ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.AEEqFun.coeFn_comp₂Measurable
theorem coeFn_comp₂Measurable (g : β → γ → δ) (hg : Measurable (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) : comp₂Measurable g hg f₁ f₂ =ᵐ[μ] fun a => g (f₁ a) (f₂ a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [MeasurableSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] [SecondCountableTopologyEither β γ] [MeasurableSpace δ] [TopologicalSpace.PseudoMetrizableSpace δ] [OpensMeasurableSpace δ] [SecondCountableTopology δ] (g : β → γ → δ) (hg : Measurable (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: ↑(MeasureTheory.AEEqFun.comp₂Measurable g hg f₁ f₂) =ᵐ[μ] fun a => g (↑f₁ a) (↑f₂ a)
-- MeasureTheory.AEEqFun.mk_toGerm
@[simp]
theorem mk_toGerm (f : α → β) (hf) : (mk f hf : α →ₘ[μ] β).toGerm = f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: (MeasureTheory.AEEqFun.mk f hf).toGerm = ↑f
-- MeasureTheory.AEEqFun.toGerm_eq
theorem toGerm_eq (f : α →ₘ[μ] β) : f.toGerm = (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (f : α →ₘ[μ] β)
Goal: f.toGerm = ↑↑f
-- MeasureTheory.AEEqFun.toGerm_injective
theorem toGerm_injective : Injective (toGerm : (α →ₘ[μ] β) → Germ (ae μ) β) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β]
Goal: Function.Injective MeasureTheory.AEEqFun.toGerm
-- MeasureTheory.AEEqFun.compQuasiMeasurePreserving_toGerm
@[simp]
theorem compQuasiMeasurePreserving_toGerm [MeasurableSpace β] {f : α → β} {ν}
    (g : β →ₘ[ν] γ) (hf : Measure.QuasiMeasurePreserving f μ ν) :
    (g.compQuasiMeasurePreserving f hf).toGerm = g.toGerm.compTendsto f hf.tendsto_ae 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {f : α → β} {ν : MeasureTheory.Measure β} (g : β →ₘ[ν] γ) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: (g.compQuasiMeasurePreserving f hf).toGerm = g.toGerm.compTendsto f ⋯
-- MeasureTheory.AEEqFun.compMeasurePreserving_toGerm
@[simp]
theorem compMeasurePreserving_toGerm [MeasurableSpace β] {f : α → β} {ν}
    (g : β →ₘ[ν] γ) (hf : MeasurePreserving f μ ν) :
    (g.compMeasurePreserving f hf).toGerm =
      g.toGerm.compTendsto f hf.quasiMeasurePreserving.tendsto_ae 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [MeasurableSpace β] {f : α → β} {ν : MeasureTheory.Measure β} (g : β →ₘ[ν] γ) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: (g.compMeasurePreserving f hf).toGerm = g.toGerm.compTendsto f ⋯
-- MeasureTheory.AEEqFun.comp_toGerm
theorem comp_toGerm (g : β → γ) (hg : Continuous g) (f : α →ₘ[μ] β) :
    (comp g hg f).toGerm = f.toGerm.map g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] (g : β → γ) (hg : Continuous g) (f : α →ₘ[μ] β)
Goal: (MeasureTheory.AEEqFun.comp g hg f).toGerm = Filter.Germ.map g f.toGerm
-- MeasureTheory.AEEqFun.compMeasurable_toGerm
theorem compMeasurable_toGerm [MeasurableSpace β] [BorelSpace β] [PseudoMetrizableSpace β]
    [PseudoMetrizableSpace γ] [SecondCountableTopology γ] [MeasurableSpace γ]
    [OpensMeasurableSpace γ] (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β) :
