ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasurableSpace.comap_le_iff_le_map
theorem comap_le_iff_le_map {f : α → β} : m'.comap f ≤ m ↔ m' ≤ m.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {m' : MeasurableSpace β} {f : α → β}
Goal: MeasurableSpace.comap f m' ≤ m ↔ m' ≤ MeasurableSpace.map f m
-- MeasurableSpace.gc_comap_map
theorem gc_comap_map (f : α → β) :
    GaloisConnection (MeasurableSpace.comap f) (MeasurableSpace.map f) 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β)
Goal: GaloisConnection (MeasurableSpace.comap f) (MeasurableSpace.map f)
-- MeasurableSpace.map_mono
theorem map_mono (h : m₁ ≤ m₂) : m₁.map f ≤ m₂.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m₁ m₂ : MeasurableSpace α} {f : α → β} (h : m₁ ≤ m₂)
Goal: MeasurableSpace.map f m₁ ≤ MeasurableSpace.map f m₂
-- MeasurableSpace.monotone_map
theorem monotone_map : Monotone (MeasurableSpace.map f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: Monotone (MeasurableSpace.map f)
-- MeasurableSpace.comap_mono
theorem comap_mono (h : m₁ ≤ m₂) : m₁.comap g ≤ m₂.comap g 
Arguments: {α : Type u_1} {β : Type u_2} {m₁ m₂ : MeasurableSpace α} {g : β → α} (h : m₁ ≤ m₂)
Goal: MeasurableSpace.comap g m₁ ≤ MeasurableSpace.comap g m₂
-- MeasurableSpace.monotone_comap
theorem monotone_comap : Monotone (MeasurableSpace.comap g) 
Arguments: {α : Type u_1} {β : Type u_2} {g : β → α}
Goal: Monotone (MeasurableSpace.comap g)
-- MeasurableSpace.comap_bot
@[simp]
theorem comap_bot : (⊥ : MeasurableSpace α).comap g = ⊥ 
Arguments: {α : Type u_1} {β : Type u_2} {g : β → α}
Goal: MeasurableSpace.comap g ⊥ = ⊥
-- MeasurableSpace.comap_sup
@[simp]
theorem comap_sup : (m₁ ⊔ m₂).comap g = m₁.comap g ⊔ m₂.comap g 
Arguments: {α : Type u_1} {β : Type u_2} {m₁ m₂ : MeasurableSpace α} {g : β → α}
Goal: MeasurableSpace.comap g (m₁ ⊔ m₂) = MeasurableSpace.comap g m₁ ⊔ MeasurableSpace.comap g m₂
-- MeasurableSpace.comap_iSup
@[simp]
theorem comap_iSup {m : ι → MeasurableSpace α} : (⨆ i, m i).comap g = ⨆ i, (m i).comap g 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort uι} {g : β → α} {m : ι → MeasurableSpace α}
Goal: MeasurableSpace.comap g (⨆ i, m i) = ⨆ i, MeasurableSpace.comap g (m i)
-- MeasurableSpace.map_top
@[simp]
theorem map_top : (⊤ : MeasurableSpace α).map f = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: MeasurableSpace.map f ⊤ = ⊤
-- MeasurableSpace.map_inf
@[simp]
theorem map_inf : (m₁ ⊓ m₂).map f = m₁.map f ⊓ m₂.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m₁ m₂ : MeasurableSpace α} {f : α → β}
Goal: MeasurableSpace.map f (m₁ ⊓ m₂) = MeasurableSpace.map f m₁ ⊓ MeasurableSpace.map f m₂
-- MeasurableSpace.map_iInf
@[simp]
theorem map_iInf {m : ι → MeasurableSpace α} : (⨅ i, m i).map f = ⨅ i, (m i).map f 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort uι} {f : α → β} {m : ι → MeasurableSpace α}
Goal: MeasurableSpace.map f (⨅ i, m i) = ⨅ i, MeasurableSpace.map f (m i)
-- MeasurableSpace.comap_map_le
theorem comap_map_le : (m.map f).comap f ≤ m 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : α → β}
Goal: MeasurableSpace.comap f (MeasurableSpace.map f m) ≤ m
-- MeasurableSpace.le_map_comap
theorem le_map_comap : m ≤ (m.comap g).map g 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {g : β → α}
Goal: m ≤ MeasurableSpace.map g (MeasurableSpace.comap g m)
-- MeasurableSpace.map_const
@[simp] theorem map_const {m} (b : β) : MeasurableSpace.map (fun _a : α ↦ b) m = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} (b : β)
Goal: MeasurableSpace.map (fun _a => b) m = ⊤
-- MeasurableSpace.comap_const
@[simp] theorem comap_const {m} (b : β) : MeasurableSpace.comap (fun _a : α => b) m = ⊥ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace β} (b : β)
