ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AEMeasurable.pow
@[aesop safe 20 apply (rule_sets := [Measurable])]
theorem AEMeasurable.pow (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun x => f x ^ g x) μ 
Arguments: {β : Type u_2} {γ : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} {g : α → γ} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun x => f x ^ g x) μ
-- Measurable.pow_const
@[fun_prop, measurability]
theorem Measurable.pow_const (hf : Measurable f) (c : γ) : Measurable fun x => f x ^ c 
Arguments: {β : Type u_2} {γ : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {f : α → β} (hf : Measurable f) (c : γ)
Goal: Measurable fun x => f x ^ c
-- AEMeasurable.pow_const
@[fun_prop, measurability]
theorem AEMeasurable.pow_const (hf : AEMeasurable f μ) (c : γ) :
    AEMeasurable (fun x => f x ^ c) μ 
Arguments: {β : Type u_2} {γ : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) (c : γ)
Goal: AEMeasurable (fun x => f x ^ c) μ
-- Measurable.const_pow
@[measurability]
theorem Measurable.const_pow (hg : Measurable g) (c : β) : Measurable fun x => c ^ g x 
Arguments: {β : Type u_2} {γ : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {g : α → γ} (hg : Measurable g) (c : β)
Goal: Measurable fun x => c ^ g x
-- AEMeasurable.const_pow
@[measurability]
theorem AEMeasurable.const_pow (hg : AEMeasurable g μ) (c : β) :
    AEMeasurable (fun x => c ^ g x) μ 
Arguments: {β : Type u_2} {γ : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {g : α → γ} (hg : AEMeasurable g μ) (c : β)
Goal: AEMeasurable (fun x => c ^ g x) μ
-- MeasurableSub.measurable_const_sub
class MeasurableSub (G : Type*) [MeasurableSpace G] [Sub G] : Prop where
  measurable_const_sub : ∀ c : G, Measurable (c - ·)
  measurable_sub_const : ∀ c : G, Measurable (· - c)
Arguments: {G : Type u_2} [MeasurableSpace G] [Sub G] [self : MeasurableSub G] (c : G)
Goal: Measurable fun x => c - x
-- MeasurableSub.measurable_sub_const
class MeasurableSub (G : Type*) [MeasurableSpace G] [Sub G] : Prop where
  measurable_const_sub : ∀ c : G, Measurable (c - ·)
  measurable_sub_const : ∀ c : G, Measurable (· - c)
Arguments: {G : Type u_2} [MeasurableSpace G] [Sub G] [self : MeasurableSub G] (c : G)
Goal: Measurable fun x => x - c
-- MeasurableSub₂.measurable_sub
class MeasurableSub₂ (G : Type*) [MeasurableSpace G] [Sub G] : Prop where
  measurable_sub : Measurable fun p : G × G => p.1 - p.2
Arguments: {G : Type u_2} [MeasurableSpace G] [Sub G] [self : MeasurableSub₂ G]
Goal: Measurable fun p => p.1 - p.2
-- MeasurableDiv.measurable_const_div
@[to_additive]
class MeasurableDiv (G₀ : Type*) [MeasurableSpace G₀] [Div G₀] : Prop where
  measurable_const_div : ∀ c : G₀, Measurable (c / ·)
  measurable_div_const : ∀ c : G₀, Measurable (· / c)
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [Div G₀] [self : MeasurableDiv G₀] (c : G₀)
Goal: Measurable fun x => c / x
-- MeasurableDiv.measurable_div_const
@[to_additive]
class MeasurableDiv (G₀ : Type*) [MeasurableSpace G₀] [Div G₀] : Prop where
  measurable_const_div : ∀ c : G₀, Measurable (c / ·)
  measurable_div_const : ∀ c : G₀, Measurable (· / c)
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [Div G₀] [self : MeasurableDiv G₀] (c : G₀)
Goal: Measurable fun x => x / c
-- MeasurableDiv₂.measurable_div
@[to_additive MeasurableSub₂]
class MeasurableDiv₂ (G₀ : Type*) [MeasurableSpace G₀] [Div G₀] : Prop where
  measurable_div : Measurable fun p : G₀ × G₀ => p.1 / p.2
