ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.StronglyMeasurable.aestronglyMeasurable
@[aesop 30% apply (rule_sets := [Measurable])]
protected theorem StronglyMeasurable.aestronglyMeasurable {α β} {_ : MeasurableSpace α}
    [TopologicalSpace β] {f : α → β} {μ : Measure α} (hf : StronglyMeasurable f) :
    AEStronglyMeasurable f μ 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {f : α → β} {μ : MeasureTheory.Measure α},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.Subsingleton.stronglyMeasurable
@[simp]
theorem Subsingleton.stronglyMeasurable {α β} [MeasurableSpace α] [TopologicalSpace β]
    [Subsingleton β] (f : α → β) : StronglyMeasurable f 
Arguments: {α : Type u_5} {β : Type u_6} [MeasurableSpace α] [TopologicalSpace β] [Subsingleton β] (f : α → β)
Goal: MeasureTheory.StronglyMeasurable f
-- MeasureTheory.SimpleFunc.stronglyMeasurable
theorem SimpleFunc.stronglyMeasurable {α β} {_ : MeasurableSpace α} [TopologicalSpace β]
    (f : α →ₛ β) : StronglyMeasurable f 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] (f : MeasureTheory.SimpleFunc α β),
  MeasureTheory.StronglyMeasurable ↑f
-- MeasureTheory.StronglyMeasurable.of_finite
@[nontriviality]
theorem StronglyMeasurable.of_finite [Finite α] {_ : MeasurableSpace α}
    [MeasurableSingletonClass α] [TopologicalSpace β]
    (f : α → β) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [Finite α]
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSingletonClass α] [inst : TopologicalSpace β] (f : α → β),
  MeasureTheory.StronglyMeasurable f
-- MeasureTheory.stronglyMeasurable_of_isEmpty
@[deprecated StronglyMeasurable.of_finite (since := "2024-02-06")]
theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} [TopologicalSpace β]
    (f : α → β) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [IsEmpty α]
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] (f : α → β), MeasureTheory.StronglyMeasurable f
-- MeasureTheory.stronglyMeasurable_const
theorem stronglyMeasurable_const {α β} {_ : MeasurableSpace α} [TopologicalSpace β] {b : β} :
    StronglyMeasurable fun _ : α => b 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {b : β}, MeasureTheory.StronglyMeasurable fun x => b
-- MeasureTheory.stronglyMeasurable_zero
@[to_additive]
theorem stronglyMeasurable_one {α β} {_ : MeasurableSpace α} [TopologicalSpace β] [One β] :
    StronglyMeasurable (1 : α → β) 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Zero β], MeasureTheory.StronglyMeasurable 0
-- MeasureTheory.stronglyMeasurable_one
@[to_additive]
theorem stronglyMeasurable_one {α β} {_ : MeasurableSpace α} [TopologicalSpace β] [One β] :
    StronglyMeasurable (1 : α → β) 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : One β], MeasureTheory.StronglyMeasurable 1
-- MeasureTheory.stronglyMeasurable_const'
theorem stronglyMeasurable_const' {α β} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
    (hf : ∀ x y, f x = f y) : StronglyMeasurable f 
Arguments: {α : Type u_5} {β : Type u_6} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β} (hf : ∀ (x y : α), f x = f y)
Goal: MeasureTheory.StronglyMeasurable f
-- MeasureTheory.Subsingleton.stronglyMeasurable'
@[simp]
theorem Subsingleton.stronglyMeasurable' {α β} [MeasurableSpace α] [TopologicalSpace β]
    [Subsingleton α] (f : α → β) : StronglyMeasurable f 
Arguments: {α : Type u_5} {β : Type u_6} [MeasurableSpace α] [TopologicalSpace β] [Subsingleton α] (f : α → β)
Goal: MeasureTheory.StronglyMeasurable f
-- MeasureTheory.StronglyMeasurable.tendsto_approx
protected theorem tendsto_approx {_ : MeasurableSpace α} (hf : StronglyMeasurable f) :
    ∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [TopologicalSpace β]
Goal: ∀ {x : MeasurableSpace α} (hf : MeasureTheory.StronglyMeasurable f) (x_1 : α),
  Filter.Tendsto (fun n => ↑(hf.approx n) x_1) Filter.atTop (𝓝 (f x_1))
-- MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le
theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
    {m : MeasurableSpace α} (hf : StronglyMeasurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
    Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) 
Arguments: {α : Type u_1} {β : Type u_5} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β] {m : MeasurableSpace α} (hf : MeasureTheory.StronglyMeasurable f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c)
Goal: Filter.Tendsto (fun n => ↑(hf.approxBounded c n) x) Filter.atTop (𝓝 (f x))
-- MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
    {m m0 : MeasurableSpace α} {μ : Measure α} (hf : StronglyMeasurable[m] f) {c : ℝ}
    (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
