ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Measurable.dite
theorem Measurable.dite [∀ x, Decidable (x ∈ s)] {f : s → β} (hf : Measurable f)
    {g : (sᶜ : Set α) → β} (hg : Measurable g) (hs : MeasurableSet s) :
    Measurable fun x => if hx : x ∈ s then f ⟨x, hx⟩ else g ⟨x, hx⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [(x : α) → Decidable (x ∈ s)] {f : ↑s → β} (hf : Measurable f) {g : ↑sᶜ → β} (hg : Measurable g) (hs : MeasurableSet s)
Goal: Measurable fun x => if hx : x ∈ s then f ⟨x, hx⟩ else g ⟨x, hx⟩
-- measurable_of_measurable_on_compl_finite
theorem measurable_of_measurable_on_compl_finite [MeasurableSingletonClass α] {f : α → β}
    (s : Set α) (hs : s.Finite) (hf : Measurable (sᶜ.restrict f)) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSingletonClass α] {f : α → β} (s : Set α) (hs : s.Finite) (hf : Measurable (sᶜ.restrict f))
Goal: Measurable f
-- measurable_of_measurable_on_compl_singleton
theorem measurable_of_measurable_on_compl_singleton [MeasurableSingletonClass α] {f : α → β} (a : α)
    (hf : Measurable ({ x | x ≠ a }.restrict f)) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSingletonClass α] {f : α → β} (a : α) (hf : Measurable ({x | x ≠ a}.restrict f))
Goal: Measurable f
-- mem_measurableAtom_self
@[simp] lemma mem_measurableAtom_self (x : β) : x ∈ measurableAtom x 
Arguments: {β : Type u_2} [MeasurableSpace β] (x : β)
Goal: x ∈ measurableAtom x
-- mem_of_mem_measurableAtom
lemma mem_of_mem_measurableAtom {x y : β} (h : y ∈ measurableAtom x) {s : Set β}
    (hs : MeasurableSet s) (hxs : x ∈ s) : y ∈ s 
Arguments: {β : Type u_2} [MeasurableSpace β] {x y : β} (h : y ∈ measurableAtom x) {s : Set β} (hs : MeasurableSet s) (hxs : x ∈ s)
Goal: y ∈ s
-- measurableAtom_subset
lemma measurableAtom_subset {s : Set β} {x : β} (hs : MeasurableSet s) (hx : x ∈ s) :
    measurableAtom x ⊆ s 
Arguments: {β : Type u_2} [MeasurableSpace β] {s : Set β} {x : β} (hs : MeasurableSet s) (hx : x ∈ s)
Goal: measurableAtom x ⊆ s
-- measurableAtom_of_measurableSingletonClass
@[simp] lemma measurableAtom_of_measurableSingletonClass [MeasurableSingletonClass β] (x : β) :
    measurableAtom x = {x} 
Arguments: {β : Type u_2} [MeasurableSpace β] [MeasurableSingletonClass β] (x : β)
Goal: measurableAtom x = {x}
-- MeasurableSet.measurableAtom_of_countable
lemma MeasurableSet.measurableAtom_of_countable [Countable β] (x : β) :
    MeasurableSet (measurableAtom x) 
Arguments: {β : Type u_2} [MeasurableSpace β] [Countable β] (x : β)
Goal: MeasurableSet (measurableAtom x)
-- measurable_fst
@[measurability]
theorem measurable_fst {_ : MeasurableSpace α} {_ : MeasurableSpace β} :
    Measurable (Prod.fst : α × β → α) 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β}, Measurable Prod.fst
-- measurable_snd
@[measurability]
theorem measurable_snd {_ : MeasurableSpace α} {_ : MeasurableSpace β} :
    Measurable (Prod.snd : α × β → β) 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β}, Measurable Prod.snd
-- Measurable.fst
@[fun_prop]
theorem Measurable.fst {f : α → β × γ} (hf : Measurable f) : Measurable fun a : α => (f a).1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : α → β × γ} (hf : Measurable f)