Goal: MeasurableSpace.comap (fun _a => b) m = ⊥
-- MeasurableSpace.comap_generateFrom
theorem comap_generateFrom {f : α → β} {s : Set (Set β)} :
    (generateFrom s).comap f = generateFrom (preimage f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set (Set β)}
Goal: MeasurableSpace.comap f (MeasurableSpace.generateFrom s) = MeasurableSpace.generateFrom (Set.preimage f '' s)
-- measurable_iff_le_map
theorem measurable_iff_le_map {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β} :
    Measurable f ↔ m₂ ≤ m₁.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β}
Goal: Measurable f ↔ m₂ ≤ MeasurableSpace.map f m₁
-- measurable_iff_comap_le
theorem measurable_iff_comap_le {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β} :
    Measurable f ↔ m₂.comap f ≤ m₁ 
Arguments: {α : Type u_1} {β : Type u_2} {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β}
Goal: Measurable f ↔ MeasurableSpace.comap f m₂ ≤ m₁
-- comap_measurable
theorem comap_measurable {m : MeasurableSpace β} (f : α → β) : Measurable[m.comap f] f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace β} (f : α → β)
Goal: Measurable f
-- Measurable.mono
theorem Measurable.mono {ma ma' : MeasurableSpace α} {mb mb' : MeasurableSpace β} {f : α → β}
    (hf : @Measurable α β ma mb f) (ha : ma ≤ ma') (hb : mb' ≤ mb) : @Measurable α β ma' mb' f 
Arguments: {α : Type u_1} {β : Type u_2} {ma ma' : MeasurableSpace α} {mb mb' : MeasurableSpace β} {f : α → β} (hf : Measurable f) (ha : ma ≤ ma') (hb : mb' ≤ mb)
Goal: Measurable f
-- measurable_id''
theorem measurable_id'' {m mα : MeasurableSpace α} (hm : m ≤ mα) : @Measurable α α mα m id 
Arguments: {α : Type u_1} {m mα : MeasurableSpace α} (hm : m ≤ mα)
Goal: Measurable id
-- measurable_from_top
@[measurability]
theorem measurable_from_top [MeasurableSpace β] {f : α → β} : Measurable[⊤] f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {f : α → β}
Goal: Measurable f
-- measurable_generateFrom
theorem measurable_generateFrom [MeasurableSpace α] {s : Set (Set β)} {f : α → β}
    (h : ∀ t ∈ s, MeasurableSet (f ⁻¹' t)) : @Measurable _ _ _ (generateFrom s) f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {s : Set (Set β)} {f : α → β} (h : ∀ t ∈ s, MeasurableSet (f ⁻¹' t))
Goal: Measurable f
-- Subsingleton.measurable
@[nontriviality, measurability]
theorem Subsingleton.measurable [Subsingleton α] : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [MeasurableSpace α] [MeasurableSpace β] [Subsingleton α]
Goal: Measurable f
-- measurable_of_subsingleton_codomain
@[nontriviality, measurability]
theorem measurable_of_subsingleton_codomain [Subsingleton β] (f : α → β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [Subsingleton β] (f : α → β)
Goal: Measurable f
-- measurable_zero
@[to_additive (attr := measurability)]
theorem measurable_one [One α] : Measurable (1 : β → α) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [Zero α]
Goal: Measurable 0
-- measurable_one
@[to_additive (attr := measurability)]
theorem measurable_one [One α] : Measurable (1 : β → α) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [One α]
Goal: Measurable 1
-- measurable_of_empty
theorem measurable_of_empty [IsEmpty α] (f : α → β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [IsEmpty α] (f : α → β)
Goal: Measurable f
-- measurable_of_empty_codomain
theorem measurable_of_empty_codomain [IsEmpty β] (f : α → β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [IsEmpty β] (f : α → β)
Goal: Measurable f
-- measurable_const'
theorem measurable_const' {f : β → α} (hf : ∀ x y, f x = f y) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : ∀ (x y : β), f x = f y)
Goal: Measurable f
-- measurable_natCast
@[measurability]