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [Div G₀] [self : MeasurableDiv₂ G₀]
Goal: Measurable fun p => p.1 / p.2
-- Measurable.const_div
@[to_additive (attr := measurability)]
theorem Measurable.const_div [MeasurableDiv G] (hf : Measurable f) (c : G) :
    Measurable fun x => c / f x 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f : α → G} [MeasurableDiv G] (hf : Measurable f) (c : G)
Goal: Measurable fun x => c / f x
-- Measurable.const_sub
@[to_additive (attr := measurability)]
theorem Measurable.const_div [MeasurableDiv G] (hf : Measurable f) (c : G) :
    Measurable fun x => c / f x 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f : α → G} [MeasurableSub G] (hf : Measurable f) (c : G)
Goal: Measurable fun x => c - f x
-- AEMeasurable.const_div
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_div [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G) :
    AEMeasurable (fun x => c / f x) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G)
Goal: AEMeasurable (fun x => c / f x) μ
-- AEMeasurable.const_sub
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_div [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G) :
    AEMeasurable (fun x => c / f x) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableSub G] (hf : AEMeasurable f μ) (c : G)
Goal: AEMeasurable (fun x => c - f x) μ
-- Measurable.sub_const
@[to_additive (attr := measurability)]
theorem Measurable.div_const [MeasurableDiv G] (hf : Measurable f) (c : G) :
    Measurable fun x => f x / c 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f : α → G} [MeasurableSub G] (hf : Measurable f) (c : G)
Goal: Measurable fun x => f x - c
-- Measurable.div_const
@[to_additive (attr := measurability)]
theorem Measurable.div_const [MeasurableDiv G] (hf : Measurable f) (c : G) :
    Measurable fun x => f x / c 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f : α → G} [MeasurableDiv G] (hf : Measurable f) (c : G)
Goal: Measurable fun x => f x / c
-- AEMeasurable.sub_const
@[to_additive (attr := measurability)]
theorem AEMeasurable.div_const [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G) :
    AEMeasurable (fun x => f x / c) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableSub G] (hf : AEMeasurable f μ) (c : G)
Goal: AEMeasurable (fun x => f x - c) μ
-- AEMeasurable.div_const
@[to_additive (attr := measurability)]
theorem AEMeasurable.div_const [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G) :
    AEMeasurable (fun x => f x / c) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G)
Goal: AEMeasurable (fun x => f x / c) μ
-- Measurable.div'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.div' [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g) :
    Measurable (f / g) 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f g : α → G} [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (f / g)
-- Measurable.sub'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.div' [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g) :
    Measurable (f / g) 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f g : α → G} [MeasurableSub₂ G] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (f - g)
-- Measurable.div
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.div [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g) :
    Measurable fun a => f a / g a 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f g : α → G} [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => f a / g a
-- Measurable.sub