Goal: Measurable fun a => (f a).1
-- Measurable.snd
@[fun_prop]
theorem Measurable.snd {f : α → β × γ} (hf : Measurable f) : Measurable fun a : α => (f a).2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : α → β × γ} (hf : Measurable f)
Goal: Measurable fun a => (f a).2
-- Measurable.prod
@[measurability]
theorem Measurable.prod {f : α → β × γ} (hf₁ : Measurable fun a => (f a).1)
    (hf₂ : Measurable fun a => (f a).2) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : α → β × γ} (hf₁ : Measurable fun a => (f a).1) (hf₂ : Measurable fun a => (f a).2)
Goal: Measurable f
-- Measurable.prod_mk
@[fun_prop]
theorem Measurable.prod_mk {β γ} {_ : MeasurableSpace β} {_ : MeasurableSpace γ} {f : α → β}
    {g : α → γ} (hf : Measurable f) (hg : Measurable g) : Measurable fun a : α => (f a, g a) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {β : Type u_6} {γ : Type u_7}
Goal: ∀ {x : MeasurableSpace β} {x_1 : MeasurableSpace γ} {f : α → β} {g : α → γ},
  Measurable f → Measurable g → Measurable fun a => (f a, g a)
-- Measurable.prod_map
theorem Measurable.prod_map [MeasurableSpace δ] {f : α → β} {g : γ → δ} (hf : Measurable f)
    (hg : Measurable g) : Measurable (Prod.map f g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace δ] {f : α → β} {g : γ → δ} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (Prod.map f g)
-- measurable_prod_mk_left
theorem measurable_prod_mk_left {x : α} : Measurable (@Prod.mk _ β x) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}
Goal: Measurable (Prod.mk x)
-- measurable_prod_mk_right
theorem measurable_prod_mk_right {y : β} : Measurable fun x : α => (x, y) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {y : β}
Goal: Measurable fun x => (x, y)
-- Measurable.of_uncurry_left
theorem Measurable.of_uncurry_left {f : α → β → γ} (hf : Measurable (uncurry f)) {x : α} :
    Measurable (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : α → β → γ} (hf : Measurable (Function.uncurry f)) {x : α}
Goal: Measurable (f x)
-- Measurable.of_uncurry_right
theorem Measurable.of_uncurry_right {f : α → β → γ} (hf : Measurable (uncurry f)) {y : β} :
    Measurable fun x => f x y 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : α → β → γ} (hf : Measurable (Function.uncurry f)) {y : β}
Goal: Measurable fun x => f x y
-- measurable_prod
theorem measurable_prod {f : α → β × γ} :
    Measurable f ↔ (Measurable fun a => (f a).1) ∧ Measurable fun a => (f a).2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : α → β × γ}
Goal: Measurable f ↔ (Measurable fun a => (f a).1) ∧ Measurable fun a => (f a).2
-- measurable_swap
@[fun_prop, measurability]
theorem measurable_swap : Measurable (Prod.swap : α × β → β × α) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: Measurable Prod.swap
-- measurable_swap_iff
theorem measurable_swap_iff {_ : MeasurableSpace γ} {f : α × β → γ} :
    Measurable (f ∘ Prod.swap) ↔ Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: ∀ {x : MeasurableSpace γ} {f : α × β → γ}, Measurable (f ∘ Prod.swap) ↔ Measurable f
-- MeasurableSet.prod
@[measurability]
protected theorem MeasurableSet.prod {s : Set α} {t : Set β} (hs : MeasurableSet s)