    ∀ᵐ x ∂μ, Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) 
Arguments: {α : Type u_1} {β : Type u_5} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.StronglyMeasurable f) {c : ℝ} (hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => ↑(hf.approxBounded c n) x) Filter.atTop (𝓝 (f x))
-- MeasureTheory.StronglyMeasurable.norm_approxBounded_le
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
    {m : MeasurableSpace α} {c : ℝ} (hf : StronglyMeasurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
    ‖hf.approxBounded c n x‖ ≤ c 
Arguments: {α : Type u_1} {β : Type u_5} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β] {m : MeasurableSpace α} {c : ℝ} (hf : MeasureTheory.StronglyMeasurable f) (hc : 0 ≤ c) (n : ℕ) (x : α)
Goal: ‖↑(hf.approxBounded c n) x‖ ≤ c
-- stronglyMeasurable_bot_iff
theorem _root_.stronglyMeasurable_bot_iff [Nonempty β] [T2Space β] :
    StronglyMeasurable[⊥] f ↔ ∃ c, f = fun _ => c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [TopologicalSpace β] [Nonempty β] [T2Space β]
Goal: MeasureTheory.StronglyMeasurable f ↔ ∃ c, f = fun x => c
-- MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite
theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
    {m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
    (ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) :
    FinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [TopologicalSpace β] [Zero β] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hf_meas : MeasureTheory.StronglyMeasurable f) {t : Set α} (ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : MeasureTheory.SigmaFinite (μ.restrict t))
Goal: MeasureTheory.FinStronglyMeasurable f μ
-- MeasureTheory.StronglyMeasurable.finStronglyMeasurable
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem finStronglyMeasurable [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α}
    (hf : StronglyMeasurable f) (μ : Measure α) [SigmaFinite μ] : FinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [TopologicalSpace β] [Zero β] {m0 : MeasurableSpace α} (hf : MeasureTheory.StronglyMeasurable f) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: MeasureTheory.FinStronglyMeasurable f μ
-- MeasureTheory.StronglyMeasurable.measurable
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem measurable {_ : MeasurableSpace α} [TopologicalSpace β] [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] (hf : StronglyMeasurable f) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_2 : MeasurableSpace β] [inst_3 : BorelSpace β], MeasureTheory.StronglyMeasurable f → Measurable f
-- MeasureTheory.StronglyMeasurable.aemeasurable
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
    [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] {μ : Measure α}
    (hf : StronglyMeasurable f) : AEMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_2 : MeasurableSpace β] [inst_3 : BorelSpace β] {μ : MeasureTheory.Measure α},
  MeasureTheory.StronglyMeasurable f → AEMeasurable f μ
-- Continuous.comp_stronglyMeasurable
theorem _root_.Continuous.comp_stronglyMeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
    [TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : StronglyMeasurable f) :
    StronglyMeasurable fun x => g (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace γ] {g : β → γ} {f : α → β},
  Continuous g → MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => g (f x)
-- MeasureTheory.StronglyMeasurable.measurableSet_support
@[to_additive]
nonrec theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
    [MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} [Zero β] [TopologicalSpace β] [TopologicalSpace.MetrizableSpace β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasurableSet (Function.support f)
-- MeasureTheory.StronglyMeasurable.measurableSet_mulSupport
@[to_additive]
nonrec theorem measurableSet_mulSupport {m : MeasurableSpace α} [One β] [TopologicalSpace β]
    [MetrizableSpace β] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} [One β] [TopologicalSpace β] [TopologicalSpace.MetrizableSpace β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasurableSet (Function.mulSupport f)
-- MeasureTheory.StronglyMeasurable.mono
protected theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
    (hf : StronglyMeasurable[m'] f) (h_mono : m' ≤ m) : StronglyMeasurable[m] f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m m' : MeasurableSpace α} [TopologicalSpace β] (hf : MeasureTheory.StronglyMeasurable f) (h_mono : m' ≤ m)
Goal: MeasureTheory.StronglyMeasurable f
-- MeasureTheory.StronglyMeasurable.prod_mk