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.div [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g) :
    Measurable fun a => f a / g a 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f g : α → G} [MeasurableSub₂ G] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => f a - g a
-- AEMeasurable.sub'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.div' [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f / g) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f g : α → G} {μ : MeasureTheory.Measure α} [MeasurableSub₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (f - g) μ
-- AEMeasurable.div'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.div' [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f / g) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f g : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (f / g) μ
-- AEMeasurable.div
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.div [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a / g a) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f g : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => f a / g a) μ
-- AEMeasurable.sub
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.div [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a / g a) μ 
Arguments: {G : Type u_2} {α : Type u_3} [MeasurableSpace G] [Sub G] {m : MeasurableSpace α} {f g : α → G} {μ : MeasureTheory.Measure α} [MeasurableSub₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => f a - g a) μ
-- measurableSet_eq_fun
@[measurability]
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [MeasurableSpace E] [AddGroup E]
    [MeasurableSingletonClass E] [MeasurableSub₂ E] {f g : α → E} (hf : Measurable f)
    (hg : Measurable g) : MeasurableSet { x | f x = g x } 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {E : Type u_4} [MeasurableSpace E] [AddGroup E] [MeasurableSingletonClass E] [MeasurableSub₂ E] {f g : α → E} (hf : Measurable f) (hg : Measurable g)
Goal: MeasurableSet {x | f x = g x}
-- measurableSet_eq_fun'
@[measurability]
lemma measurableSet_eq_fun' {β : Type*} [CanonicallyOrderedAddCommMonoid β] [Sub β] [OrderedSub β]
    {_ : MeasurableSpace β} [MeasurableSub₂ β] [MeasurableSingletonClass β]
    {f g : α → β} (hf : Measurable f) (hg : Measurable g) :
    MeasurableSet {x | f x = g x} 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {β : Type u_4} [CanonicallyOrderedAddCommMonoid β] [Sub β] [OrderedSub β]
Goal: ∀ {x : MeasurableSpace β} [inst : MeasurableSub₂ β] [inst : MeasurableSingletonClass β] {f g : α → β},
  Measurable f → Measurable g → MeasurableSet {x | f x = g x}
-- nullMeasurableSet_eq_fun
theorem nullMeasurableSet_eq_fun {E} [MeasurableSpace E] [AddGroup E] [MeasurableSingletonClass E]
    [MeasurableSub₂ E] {f g : α → E} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    NullMeasurableSet { x | f x = g x } μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_4} [MeasurableSpace E] [AddGroup E] [MeasurableSingletonClass E] [MeasurableSub₂ E] {f g : α → E} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: MeasureTheory.NullMeasurableSet {x | f x = g x} μ
-- measurableSet_eq_fun_of_countable
theorem measurableSet_eq_fun_of_countable {m : MeasurableSpace α} {E} [MeasurableSpace E]
    [MeasurableSingletonClass E] [Countable E] {f g : α → E} (hf : Measurable f)
    (hg : Measurable g) : MeasurableSet { x | f x = g x } 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {E : Type u_4} [MeasurableSpace E] [MeasurableSingletonClass E] [Countable E] {f g : α → E} (hf : Measurable f) (hg : Measurable g)