    (ht : MeasurableSet t) : MeasurableSet (s ×ˢ t) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set α} {t : Set β} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: MeasurableSet (s ×ˢ t)
-- measurableSet_prod_of_nonempty
theorem measurableSet_prod_of_nonempty {s : Set α} {t : Set β} (h : (s ×ˢ t).Nonempty) :
    MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set α} {t : Set β} (h : (s ×ˢ t).Nonempty)
Goal: MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t
-- measurableSet_prod
theorem measurableSet_prod {s : Set α} {t : Set β} :
    MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set α} {t : Set β}
Goal: MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅
-- measurableSet_swap_iff
theorem measurableSet_swap_iff {s : Set (α × β)} :
    MeasurableSet (Prod.swap ⁻¹' s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set (α × β)}
Goal: MeasurableSet (Prod.swap ⁻¹' s) ↔ MeasurableSet s
-- measurable_from_prod_countable'
theorem measurable_from_prod_countable' [Countable β]
    {_ : MeasurableSpace γ} {f : α × β → γ} (hf : ∀ y, Measurable fun x => f (x, y))
    (h'f : ∀ y y' x, y' ∈ measurableAtom y → f (x, y') = f (x, y)) :
    Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable β]
Goal: ∀ {x : MeasurableSpace γ} {f : α × β → γ},
  (∀ (y : β), Measurable fun x => f (x, y)) →
    (∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)) → Measurable f
-- measurable_from_prod_countable
theorem measurable_from_prod_countable [Countable β] [MeasurableSingletonClass β]
    {_ : MeasurableSpace γ} {f : α × β → γ} (hf : ∀ y, Measurable fun x => f (x, y)) :
    Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable β] [MeasurableSingletonClass β]
Goal: ∀ {x : MeasurableSpace γ} {f : α × β → γ}, (∀ (y : β), Measurable fun x => f (x, y)) → Measurable f
-- Measurable.find
@[measurability]
theorem Measurable.find {_ : MeasurableSpace α} {f : ℕ → α → β} {p : ℕ → α → Prop}
    [∀ n, DecidablePred (p n)] (hf : ∀ n, Measurable (f n)) (hp : ∀ n, MeasurableSet { x | p n x })
    (h : ∀ x, ∃ n, p n x) : Measurable fun x => f (Nat.find (h x)) x 
Arguments: {α : Type u_1} {β : Type u_2} {mβ : MeasurableSpace β}
Goal: ∀ {x : MeasurableSpace α} {f : ℕ → α → β} {p : ℕ → α → Prop} [inst : (n : ℕ) → DecidablePred (p n)],
  (∀ (n : ℕ), Measurable (f n)) →
    (∀ (n : ℕ), MeasurableSet {x | p n x}) → ∀ (h : ∀ (x : α), ∃ n, p n x), Measurable fun x => f (Nat.find ⋯) x
-- measurable_iUnionLift
theorem measurable_iUnionLift [Countable ι] {t : ι → Set α} {f : ∀ i, t i → β}
    (htf : ∀ (i j) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩)
    {T : Set α} (hT : T ⊆ ⋃ i, t i) (htm : ∀ i, MeasurableSet (t i)) (hfm : ∀ i, Measurable (f i)) :
    Measurable (iUnionLift t f htf T hT) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort uι} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] {t : ι → Set α} {f : (i : ι) → ↑(t i) → β} (htf : ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩) {T : Set α} (hT : T ⊆ ⋃ i, t i) (htm : ∀ (i : ι), MeasurableSet (t i)) (hfm : ∀ (i : ι), Measurable (f i))
Goal: Measurable (Set.iUnionLift t f htf T hT)
-- measurable_liftCover
