Type u_3) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: MeasurableSet (Set.enumerateCountable ⋯ ∅ n)
-- MeasurableSpace.finite_countablePartition
lemma finite_countablePartition (α : Type*) [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
    Set.Finite (countablePartition α n) 
Arguments: (α : Type u_3) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: (MeasurableSpace.countablePartition α n).Finite
-- MeasurableSpace.disjoint_countablePartition
lemma disjoint_countablePartition {n : ℕ} {s t : Set α}
    (hs : s ∈ countablePartition α n) (ht : t ∈ countablePartition α n) (hst : s ≠ t) :
    Disjoint s t 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] {n : ℕ} {s t : Set α} (hs : s ∈ MeasurableSpace.countablePartition α n) (ht : t ∈ MeasurableSpace.countablePartition α n) (hst : s ≠ t)
Goal: Disjoint s t
-- MeasurableSpace.sUnion_countablePartition
lemma sUnion_countablePartition (α : Type*) [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
    ⋃₀ countablePartition α n = univ 
Arguments: (α : Type u_3) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: ⋃₀ MeasurableSpace.countablePartition α n = Set.univ
-- MeasurableSpace.measurableSet_generateFrom_countablePartition_iff
lemma measurableSet_generateFrom_countablePartition_iff (n : ℕ) (s : Set α) :
    MeasurableSet[generateFrom (countablePartition α n)] s
      ↔ ∃ S : Finset (Set α), ↑S ⊆ countablePartition α n ∧ s = ⋃₀ S 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] (n : ℕ) (s : Set α)
Goal: MeasurableSet s ↔ ∃ S, ↑S ⊆ MeasurableSpace.countablePartition α n ∧ s = ⋃₀ ↑S
-- MeasurableSpace.measurableSet_succ_countablePartition
lemma measurableSet_succ_countablePartition (n : ℕ) {s : Set α} (hs : s ∈ countablePartition α n) :
    MeasurableSet[generateFrom (countablePartition α (n + 1))] s 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] (n : ℕ) {s : Set α} (hs : s ∈ MeasurableSpace.countablePartition α n)
Goal: MeasurableSet s
-- MeasurableSpace.generateFrom_countablePartition_le_succ
lemma generateFrom_countablePartition_le_succ (α : Type*) [MeasurableSpace α] [CountablyGenerated α]
    (n : ℕ) :
    generateFrom (countablePartition α n) ≤ generateFrom (countablePartition α (n + 1)) 
Arguments: (α : Type u_3) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: MeasurableSpace.generateFrom (MeasurableSpace.countablePartition α n) ≤
  MeasurableSpace.generateFrom (MeasurableSpace.countablePartition α (n + 1))
-- MeasurableSpace.generateFrom_iUnion_countablePartition
lemma generateFrom_iUnion_countablePartition (α : Type*) [m : MeasurableSpace α]
    [CountablyGenerated α] :
    generateFrom (⋃ n, countablePartition α n) = m 
Arguments: (α : Type u_3) [m : MeasurableSpace α] [MeasurableSpace.CountablyGenerated α]
Goal: MeasurableSpace.generateFrom (⋃ n, MeasurableSpace.countablePartition α n) = m
-- MeasurableSpace.generateFrom_countablePartition_le
lemma generateFrom_countablePartition_le (α : Type*) [m : MeasurableSpace α] [CountablyGenerated α]
    (n : ℕ) :
    generateFrom (countablePartition α n) ≤ m 
Arguments: (α : Type u_3) [m : MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: MeasurableSpace.generateFrom (MeasurableSpace.countablePartition α n) ≤ m
-- MeasurableSpace.measurableSet_countablePartition
lemma measurableSet_countablePartition (n : ℕ) {s : Set α} (hs : s ∈ countablePartition α n) :
    MeasurableSet s 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] (n : ℕ) {s : Set α} (hs : s ∈ MeasurableSpace.countablePartition α n)
Goal: MeasurableSet s
-- MeasurableSpace.countablePartitionSet_mem
lemma countablePartitionSet_mem (n : ℕ) (a : α) :
    countablePartitionSet n a ∈ countablePartition α n 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] (n : ℕ) (a : α)
Goal: MeasurableSpace.countablePartitionSet n a ∈ MeasurableSpace.countablePartition α n
-- MeasurableSpace.mem_countablePartitionSet
lemma mem_countablePartitionSet (n : ℕ) (a : α) : a ∈ countablePartitionSet n a 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] (n : ℕ) (a : α)
Goal: a ∈ MeasurableSpace.countablePartitionSet n a