Goal: MeasurableSet {x | f x = g x}
-- ae_eq_trim_of_measurable
theorem ae_eq_trim_of_measurable {α E} {m m0 : MeasurableSpace α} {μ : Measure α}
    [MeasurableSpace E] [AddGroup E] [MeasurableSingletonClass E] [MeasurableSub₂ E]
    (hm : m ≤ m0) {f g : α → E} (hf : Measurable[m] f) (hg : Measurable[m] g) (hfg : f =ᵐ[μ] g) :
    f =ᵐ[μ.trim hm] g 
Arguments: {α : Type u_4} {E : Type u_5} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace E] [AddGroup E] [MeasurableSingletonClass E] [MeasurableSub₂ E] (hm : m ≤ m0) {f g : α → E} (hf : Measurable f) (hg : Measurable g) (hfg : f =ᵐ[μ] g)
Goal: f =ᵐ[μ.trim hm] g
-- MeasurableNeg.measurable_neg
class MeasurableNeg (G : Type*) [Neg G] [MeasurableSpace G] : Prop where
  measurable_neg : Measurable (Neg.neg : G → G)
Arguments: {G : Type u_2} [Neg G] [MeasurableSpace G] [self : MeasurableNeg G]
Goal: Measurable Neg.neg
-- MeasurableInv.measurable_inv
@[to_additive]
class MeasurableInv (G : Type*) [Inv G] [MeasurableSpace G] : Prop where
  measurable_inv : Measurable (Inv.inv : G → G)
Arguments: {G : Type u_2} [Inv G] [MeasurableSpace G] [self : MeasurableInv G]
Goal: Measurable Inv.inv
-- Measurable.neg
@[to_additive (attr := fun_prop, measurability)]
theorem Measurable.inv (hf : Measurable f) : Measurable fun x => (f x)⁻¹ 
Arguments: {G : Type u_2} {α : Type u_3} [Neg G] [MeasurableSpace G] [MeasurableNeg G] {m : MeasurableSpace α} {f : α → G} (hf : Measurable f)
Goal: Measurable fun x => -f x
-- Measurable.inv
@[to_additive (attr := fun_prop, measurability)]
theorem Measurable.inv (hf : Measurable f) : Measurable fun x => (f x)⁻¹ 
Arguments: {G : Type u_2} {α : Type u_3} [Inv G] [MeasurableSpace G] [MeasurableInv G] {m : MeasurableSpace α} {f : α → G} (hf : Measurable f)
Goal: Measurable fun x => (f x)⁻¹
-- AEMeasurable.neg
@[to_additive (attr := fun_prop, measurability)]
theorem AEMeasurable.inv (hf : AEMeasurable f μ) : AEMeasurable (fun x => (f x)⁻¹) μ 
Arguments: {G : Type u_2} {α : Type u_3} [Neg G] [MeasurableSpace G] [MeasurableNeg G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => -f x) μ
-- AEMeasurable.inv
@[to_additive (attr := fun_prop, measurability)]
theorem AEMeasurable.inv (hf : AEMeasurable f μ) : AEMeasurable (fun x => (f x)⁻¹) μ 
Arguments: {G : Type u_2} {α : Type u_3} [Inv G] [MeasurableSpace G] [MeasurableInv G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => (f x)⁻¹) μ
-- measurable_inv_iff
@[to_additive (attr := simp)]
theorem measurable_inv_iff {G : Type*} [Group G] [MeasurableSpace G] [MeasurableInv G]
    {f : α → G} : (Measurable fun x => (f x)⁻¹) ↔ Measurable f 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {G : Type u_4} [Group G] [MeasurableSpace G] [MeasurableInv G] {f : α → G}
Goal: (Measurable fun x => (f x)⁻¹) ↔ Measurable f
-- measurable_neg_iff
@[to_additive (attr := simp)]
theorem measurable_inv_iff {G : Type*} [Group G] [MeasurableSpace G] [MeasurableInv G]
    {f : α → G} : (Measurable fun x => (f x)⁻¹) ↔ Measurable f 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {G : Type u_4} [AddGroup G] [MeasurableSpace G] [MeasurableNeg G] {f : α → G}
Goal: (Measurable fun x => -f x) ↔ Measurable f
-- aemeasurable_neg_iff
@[to_additive (attr := simp)]
theorem aemeasurable_inv_iff {G : Type*} [Group G] [MeasurableSpace G] [MeasurableInv G]
    {f : α → G} : AEMeasurable (fun x => (f x)⁻¹) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G : Type u_4} [AddGroup G] [MeasurableSpace G] [MeasurableNeg G] {f : α → G}