theorem measurable_natCast [NatCast α] (n : ℕ) : Measurable (n : β → α) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [NatCast α] (n : ℕ)
Goal: Measurable ↑n
-- measurable_intCast
@[measurability]
theorem measurable_intCast [IntCast α] (n : ℤ) : Measurable (n : β → α) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [IntCast α] (n : ℤ)
Goal: Measurable ↑n
-- measurable_of_countable
theorem measurable_of_countable [Countable α] [MeasurableSingletonClass α] (f : α → β) :
    Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [Countable α] [MeasurableSingletonClass α] (f : α → β)
Goal: Measurable f
-- measurable_of_finite
theorem measurable_of_finite [Finite α] [MeasurableSingletonClass α] (f : α → β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [Finite α] [MeasurableSingletonClass α] (f : α → β)
Goal: Measurable f
-- Measurable.iterate
@[measurability]
theorem Measurable.iterate {f : α → α} (hf : Measurable f) : ∀ n, Measurable f^[n]
  
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → α} (hf : Measurable f) (n : ℕ)
Goal: Measurable f^[n]
-- measurableSet_preimage
@[measurability]
theorem measurableSet_preimage {t : Set β} (hf : Measurable f) (ht : MeasurableSet t) :
    MeasurableSet (f ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {t : Set β} (hf : Measurable f) (ht : MeasurableSet t)
Goal: MeasurableSet (f ⁻¹' t)
-- MeasurableSet.preimage
protected theorem MeasurableSet.preimage {t : Set β} (ht : MeasurableSet t) (hf : Measurable f) :
    MeasurableSet (f ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {t : Set β} (ht : MeasurableSet t) (hf : Measurable f)
Goal: MeasurableSet (f ⁻¹' t)
-- Measurable.piecewise
@[measurability]
protected theorem Measurable.piecewise {_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s)
    (hf : Measurable f) (hg : Measurable g) : Measurable (piecewise s f g) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {f g : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: ∀ {x : DecidablePred fun x => x ∈ s}, MeasurableSet s → Measurable f → Measurable g → Measurable (s.piecewise f g)
-- Measurable.ite
theorem Measurable.ite {p : α → Prop} {_ : DecidablePred p} (hp : MeasurableSet { a : α | p a })
    (hf : Measurable f) (hg : Measurable g) : Measurable fun x => ite (p x) (f x) (g x) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {p : α → Prop}
Goal: ∀ {x : DecidablePred p},
  MeasurableSet {a | p a} → Measurable f → Measurable g → Measurable fun x_1 => if p x_1 then f x_1 else g x_1
-- Measurable.indicator
@[measurability, fun_prop]
theorem Measurable.indicator [Zero β] (hf : Measurable f) (hs : MeasurableSet s) :
    Measurable (s.indicator f) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Zero β] (hf : Measurable f) (hs : MeasurableSet s)
Goal: Measurable (s.indicator f)
-- measurable_indicator_const_iff
lemma measurable_indicator_const_iff [Zero β] [MeasurableSingletonClass β] (b : β) [NeZero b] :
    Measurable (s.indicator (fun (_ : α) ↦ b)) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Zero β] [MeasurableSingletonClass β] (b : β) [NeZero b]
Goal: Measurable (s.indicator fun x => b) ↔ MeasurableSet s
-- measurableSet_mulSupport
@[to_additive (attr := measurability)]
theorem measurableSet_mulSupport [One β] [MeasurableSingletonClass β] (hf : Measurable f) :
    MeasurableSet (mulSupport f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [One β] [MeasurableSingletonClass β] (hf : Measurable f)
Goal: MeasurableSet (Function.mulSupport f)
-- measurableSet_support
@[to_additive (attr := measurability)]
theorem measurableSet_mulSupport [One β] [MeasurableSingletonClass β] (hf : Measurable f) :
    MeasurableSet (mulSupport f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Zero β] [MeasurableSingletonClass β] (hf : Measurable f)
Goal: MeasurableSet (Function.support f)