theorem measurable_liftCover [Countable ι] (t : ι → Set α) (htm : ∀ i, MeasurableSet (t i))
    (f : ∀ i, t i → β) (hfm : ∀ i, Measurable (f i))
    (hf : ∀ (i j) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩)
    (htU : ⋃ i, t i = univ) :
    Measurable (liftCover t f hf htU) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort uι} {m : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (t : ι → Set α) (htm : ∀ (i : ι), MeasurableSet (t i)) (f : (i : ι) → ↑(t i) → β) (hfm : ∀ (i : ι), Measurable (f i)) (hf : ∀ (i j : ι) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩) (htU : ⋃ i, t i = Set.univ)
Goal: Measurable (Set.liftCover t f hf htU)
-- exists_measurable_piecewise
theorem exists_measurable_piecewise {ι} [Countable ι] [Nonempty ι] (t : ι → Set α)
    (t_meas : ∀ n, MeasurableSet (t n)) (g : ι → α → β) (hg : ∀ n, Measurable (g n))
    (ht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)) :
    ∃ f : α → β, Measurable f ∧ ∀ n, EqOn f (g n) (t n) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {ι : Type u_6} [Countable ι] [Nonempty ι] (t : ι → Set α) (t_meas : ∀ (n : ι), MeasurableSet (t n)) (g : ι → α → β) (hg : ∀ (n : ι), Measurable (g n)) (ht : Pairwise fun i j => Set.EqOn (g i) (g j) (t i ∩ t j))
Goal: ∃ f, Measurable f ∧ ∀ (n : ι), Set.EqOn f (g n) (t n)
-- measurable_pi_iff
theorem measurable_pi_iff {g : α → ∀ a, π a} : Measurable g ↔ ∀ a, Measurable fun x => g x a 
Arguments: {α : Type u_1} {δ : Type u_4} {π : δ → Type u_6} [MeasurableSpace α] [(a : δ) → MeasurableSpace (π a)] {g : α → (a : δ) → π a}
Goal: Measurable g ↔ ∀ (a : δ), Measurable fun x => g x a
-- measurable_pi_apply
@[fun_prop, aesop safe 100 apply (rule_sets := [Measurable])]
theorem measurable_pi_apply (a : δ) : Measurable fun f : ∀ a, π a => f a 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] (a : δ)
Goal: Measurable fun f => f a
-- Measurable.eval
@[aesop safe 100 apply (rule_sets := [Measurable])]
theorem Measurable.eval {a : δ} {g : α → ∀ a, π a} (hg : Measurable g) :
    Measurable fun x => g x a 
Arguments: {α : Type u_1} {δ : Type u_4} {π : δ → Type u_6} [MeasurableSpace α] [(a : δ) → MeasurableSpace (π a)] {a : δ} {g : α → (a : δ) → π a} (hg : Measurable g)
Goal: Measurable fun x => g x a
-- measurable_pi_lambda
@[fun_prop, aesop safe 100 apply (rule_sets := [Measurable])]
theorem measurable_pi_lambda (f : α → ∀ a, π a) (hf : ∀ a, Measurable fun c => f c a) :
    Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} {π : δ → Type u_6} [MeasurableSpace α] [(a : δ) → MeasurableSpace (π a)] (f : α → (a : δ) → π a) (hf : ∀ (a : δ), Measurable fun c => f c a)
Goal: Measurable f
-- measurable_update'
theorem measurable_update'  {a : δ} [DecidableEq δ] :
    Measurable (fun p : (∀ i, π i) × π a ↦ update p.1 a p.2) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] {a : δ} [DecidableEq δ]
Goal: Measurable fun p => Function.update p.1 a p.2
-- measurable_uniqueElim
theorem measurable_uniqueElim [Unique δ] [∀ i, MeasurableSpace (π i)] :
    Measurable (uniqueElim : π (default : δ) → ∀ i, π i) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [Unique δ] [(i : δ) → MeasurableSpace (π i)]
Goal: Measurable uniqueElim