Goal: AEMeasurable (fun x => -f x) μ ↔ AEMeasurable f μ
-- aemeasurable_inv_iff
@[to_additive (attr := simp)]
theorem aemeasurable_inv_iff {G : Type*} [Group G] [MeasurableSpace G] [MeasurableInv G]
    {f : α → G} : AEMeasurable (fun x => (f x)⁻¹) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G : Type u_4} [Group G] [MeasurableSpace G] [MeasurableInv G] {f : α → G}
Goal: AEMeasurable (fun x => (f x)⁻¹) μ ↔ AEMeasurable f μ
-- measurable_inv_iff₀
@[simp]
theorem measurable_inv_iff₀ {G₀ : Type*} [GroupWithZero G₀] [MeasurableSpace G₀]
    [MeasurableInv G₀] {f : α → G₀} : (Measurable fun x => (f x)⁻¹) ↔ Measurable f 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {G₀ : Type u_4} [GroupWithZero G₀] [MeasurableSpace G₀] [MeasurableInv G₀] {f : α → G₀}
Goal: (Measurable fun x => (f x)⁻¹) ↔ Measurable f
-- aemeasurable_inv_iff₀
@[simp]
theorem aemeasurable_inv_iff₀ {G₀ : Type*} [GroupWithZero G₀] [MeasurableSpace G₀]
    [MeasurableInv G₀] {f : α → G₀} : AEMeasurable (fun x => (f x)⁻¹) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G₀ : Type u_4} [GroupWithZero G₀] [MeasurableSpace G₀] [MeasurableInv G₀] {f : α → G₀}
Goal: AEMeasurable (fun x => (f x)⁻¹) μ ↔ AEMeasurable f μ
-- MeasurableSet.inv
@[to_additive]
theorem MeasurableSet.inv {s : Set G} (hs : MeasurableSet s) : MeasurableSet s⁻¹ 
Arguments: {G : Type u_2} [Inv G] [MeasurableSpace G] [MeasurableInv G] {s : Set G} (hs : MeasurableSet s)
Goal: MeasurableSet s⁻¹
-- MeasurableSet.neg
@[to_additive]
theorem MeasurableSet.inv {s : Set G} (hs : MeasurableSet s) : MeasurableSet s⁻¹ 
Arguments: {G : Type u_2} [Neg G] [MeasurableSpace G] [MeasurableNeg G] {s : Set G} (hs : MeasurableSet s)
Goal: MeasurableSet (-s)
-- measurableEmbedding_inv
@[to_additive]
theorem measurableEmbedding_inv [InvolutiveInv α] [MeasurableInv α] :
    MeasurableEmbedding (Inv.inv (α := α)) 
Arguments: {α : Type u_3} {m : MeasurableSpace α} [InvolutiveInv α] [MeasurableInv α]
Goal: MeasurableEmbedding Inv.inv
-- measurableEmbedding_neg
@[to_additive]
theorem measurableEmbedding_inv [InvolutiveInv α] [MeasurableInv α] :
    MeasurableEmbedding (Inv.inv (α := α)) 
Arguments: {α : Type u_3} {m : MeasurableSpace α} [InvolutiveNeg α] [MeasurableNeg α]
Goal: MeasurableEmbedding Neg.neg
-- Measurable.mul_iff_right
@[to_additive]
theorem Measurable.mul_iff_right {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {f g : α → G} (hf : Measurable f) :
    Measurable (f * g) ↔ Measurable g 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G] [MeasurableMul₂ G] [MeasurableInv G] {f g : α → G} (hf : Measurable f)
Goal: Measurable (f * g) ↔ Measurable g
-- Measurable.add_iff_right
@[to_additive]
theorem Measurable.mul_iff_right {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {f g : α → G} (hf : Measurable f) :
    Measurable (f * g) ↔ Measurable g 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [AddCommGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] {f g : α → G} (hf : Measurable f)
Goal: Measurable (f + g) ↔ Measurable g
-- AEMeasurable.mul_iff_right
@[to_additive]
theorem AEMeasurable.mul_iff_right {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {μ : Measure α} {f g : α → G} (hf : AEMeasurable f μ) :
    AEMeasurable (f * g) μ ↔ AEMeasurable g μ 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G] [MeasurableMul₂ G] [MeasurableInv G] {μ : MeasureTheory.Measure α} {f g : α → G} (hf : AEMeasurable f μ)
Goal: AEMeasurable (f * g) μ ↔ AEMeasurable g μ
-- AEMeasurable.add_iff_right