    (compMeasurable g hg f).toGerm = f.toGerm.map g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [MeasurableSpace β] [BorelSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [TopologicalSpace.PseudoMetrizableSpace γ] [SecondCountableTopology γ] [MeasurableSpace γ] [OpensMeasurableSpace γ] (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β)
Goal: (MeasureTheory.AEEqFun.compMeasurable g hg f).toGerm = Filter.Germ.map g f.toGerm
-- MeasureTheory.AEEqFun.comp₂_toGerm
theorem comp₂_toGerm (g : β → γ → δ) (hg : Continuous (uncurry g)) (f₁ : α →ₘ[μ] β)
    (f₂ : α →ₘ[μ] γ) : (comp₂ g hg f₁ f₂).toGerm = f₁.toGerm.map₂ g f₂.toGerm 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (g : β → γ → δ) (hg : Continuous (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: (MeasureTheory.AEEqFun.comp₂ g hg f₁ f₂).toGerm = Filter.Germ.map₂ g f₁.toGerm f₂.toGerm
-- MeasureTheory.AEEqFun.comp₂Measurable_toGerm
theorem comp₂Measurable_toGerm [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
    [PseudoMetrizableSpace γ] [SecondCountableTopologyEither β γ]
    [MeasurableSpace γ] [BorelSpace γ] [PseudoMetrizableSpace δ] [SecondCountableTopology δ]
    [MeasurableSpace δ] [OpensMeasurableSpace δ] (g : β → γ → δ) (hg : Measurable (uncurry g))
    (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ) :
    (comp₂Measurable g hg f₁ f₂).toGerm = f₁.toGerm.map₂ g f₂.toGerm 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] [TopologicalSpace.PseudoMetrizableSpace γ] [SecondCountableTopologyEither β γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.PseudoMetrizableSpace δ] [SecondCountableTopology δ] [MeasurableSpace δ] [OpensMeasurableSpace δ] (g : β → γ → δ) (hg : Measurable (Function.uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ)
Goal: (MeasureTheory.AEEqFun.comp₂Measurable g hg f₁ f₂).toGerm = Filter.Germ.map₂ g f₁.toGerm f₂.toGerm
-- MeasureTheory.AEEqFun.liftRel_mk_mk
theorem liftRel_mk_mk {r : β → γ → Prop} {f : α → β} {g : α → γ} {hf hg} :
    LiftRel r (mk f hf : α →ₘ[μ] β) (mk g hg) ↔ ∀ᵐ a ∂μ, r (f a) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] {r : β → γ → Prop} {f : α → β} {g : α → γ} {hf : MeasureTheory.AEStronglyMeasurable f μ} {hg : MeasureTheory.AEStronglyMeasurable g μ}
Goal: MeasureTheory.AEEqFun.LiftRel r (MeasureTheory.AEEqFun.mk f hf) (MeasureTheory.AEEqFun.mk g hg) ↔
  ∀ᵐ (a : α) ∂μ, r (f a) (g a)
-- MeasureTheory.AEEqFun.liftRel_iff_coeFn
theorem liftRel_iff_coeFn {r : β → γ → Prop} {f : α →ₘ[μ] β} {g : α →ₘ[μ] γ} :
    LiftRel r f g ↔ ∀ᵐ a ∂μ, r (f a) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] {r : β → γ → Prop} {f : α →ₘ[μ] β} {g : α →ₘ[μ] γ}
Goal: MeasureTheory.AEEqFun.LiftRel r f g ↔ ∀ᵐ (a : α) ∂μ, r (↑f a) (↑g a)
-- MeasureTheory.AEEqFun.mk_le_mk
@[simp]
theorem mk_le_mk [Preorder β] {f g : α → β} (hf hg) : (mk f hf : α →ₘ[μ] β) ≤ mk g hg ↔ f ≤ᵐ[μ] g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Preorder β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEEqFun.mk f hf ≤ MeasureTheory.AEEqFun.mk g hg ↔ f ≤ᵐ[μ] g
-- MeasureTheory.AEEqFun.coeFn_le
@[simp, norm_cast]