protected theorem prod_mk {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace γ]
    {f : α → β} {g : α → γ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => (f x, g x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : α → γ} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable fun x => (f x, g x)
-- MeasureTheory.StronglyMeasurable.comp_measurable
theorem comp_measurable [TopologicalSpace β] {_ : MeasurableSpace α} {_ : MeasurableSpace γ}
    {f : α → β} {g : γ → α} (hf : StronglyMeasurable f) (hg : Measurable g) :
    StronglyMeasurable (f ∘ g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β]
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → β} {g : γ → α},
  MeasureTheory.StronglyMeasurable f → Measurable g → MeasureTheory.StronglyMeasurable (f ∘ g)
-- MeasureTheory.StronglyMeasurable.of_uncurry_left
theorem of_uncurry_left [TopologicalSpace β] {_ : MeasurableSpace α} {_ : MeasurableSpace γ}
    {f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {x : α} : StronglyMeasurable (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β]
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → γ → β},
  MeasureTheory.StronglyMeasurable (Function.uncurry f) → ∀ {x : α}, MeasureTheory.StronglyMeasurable (f x)
-- MeasureTheory.StronglyMeasurable.of_uncurry_right
theorem of_uncurry_right [TopologicalSpace β] {_ : MeasurableSpace α} {_ : MeasurableSpace γ}
    {f : α → γ → β} (hf : StronglyMeasurable (uncurry f)) {y : γ} :
    StronglyMeasurable fun x => f x y 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β]
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → γ → β},
  MeasureTheory.StronglyMeasurable (Function.uncurry f) → ∀ {y : γ}, MeasureTheory.StronglyMeasurable fun x => f x y
-- MeasureTheory.StronglyMeasurable.add
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f * g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Add β] [ContinuousAdd β] (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable (f + g)
-- MeasureTheory.StronglyMeasurable.mul
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f * g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Mul β] [ContinuousMul β] (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable (f * g)
-- MeasureTheory.StronglyMeasurable.add_const
@[to_additive (attr := measurability)]
theorem mul_const [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => f x * c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Add β] [ContinuousAdd β] (hf : MeasureTheory.StronglyMeasurable f) (c : β)
Goal: MeasureTheory.StronglyMeasurable fun x => f x + c
-- MeasureTheory.StronglyMeasurable.mul_const
@[to_additive (attr := measurability)]
theorem mul_const [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => f x * c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Mul β] [ContinuousMul β] (hf : MeasureTheory.StronglyMeasurable f) (c : β)
Goal: MeasureTheory.StronglyMeasurable fun x => f x * c
-- MeasureTheory.StronglyMeasurable.const_mul
@[to_additive (attr := measurability)]
theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => c * f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Mul β] [ContinuousMul β] (hf : MeasureTheory.StronglyMeasurable f) (c : β)
Goal: MeasureTheory.StronglyMeasurable fun x => c * f x
-- MeasureTheory.StronglyMeasurable.const_add
@[to_additive (attr := measurability)]
theorem const_mul [Mul β] [ContinuousMul β] (hf : StronglyMeasurable f) (c : β) :
    StronglyMeasurable fun x => c * f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Add β] [ContinuousAdd β] (hf : MeasureTheory.StronglyMeasurable f) (c : β)
Goal: MeasureTheory.StronglyMeasurable fun x => c + f x
-- MeasureTheory.StronglyMeasurable.pow
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : StronglyMeasurable f) (n : ℕ) :
    StronglyMeasurable (f ^ n) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Monoid β] [ContinuousMul β] (hf : MeasureTheory.StronglyMeasurable f) (n : ℕ)
Goal: MeasureTheory.StronglyMeasurable (f ^ n)
-- MeasureTheory.StronglyMeasurable.const_nsmul
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : StronglyMeasurable f) (n : ℕ) :
    StronglyMeasurable (f ^ n) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [AddMonoid β] [ContinuousAdd β] (hf : MeasureTheory.StronglyMeasurable f) (n : ℕ)
Goal: MeasureTheory.StronglyMeasurable (n • f)
-- MeasureTheory.StronglyMeasurable.inv
@[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
    StronglyMeasurable f⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Inv β] [ContinuousInv β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable f⁻¹