-- measurable_updateFinset
theorem measurable_updateFinset [DecidableEq δ] {s : Finset δ} {x : ∀ i, π i} :
    Measurable (updateFinset x s) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] [DecidableEq δ] {s : Finset δ} {x : (i : δ) → π i}
Goal: Measurable (Function.updateFinset x s)
-- measurable_update
@[measurability]
theorem measurable_update (f : ∀ a : δ, π a) {a : δ} [DecidableEq δ] : Measurable (update f a) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] (f : (a : δ) → π a) {a : δ} [DecidableEq δ]
Goal: Measurable (Function.update f a)
-- measurable_update_left
theorem measurable_update_left {a : δ} [DecidableEq δ] {x : π a} :
    Measurable (update · a x) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] {a : δ} [DecidableEq δ] {x : π a}
Goal: Measurable fun x_1 => Function.update x_1 a x
-- measurable_eq_mp
theorem measurable_eq_mp {i i' : δ} (h : i = i') : Measurable (congr_arg π h).mp 
Arguments: {δ : Type u_4} (π : δ → Type u_6) [(a : δ) → MeasurableSpace (π a)] {i i' : δ} (h : i = i')
Goal: Measurable ⋯.mp
-- Measurable.eq_mp
theorem Measurable.eq_mp {β} [MeasurableSpace β] {i i' : δ} (h : i = i') {f : β → π i}
    (hf : Measurable f) : Measurable fun x => (congr_arg π h).mp (f x) 
Arguments: {δ : Type u_4} (π : δ → Type u_6) [(a : δ) → MeasurableSpace (π a)] {β : Type u_7} [MeasurableSpace β] {i i' : δ} (h : i = i') {f : β → π i} (hf : Measurable f)
Goal: Measurable fun x => ⋯.mp (f x)
-- measurable_piCongrLeft
theorem measurable_piCongrLeft (f : δ' ≃ δ) : Measurable (piCongrLeft π f) 
Arguments: {δ : Type u_4} {δ' : Type u_5} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] (f : δ' ≃ δ)
Goal: Measurable ⇑(Equiv.piCongrLeft π f)
-- MeasurableSet.pi
@[measurability]
protected theorem MeasurableSet.pi {s : Set δ} {t : ∀ i : δ, Set (π i)} (hs : s.Countable)
    (ht : ∀ i ∈ s, MeasurableSet (t i)) : MeasurableSet (s.pi t) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] {s : Set δ} {t : (i : δ) → Set (π i)} (hs : s.Countable) (ht : ∀ i ∈ s, MeasurableSet (t i))
Goal: MeasurableSet (s.pi t)
-- MeasurableSet.univ_pi
protected theorem MeasurableSet.univ_pi [Countable δ] {t : ∀ i : δ, Set (π i)}
    (ht : ∀ i, MeasurableSet (t i)) : MeasurableSet (pi univ t) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] [Countable δ] {t : (i : δ) → Set (π i)} (ht : ∀ (i : δ), MeasurableSet (t i))
Goal: MeasurableSet (Set.univ.pi t)
-- measurableSet_pi_of_nonempty
theorem measurableSet_pi_of_nonempty {s : Set δ} {t : ∀ i, Set (π i)} (hs : s.Countable)
    (h : (pi s t).Nonempty) : MeasurableSet (pi s t) ↔ ∀ i ∈ s, MeasurableSet (t i) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] {s : Set δ} {t : (i : δ) → Set (π i)} (hs : s.Countable) (h : (s.pi t).Nonempty)
Goal: MeasurableSet (s.pi t) ↔ ∀ i ∈ s, MeasurableSet (t i)
-- measurableSet_pi
theorem measurableSet_pi {s : Set δ} {t : ∀ i, Set (π i)} (hs : s.Countable) :
    MeasurableSet (pi s t) ↔ (∀ i ∈ s, MeasurableSet (t i)) ∨ pi s t = ∅ 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(a : δ) → MeasurableSpace (π a)] {s : Set δ} {t : (i : δ) → Set (π i)} (hs : s.Countable)
Goal: MeasurableSet (s.pi t) ↔ (∀ i ∈ s, MeasurableSet (t i)) ∨ s.pi t = ∅