-- Measurable.measurable_of_countable_ne
theorem Measurable.measurable_of_countable_ne [MeasurableSingletonClass α] (hf : Measurable f)
    (h : Set.Countable { x | f x ≠ g x }) : Measurable g 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSingletonClass α] (hf : Measurable f) (h : {x | f x ≠ g x}.Countable)
Goal: Measurable g
-- measurable_to_countable
theorem measurable_to_countable [MeasurableSpace α] [Countable α] [MeasurableSpace β] {f : β → α}
    (h : ∀ y, MeasurableSet (f ⁻¹' {f y})) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Countable α] [MeasurableSpace β] {f : β → α} (h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}))
Goal: Measurable f
-- measurable_to_countable'
theorem measurable_to_countable' [MeasurableSpace α] [Countable α] [MeasurableSpace β] {f : β → α}
    (h : ∀ x, MeasurableSet (f ⁻¹' {x})) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Countable α] [MeasurableSpace β] {f : β → α} (h : ∀ (x : α), MeasurableSet (f ⁻¹' {x}))
Goal: Measurable f
-- measurable_unit
@[measurability]
theorem measurable_unit [MeasurableSpace α] (f : Unit → α) : Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : Unit → α)
Goal: Measurable f
-- measurable_down
lemma measurable_down : Measurable (ULift.down : ULift α → α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Measurable ULift.down
-- measurable_up
lemma measurable_up : Measurable (ULift.up : α → ULift α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Measurable ULift.up
-- measurableSet_preimage_down
@[simp] lemma measurableSet_preimage_down {s : Set α} :
    MeasurableSet (ULift.down ⁻¹' s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α}
Goal: MeasurableSet (ULift.down ⁻¹' s) ↔ MeasurableSet s
-- measurableSet_preimage_up
@[simp] lemma measurableSet_preimage_up {s : Set (ULift α)} :
    MeasurableSet (ULift.up ⁻¹' s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set (ULift.{u_6, u_1} α)}
Goal: MeasurableSet (ULift.up ⁻¹' s) ↔ MeasurableSet s
-- measurable_from_nat
@[measurability]
theorem measurable_from_nat {f : ℕ → α} : Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : ℕ → α}
Goal: Measurable f
-- measurable_to_nat
theorem measurable_to_nat {f : α → ℕ} : (∀ y, MeasurableSet (f ⁻¹' {f y})) → Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℕ}
Goal: (∀ (y : α), MeasurableSet (f ⁻¹' {f y})) → Measurable f
-- measurable_to_bool
theorem measurable_to_bool {f : α → Bool} (h : MeasurableSet (f ⁻¹' {true})) : Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → Bool} (h : MeasurableSet (f ⁻¹' {true}))
Goal: Measurable f
-- measurable_to_prop
theorem measurable_to_prop {f : α → Prop} (h : MeasurableSet (f ⁻¹' {True})) : Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → Prop} (h : MeasurableSet (f ⁻¹' {True}))
Goal: Measurable f
-- measurable_findGreatest'
theorem measurable_findGreatest' {p : α → ℕ → Prop} [∀ x, DecidablePred (p x)] {N : ℕ}
    (hN : ∀ k ≤ N, MeasurableSet { x | Nat.findGreatest (p x) N = k }) :
    Measurable fun x => Nat.findGreatest (p x) N 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : α → ℕ → Prop} [(x : α) → DecidablePred (p x)] {N : ℕ} (hN : ∀ k ≤ N, MeasurableSet {x | Nat.findGreatest (p x) N = k})
Goal: Measurable fun x => Nat.findGreatest (p x) N
-- measurable_findGreatest
theorem measurable_findGreatest {p : α → ℕ → Prop} [∀ x, DecidablePred (p x)] {N}
    (hN : ∀ k ≤ N, MeasurableSet { x | p x k }) : Measurable fun x => Nat.findGreatest (p x) N 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : α → ℕ → Prop} [(x : α) → DecidablePred (p x)] {N : ℕ} (hN : ∀ k ≤ N, MeasurableSet {x | p x k})
Goal: Measurable fun x => Nat.findGreatest (p x) N
-- measurable_find
theorem measurable_find {p : α → ℕ → Prop} [∀ x, DecidablePred (p x)] (hp : ∀ x, ∃ N, p x N)