theorem coeFn_le [Preorder β] {f g : α →ₘ[μ] β} : (f : α → β) ≤ᵐ[μ] g ↔ f ≤ g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Preorder β] {f g : α →ₘ[μ] β}
Goal: ↑f ≤ᵐ[μ] ↑g ↔ f ≤ g
-- MeasureTheory.AEEqFun.coeFn_sup
theorem coeFn_sup (f g : α →ₘ[μ] β) : ⇑(f ⊔ g) =ᵐ[μ] fun x => f x ⊔ g x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] (f g : α →ₘ[μ] β)
Goal: ↑(f ⊔ g) =ᵐ[μ] fun x => ↑f x ⊔ ↑g x
-- MeasureTheory.AEEqFun.le_sup_left
protected theorem le_sup_left (f g : α →ₘ[μ] β) : f ≤ f ⊔ g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] (f g : α →ₘ[μ] β)
Goal: f ≤ f ⊔ g
-- MeasureTheory.AEEqFun.le_sup_right
protected theorem le_sup_right (f g : α →ₘ[μ] β) : g ≤ f ⊔ g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] (f g : α →ₘ[μ] β)
Goal: g ≤ f ⊔ g
-- MeasureTheory.AEEqFun.sup_le
protected theorem sup_le (f g f' : α →ₘ[μ] β) (hf : f ≤ f') (hg : g ≤ f') : f ⊔ g ≤ f' 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] (f g f' : α →ₘ[μ] β) (hf : f ≤ f') (hg : g ≤ f')
Goal: f ⊔ g ≤ f'
-- MeasureTheory.AEEqFun.coeFn_inf
theorem coeFn_inf (f g : α →ₘ[μ] β) : ⇑(f ⊓ g) =ᵐ[μ] fun x => f x ⊓ g x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeInf β] [ContinuousInf β] (f g : α →ₘ[μ] β)
Goal: ↑(f ⊓ g) =ᵐ[μ] fun x => ↑f x ⊓ ↑g x
-- MeasureTheory.AEEqFun.inf_le_left
protected theorem inf_le_left (f g : α →ₘ[μ] β) : f ⊓ g ≤ f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeInf β] [ContinuousInf β] (f g : α →ₘ[μ] β)
Goal: f ⊓ g ≤ f
-- MeasureTheory.AEEqFun.inf_le_right
protected theorem inf_le_right (f g : α →ₘ[μ] β) : f ⊓ g ≤ g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeInf β] [ContinuousInf β] (f g : α →ₘ[μ] β)
Goal: f ⊓ g ≤ g
-- MeasureTheory.AEEqFun.le_inf
protected theorem le_inf (f' f g : α →ₘ[μ] β) (hf : f' ≤ f) (hg : f' ≤ g) : f' ≤ f ⊓ g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SemilatticeInf β] [ContinuousInf β] (f' f g : α →ₘ[μ] β) (hf : f' ≤ f) (hg : f' ≤ g)
Goal: f' ≤ f ⊓ g
-- MeasureTheory.AEEqFun.coeFn_const
theorem coeFn_const (b : β) : (const α b : α →ₘ[μ] β) =ᵐ[μ] Function.const α b 
Arguments: (α : Type u_1) {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] (b : β)
Goal: ↑(MeasureTheory.AEEqFun.const α b) =ᵐ[μ] Function.const α b
-- MeasureTheory.AEEqFun.zero_def
@[to_additive]
theorem one_def [One β] : (1 : α →ₘ[μ] β) = mk (fun _ : α => 1) aestronglyMeasurable_const 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Zero β]
Goal: 0 = MeasureTheory.AEEqFun.mk (fun x => 0) ⋯
-- MeasureTheory.AEEqFun.one_def
@[to_additive]
theorem one_def [One β] : (1 : α →ₘ[μ] β) = mk (fun _ : α => 1) aestronglyMeasurable_const 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [One β]
Goal: 1 = MeasureTheory.AEEqFun.mk (fun x => 1) ⋯
-- MeasureTheory.AEEqFun.coeFn_zero
@[to_additive]
theorem coeFn_one [One β] : ⇑(1 : α →ₘ[μ] β) =ᵐ[μ] 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Zero β]
Goal: ↑0 =ᵐ[μ] 0
-- MeasureTheory.AEEqFun.coeFn_one
@[to_additive]
theorem coeFn_one [One β] : ⇑(1 : α →ₘ[μ] β) =ᵐ[μ] 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [One β]
Goal: ↑1 =ᵐ[μ] 1
-- MeasureTheory.AEEqFun.one_toGerm