@[to_additive]
theorem AEMeasurable.mul_iff_right {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {μ : Measure α} {f g : α → G} (hf : AEMeasurable f μ) :
    AEMeasurable (f * g) μ ↔ AEMeasurable g μ 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [AddCommGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] {μ : MeasureTheory.Measure α} {f g : α → G} (hf : AEMeasurable f μ)
Goal: AEMeasurable (f + g) μ ↔ AEMeasurable g μ
-- Measurable.mul_iff_left
@[to_additive]
theorem Measurable.mul_iff_left {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {f g : α → G} (hf : Measurable f) :
    Measurable (g * f) ↔ Measurable g 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G] [MeasurableMul₂ G] [MeasurableInv G] {f g : α → G} (hf : Measurable f)
Goal: Measurable (g * f) ↔ Measurable g
-- Measurable.add_iff_left
@[to_additive]
theorem Measurable.mul_iff_left {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {f g : α → G} (hf : Measurable f) :
    Measurable (g * f) ↔ Measurable g 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [AddCommGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] {f g : α → G} (hf : Measurable f)
Goal: Measurable (g + f) ↔ Measurable g
-- AEMeasurable.add_iff_left
@[to_additive]
theorem AEMeasurable.mul_iff_left {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {μ : Measure α} {f g : α → G} (hf : AEMeasurable f μ) :
    AEMeasurable (g * f) μ ↔ AEMeasurable g μ 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [AddCommGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] {μ : MeasureTheory.Measure α} {f g : α → G} (hf : AEMeasurable f μ)
Goal: AEMeasurable (g + f) μ ↔ AEMeasurable g μ
-- AEMeasurable.mul_iff_left
@[to_additive]
theorem AEMeasurable.mul_iff_left {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
    [MeasurableMul₂ G] [MeasurableInv G] {μ : Measure α} {f g : α → G} (hf : AEMeasurable f μ) :
    AEMeasurable (g * f) μ ↔ AEMeasurable g μ 
Arguments: {α : Type u_1} {G : Type u_2} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G] [MeasurableMul₂ G] [MeasurableInv G] {μ : MeasureTheory.Measure α} {f g : α → G} (hf : AEMeasurable f μ)
Goal: AEMeasurable (g * f) μ ↔ AEMeasurable g μ
-- MeasurableVAdd.measurable_const_vadd
class MeasurableVAdd (M α : Type*) [VAdd M α] [MeasurableSpace M] [MeasurableSpace α] :
    Prop where
  measurable_const_vadd : ∀ c : M, Measurable (c +ᵥ · : α → α)
  measurable_vadd_const : ∀ x : α, Measurable (· +ᵥ x : M → α)
Arguments: {M : Type u_2} {α : Type u_3} [VAdd M α] [MeasurableSpace M] [MeasurableSpace α] [self : MeasurableVAdd M α] (c : M)
Goal: Measurable fun x => c +ᵥ x
-- MeasurableVAdd.measurable_vadd_const
class MeasurableVAdd (M α : Type*) [VAdd M α] [MeasurableSpace M] [MeasurableSpace α] :
    Prop where
  measurable_const_vadd : ∀ c : M, Measurable (c +ᵥ · : α → α)
  measurable_vadd_const : ∀ x : α, Measurable (· +ᵥ x : M → α)
Arguments: {M : Type u_2} {α : Type u_3} [VAdd M α] [MeasurableSpace M] [MeasurableSpace α] [self : MeasurableVAdd M α] (x : α)
Goal: Measurable fun x_1 => x_1 +ᵥ x
-- MeasurableSMul.measurable_smul_const
@[to_additive]
class MeasurableSMul (M α : Type*) [SMul M α] [MeasurableSpace M] [MeasurableSpace α] :
    Prop where
  measurable_const_smul : ∀ c : M, Measurable (c • · : α → α)
  measurable_smul_const : ∀ x : α, Measurable (· • x : M → α)
Arguments: {M : Type u_2} {α : Type u_3} [SMul M α] [MeasurableSpace M] [MeasurableSpace α] [self : MeasurableSMul M α] (x : α)
Goal: Measurable fun x_1 => x_1 • x