-- MeasureTheory.StronglyMeasurable.neg
@[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : StronglyMeasurable f) :
    StronglyMeasurable f⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Neg β] [ContinuousNeg β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable (-f)
-- MeasureTheory.StronglyMeasurable.div
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f / g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Div β] [ContinuousDiv β] (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable (f / g)
-- MeasureTheory.StronglyMeasurable.sub
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Div β] [ContinuousDiv β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f / g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [Sub β] [ContinuousSub β] (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable (f - g)
-- MeasureTheory.StronglyMeasurable.mul_iff_right
@[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (f * g) ↔ StronglyMeasurable g 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [CommGroup β] [TopologicalGroup β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable (f * g) ↔ MeasureTheory.StronglyMeasurable g
-- MeasureTheory.StronglyMeasurable.add_iff_right
@[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (f * g) ↔ StronglyMeasurable g 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [AddCommGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable (f + g) ↔ MeasureTheory.StronglyMeasurable g
-- MeasureTheory.StronglyMeasurable.mul_iff_left
@[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (g * f) ↔ StronglyMeasurable g 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [CommGroup β] [TopologicalGroup β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable (g * f) ↔ MeasureTheory.StronglyMeasurable g
-- MeasureTheory.StronglyMeasurable.add_iff_left
@[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : StronglyMeasurable f) :
    StronglyMeasurable (g * f) ↔ StronglyMeasurable g 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] [AddCommGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable (g + f) ↔ MeasureTheory.StronglyMeasurable g
-- MeasureTheory.StronglyMeasurable.vadd
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => f x • g x 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [VAdd 𝕜 β] [ContinuousVAdd 𝕜 β] {f : α → 𝕜} {g : α → β} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable fun x => f x +ᵥ g x
-- MeasureTheory.StronglyMeasurable.smul
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => f x • g x 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜} {g : α → β} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable fun x => f x • g x
-- MeasureTheory.StronglyMeasurable.const_smul
@[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable (c • f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : MeasureTheory.StronglyMeasurable f) (c : 𝕜)
Goal: MeasureTheory.StronglyMeasurable (c • f)
-- MeasureTheory.StronglyMeasurable.const_vadd
@[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable (c • f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [VAdd 𝕜 β] [ContinuousConstVAdd 𝕜 β] (hf : MeasureTheory.StronglyMeasurable f) (c : 𝕜)
Goal: MeasureTheory.StronglyMeasurable (c +ᵥ f)
-- MeasureTheory.StronglyMeasurable.const_smul'
@[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable fun x => c • f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : MeasureTheory.StronglyMeasurable f) (c : 𝕜)
Goal: MeasureTheory.StronglyMeasurable fun x => c • f x
-- MeasureTheory.StronglyMeasurable.const_vadd'
@[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : StronglyMeasurable f)
    (c : 𝕜) : StronglyMeasurable fun x => c • f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [VAdd 𝕜 β] [ContinuousConstVAdd 𝕜 β] (hf : MeasureTheory.StronglyMeasurable f) (c : 𝕜)
Goal: MeasureTheory.StronglyMeasurable fun x => c +ᵥ f x
-- MeasureTheory.StronglyMeasurable.vadd_const
@[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : StronglyMeasurable f) (c : β) : StronglyMeasurable fun x => f x • c 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [VAdd 𝕜 β] [ContinuousVAdd 𝕜 β] {f : α → 𝕜} (hf : MeasureTheory.StronglyMeasurable f) (c : β)
Goal: MeasureTheory.StronglyMeasurable fun x => f x +ᵥ c
-- MeasureTheory.StronglyMeasurable.smul_const
@[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : StronglyMeasurable f) (c : β) : StronglyMeasurable fun x => f x • c 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜} (hf : MeasureTheory.StronglyMeasurable f) (c : β)
Goal: MeasureTheory.StronglyMeasurable fun x => f x • c