    (hm : ∀ k, MeasurableSet { x | p x k }) : Measurable fun x => Nat.find (hp x) 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : α → ℕ → Prop} [(x : α) → DecidablePred (p x)] (hp : ∀ (x : α), ∃ N, p x N) (hm : ∀ (k : ℕ), MeasurableSet {x | p x k})
Goal: Measurable fun x => Nat.find ⋯
-- measurableSet_quotient
theorem measurableSet_quotient {s : Setoid α} {t : Set (Quotient s)} :
    MeasurableSet t ↔ MeasurableSet (Quotient.mk'' ⁻¹' t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Setoid α} {t : Set (Quotient s)}
Goal: MeasurableSet t ↔ MeasurableSet (Quotient.mk'' ⁻¹' t)
-- measurable_from_quotient
theorem measurable_from_quotient {s : Setoid α} {f : Quotient s → β} :
    Measurable f ↔ Measurable (f ∘ Quotient.mk'') 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {s : Setoid α} {f : Quotient s → β}
Goal: Measurable f ↔ Measurable (f ∘ Quotient.mk'')
-- measurable_quotient_mk'
@[measurability]
theorem measurable_quotient_mk' [s : Setoid α] : Measurable (Quotient.mk' : α → Quotient s) 
Arguments: {α : Type u_1} [MeasurableSpace α] [s : Setoid α]
Goal: Measurable Quotient.mk'
-- measurable_quotient_mk''
@[measurability]
theorem measurable_quotient_mk'' {s : Setoid α} : Measurable (Quotient.mk'' : α → Quotient s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Setoid α}
Goal: Measurable Quotient.mk''
-- measurable_quot_mk
@[measurability]
theorem measurable_quot_mk {r : α → α → Prop} : Measurable (Quot.mk r) 
Arguments: {α : Type u_1} [MeasurableSpace α] {r : α → α → Prop}
Goal: Measurable (Quot.mk r)
-- QuotientGroup.measurable_coe
@[to_additive (attr := measurability)]
theorem QuotientGroup.measurable_coe {G} [Group G] [MeasurableSpace G] {S : Subgroup G} :
    Measurable ((↑) : G → G ⧸ S) 
Arguments: {G : Type u_6} [Group G] [MeasurableSpace G] {S : Subgroup G}
Goal: Measurable QuotientGroup.mk
-- QuotientAddGroup.measurable_coe
@[to_additive (attr := measurability)]
theorem QuotientGroup.measurable_coe {G} [Group G] [MeasurableSpace G] {S : Subgroup G} :
    Measurable ((↑) : G → G ⧸ S) 
Arguments: {G : Type u_6} [AddGroup G] [MeasurableSpace G] {S : AddSubgroup G}
Goal: Measurable QuotientAddGroup.mk
-- QuotientGroup.measurable_from_quotient
@[to_additive]
nonrec theorem QuotientGroup.measurable_from_quotient {G} [Group G] [MeasurableSpace G]
    {S : Subgroup G} {f : G ⧸ S → α} : Measurable f ↔ Measurable (f ∘ ((↑) : G → G ⧸ S)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {G : Type u_6} [Group G] [MeasurableSpace G] {S : Subgroup G} {f : G ⧸ S → α}
Goal: Measurable f ↔ Measurable (f ∘ QuotientGroup.mk)
-- QuotientAddGroup.measurable_from_quotient
@[to_additive]
nonrec theorem QuotientGroup.measurable_from_quotient {G} [Group G] [MeasurableSpace G]
    {S : Subgroup G} {f : G ⧸ S → α} : Measurable f ↔ Measurable (f ∘ ((↑) : G → G ⧸ S)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {G : Type u_6} [AddGroup G] [MeasurableSpace G] {S : AddSubgroup G} {f : G ⧸ S → α}
Goal: Measurable f ↔ Measurable (f ∘ QuotientAddGroup.mk)
-- measurable_subtype_coe
@[measurability]
theorem measurable_subtype_coe {p : α → Prop} : Measurable ((↑) : Subtype p → α) 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : α → Prop}
Goal: Measurable Subtype.val
-- MeasurableSet.of_subtype_image
theorem MeasurableSet.of_subtype_image {s : Set α} {t : Set s}
    (h : MeasurableSet (Subtype.val '' t)) : MeasurableSet t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : Set α} {t : Set ↑s} (h : MeasurableSet (Subtype.val '' t))
Goal: MeasurableSet t
-- MeasurableSet.subtype_image
theorem MeasurableSet.subtype_image {s : Set α} {t : Set s} (hs : MeasurableSet s) :