-- measurable_piEquivPiSubtypeProd_symm
@[measurability]
theorem measurable_piEquivPiSubtypeProd_symm (p : δ → Prop) [DecidablePred p] :
    Measurable (Equiv.piEquivPiSubtypeProd p π).symm 
Arguments: {δ : Type u_4} (π : δ → Type u_6) [(a : δ) → MeasurableSpace (π a)] (p : δ → Prop) [DecidablePred p]
Goal: Measurable ⇑(Equiv.piEquivPiSubtypeProd p π).symm
-- measurable_piEquivPiSubtypeProd
@[measurability]
theorem measurable_piEquivPiSubtypeProd (p : δ → Prop) [DecidablePred p] :
    Measurable (Equiv.piEquivPiSubtypeProd p π) 
Arguments: {δ : Type u_4} (π : δ → Type u_6) [(a : δ) → MeasurableSpace (π a)] (p : δ → Prop) [DecidablePred p]
Goal: Measurable ⇑(Equiv.piEquivPiSubtypeProd p π)
-- measurable_tProd_mk
theorem measurable_tProd_mk (l : List δ) : Measurable (@TProd.mk δ π l) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(x : δ) → MeasurableSpace (π x)] (l : List δ)
Goal: Measurable (List.TProd.mk l)
-- measurable_tProd_elim
theorem measurable_tProd_elim [DecidableEq δ] :
    ∀ {l : List δ} {i : δ} (hi : i ∈ l), Measurable fun v : TProd π l => v.elim hi
  
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(x : δ) → MeasurableSpace (π x)] [DecidableEq δ] {l : List δ} {i : δ} (hi : i ∈ l)
Goal: Measurable fun v => v.elim hi
-- measurable_tProd_elim'
theorem measurable_tProd_elim' [DecidableEq δ] {l : List δ} (h : ∀ i, i ∈ l) :
    Measurable (TProd.elim' h : TProd π l → ∀ i, π i) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(x : δ) → MeasurableSpace (π x)] [DecidableEq δ] {l : List δ} (h : ∀ (i : δ), i ∈ l)
Goal: Measurable (List.TProd.elim' h)
-- MeasurableSet.tProd
theorem MeasurableSet.tProd (l : List δ) {s : ∀ i, Set (π i)} (hs : ∀ i, MeasurableSet (s i)) :
    MeasurableSet (Set.tprod l s) 
Arguments: {δ : Type u_4} {π : δ → Type u_6} [(x : δ) → MeasurableSpace (π x)] (l : List δ) {s : (i : δ) → Set (π i)} (hs : ∀ (i : δ), MeasurableSet (s i))
Goal: MeasurableSet (Set.tprod l s)
-- measurable_inl
@[measurability]
theorem measurable_inl [MeasurableSpace α] [MeasurableSpace β] : Measurable (@Sum.inl α β) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β]
Goal: Measurable Sum.inl
-- measurable_inr
@[measurability]
theorem measurable_inr [MeasurableSpace α] [MeasurableSpace β] : Measurable (@Sum.inr α β) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β]
Goal: Measurable Sum.inr
-- measurableSet_sum_iff
theorem measurableSet_sum_iff {s : Set (α ⊕ β)} :
    MeasurableSet s ↔ MeasurableSet (Sum.inl ⁻¹' s) ∧ MeasurableSet (Sum.inr ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set (α ⊕ β)}
Goal: MeasurableSet s ↔ MeasurableSet (Sum.inl ⁻¹' s) ∧ MeasurableSet (Sum.inr ⁻¹' s)
-- measurable_sum
theorem measurable_sum {_ : MeasurableSpace γ} {f : α ⊕ β → γ} (hl : Measurable (f ∘ Sum.inl))
    (hr : Measurable (f ∘ Sum.inr)) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: ∀ {x : MeasurableSpace γ} {f : α ⊕ β → γ}, Measurable (f ∘ Sum.inl) → Measurable (f ∘ Sum.inr) → Measurable f
-- Measurable.sumElim
@[measurability]
theorem Measurable.sumElim {_ : MeasurableSpace γ} {f : α → γ} {g : β → γ} (hf : Measurable f)