    MeasurableSet t → MeasurableSet (((↑) : s → α) '' t) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : Set α} {t : Set ↑s} (hs : MeasurableSet s)
Goal: MeasurableSet t → MeasurableSet (Subtype.val '' t)
-- Measurable.subtype_coe
@[measurability]
theorem Measurable.subtype_coe {p : β → Prop} {f : α → Subtype p} (hf : Measurable f) :
    Measurable fun a : α => (f a : β) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {p : β → Prop} {f : α → Subtype p} (hf : Measurable f)
Goal: Measurable fun a => ↑(f a)
-- Measurable.subtype_mk
@[measurability]
theorem Measurable.subtype_mk {p : β → Prop} {f : α → β} (hf : Measurable f) {h : ∀ x, p (f x)} :
    Measurable fun x => (⟨f x, h x⟩ : Subtype p) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {p : β → Prop} {f : α → β} (hf : Measurable f) {h : ∀ (x : α), p (f x)}
Goal: Measurable fun x => ⟨f x, ⋯⟩
-- Measurable.rangeFactorization
@[measurability]
protected theorem Measurable.rangeFactorization {f : α → β} (hf : Measurable f) :
    Measurable (rangeFactorization f) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (hf : Measurable f)
Goal: Measurable (Set.rangeFactorization f)
-- Measurable.subtype_map
theorem Measurable.subtype_map {f : α → β} {p : α → Prop} {q : β → Prop} (hf : Measurable f)
    (hpq : ∀ x, p x → q (f x)) : Measurable (Subtype.map f hpq) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} {p : α → Prop} {q : β → Prop} (hf : Measurable f) (hpq : ∀ (x : α), p x → q (f x))
Goal: Measurable (Subtype.map f hpq)
-- measurable_inclusion
theorem measurable_inclusion {s t : Set α} (h : s ⊆ t) : Measurable (inclusion h) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : Set α} (h : s ⊆ t)
Goal: Measurable (Set.inclusion h)
-- MeasurableSet.image_inclusion'
theorem MeasurableSet.image_inclusion' {s t : Set α} (h : s ⊆ t) {u : Set s}
    (hs : MeasurableSet (Subtype.val ⁻¹' s : Set t)) (hu : MeasurableSet u) :
    MeasurableSet (inclusion h '' u) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : Set α} (h : s ⊆ t) {u : Set ↑s} (hs : MeasurableSet (Subtype.val ⁻¹' s)) (hu : MeasurableSet u)
Goal: MeasurableSet (Set.inclusion h '' u)
-- MeasurableSet.image_inclusion
theorem MeasurableSet.image_inclusion {s t : Set α} (h : s ⊆ t) {u : Set s}
    (hs : MeasurableSet s) (hu : MeasurableSet u) :
    MeasurableSet (inclusion h '' u) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : Set α} (h : s ⊆ t) {u : Set ↑s} (hs : MeasurableSet s) (hu : MeasurableSet u)
Goal: MeasurableSet (Set.inclusion h '' u)
-- MeasurableSet.of_union_cover
theorem MeasurableSet.of_union_cover {s t u : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t)
    (h : univ ⊆ s ∪ t) (hsu : MeasurableSet (((↑) : s → α) ⁻¹' u))
    (htu : MeasurableSet (((↑) : t → α) ⁻¹' u)) : MeasurableSet u 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t u : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (h : Set.univ ⊆ s ∪ t) (hsu : MeasurableSet (Subtype.val ⁻¹' u)) (htu : MeasurableSet (Subtype.val ⁻¹' u))
Goal: MeasurableSet u
-- measurable_of_measurable_union_cover
theorem measurable_of_measurable_union_cover {f : α → β} (s t : Set α) (hs : MeasurableSet s)
    (ht : MeasurableSet t) (h : univ ⊆ s ∪ t) (hc : Measurable fun a : s => f a)
    (hd : Measurable fun a : t => f a) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t) (h : Set.univ ⊆ s ∪ t) (hc : Measurable fun a => f ↑a) (hd : Measurable fun a => f ↑a)
Goal: Measurable f
-- measurable_of_restrict_of_restrict_compl
theorem measurable_of_restrict_of_restrict_compl {f : α → β} {s : Set α} (hs : MeasurableSet s)
    (h₁ : Measurable (s.restrict f)) (h₂ : Measurable (sᶜ.restrict f)) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} {s : Set α} (hs : MeasurableSet s) (h₁ : Measurable (s.restrict f)) (h₂ : Measurable (sᶜ.restrict f))
Goal: Measurable f