    (hg : Measurable g) : Measurable (Sum.elim f g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: ∀ {x : MeasurableSpace γ} {f : α → γ} {g : β → γ}, Measurable f → Measurable g → Measurable (Sum.elim f g)
-- Measurable.sumMap
theorem Measurable.sumMap {_ : MeasurableSpace γ} {_ : MeasurableSpace δ} {f : α → β} {g : γ → δ}
    (hf : Measurable f) (hg : Measurable g) : Measurable (Sum.map f g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {m : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace δ} {f : α → β} {g : γ → δ},
  Measurable f → Measurable g → Measurable (Sum.map f g)
-- measurableSet_inl_image
@[simp] theorem measurableSet_inl_image {s : Set α} :
    MeasurableSet (Sum.inl '' s : Set (α ⊕ β)) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set α}
Goal: MeasurableSet (Sum.inl '' s) ↔ MeasurableSet s
-- measurableSet_inr_image
@[simp] theorem measurableSet_inr_image {s : Set β} :
    MeasurableSet (Sum.inr '' s : Set (α ⊕ β)) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {s : Set β}
Goal: MeasurableSet (Sum.inr '' s) ↔ MeasurableSet s
-- measurableSet_range_inl
theorem measurableSet_range_inl [MeasurableSpace α] :
    MeasurableSet (range Sum.inl : Set (α ⊕ β)) 
Arguments: {α : Type u_1} {β : Type u_2} {mβ : MeasurableSpace β} [MeasurableSpace α]
Goal: MeasurableSet (Set.range Sum.inl)
-- measurableSet_range_inr
theorem measurableSet_range_inr [MeasurableSpace α] :
    MeasurableSet (range Sum.inr : Set (α ⊕ β)) 
Arguments: {α : Type u_1} {β : Type u_2} {mβ : MeasurableSpace β} [MeasurableSpace α]
Goal: MeasurableSet (Set.range Sum.inr)
-- measurableSet_setOf
@[simp] theorem measurableSet_setOf : MeasurableSet {a | p a} ↔ Measurable p 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : α → Prop}
Goal: MeasurableSet {a | p a} ↔ Measurable p
-- measurable_mem
@[simp] theorem measurable_mem : Measurable (· ∈ s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [MeasurableSpace α]
Goal: (Measurable fun x => x ∈ s) ↔ MeasurableSet s
-- Measurable.not
lemma Measurable.not (hp : Measurable p) : Measurable (¬ p ·) 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : α → Prop} (hp : Measurable p)
Goal: Measurable fun x => ¬p x
-- Measurable.and
lemma Measurable.and (hp : Measurable p) (hq : Measurable q) : Measurable fun a ↦ p a ∧ q a 
Arguments: {α : Type u_1} [MeasurableSpace α] {p q : α → Prop} (hp : Measurable p) (hq : Measurable q)
Goal: Measurable fun a => p a ∧ q a
-- Measurable.or
lemma Measurable.or (hp : Measurable p) (hq : Measurable q) : Measurable fun a ↦ p a ∨ q a 
Arguments: {α : Type u_1} [MeasurableSpace α] {p q : α → Prop} (hp : Measurable p) (hq : Measurable q)
Goal: Measurable fun a => p a ∨ q a
-- Measurable.imp
lemma Measurable.imp (hp : Measurable p) (hq : Measurable q) : Measurable fun a ↦ p a → q a 
Arguments: {α : Type u_1} [MeasurableSpace α] {p q : α → Prop} (hp : Measurable p) (hq : Measurable q)
Goal: Measurable fun a => p a → q a
-- Measurable.iff
lemma Measurable.iff (hp : Measurable p) (hq : Measurable q) : Measurable fun a ↦ p a ↔ q a 
Arguments: {α : Type u_1} [MeasurableSpace α] {p q : α → Prop} (hp : Measurable p) (hq : Measurable q)
Goal: Measurable fun a => p a ↔ q a