@[to_additive (attr := simp)]
theorem one_toGerm [One β] : (1 : α →ₘ[μ] β).toGerm = 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [One β]
Goal: MeasureTheory.AEEqFun.toGerm 1 = 1
-- MeasureTheory.AEEqFun.zero_toGerm
@[to_additive (attr := simp)]
theorem one_toGerm [One β] : (1 : α →ₘ[μ] β).toGerm = 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Zero β]
Goal: MeasureTheory.AEEqFun.toGerm 0 = 0
-- MeasureTheory.AEEqFun.smul_mk
@[simp]
theorem smul_mk (c : 𝕜) (f : α → γ) (hf : AEStronglyMeasurable f μ) :
    c • (mk f hf : α →ₘ[μ] γ) = mk (c • f) (hf.const_smul _) 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] {𝕜 : Type u_5} [SMul 𝕜 γ] [ContinuousConstSMul 𝕜 γ] (c : 𝕜) (f : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: c • MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (c • f) ⋯
-- MeasureTheory.AEEqFun.coeFn_smul
theorem coeFn_smul (c : 𝕜) (f : α →ₘ[μ] γ) : ⇑(c • f) =ᵐ[μ] c • ⇑f 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] {𝕜 : Type u_5} [SMul 𝕜 γ] [ContinuousConstSMul 𝕜 γ] (c : 𝕜) (f : α →ₘ[μ] γ)
Goal: ↑(c • f) =ᵐ[μ] c • ↑f
-- MeasureTheory.AEEqFun.smul_toGerm
theorem smul_toGerm (c : 𝕜) (f : α →ₘ[μ] γ) : (c • f).toGerm = c • f.toGerm 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] {𝕜 : Type u_5} [SMul 𝕜 γ] [ContinuousConstSMul 𝕜 γ] (c : 𝕜) (f : α →ₘ[μ] γ)
Goal: (c • f).toGerm = c • f.toGerm
-- MeasureTheory.AEEqFun.mk_mul_mk
@[to_additive (attr := simp)]
theorem mk_mul_mk (f g : α → γ) (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    (mk f hf : α →ₘ[μ] γ) * mk g hg = mk (f * g) (hf.mul hg) 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Mul γ] [ContinuousMul γ] (f g : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEEqFun.mk f hf * MeasureTheory.AEEqFun.mk g hg = MeasureTheory.AEEqFun.mk (f * g) ⋯
-- MeasureTheory.AEEqFun.mk_add_mk
@[to_additive (attr := simp)]
theorem mk_mul_mk (f g : α → γ) (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    (mk f hf : α →ₘ[μ] γ) * mk g hg = mk (f * g) (hf.mul hg) 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Add γ] [ContinuousAdd γ] (f g : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEEqFun.mk f hf + MeasureTheory.AEEqFun.mk g hg = MeasureTheory.AEEqFun.mk (f + g) ⋯
-- MeasureTheory.AEEqFun.coeFn_add
@[to_additive]
theorem coeFn_mul (f g : α →ₘ[μ] γ) : ⇑(f * g) =ᵐ[μ] f * g 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Add γ] [ContinuousAdd γ] (f g : α →ₘ[μ] γ)
Goal: ↑(f + g) =ᵐ[μ] ↑f + ↑g
-- MeasureTheory.AEEqFun.coeFn_mul
@[to_additive]
theorem coeFn_mul (f g : α →ₘ[μ] γ) : ⇑(f * g) =ᵐ[μ] f * g 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Mul γ] [ContinuousMul γ] (f g : α →ₘ[μ] γ)
Goal: ↑(f * g) =ᵐ[μ] ↑f * ↑g
-- MeasureTheory.AEEqFun.add_toGerm
@[to_additive (attr := simp)]
theorem mul_toGerm (f g : α →ₘ[μ] γ) : (f * g).toGerm = f.toGerm * g.toGerm 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Add γ] [ContinuousAdd γ] (f g : α →ₘ[μ] γ)
Goal: (f + g).toGerm = f.toGerm + g.toGerm
-- MeasureTheory.AEEqFun.mul_toGerm
@[to_additive (attr := simp)]
theorem mul_toGerm (f g : α →ₘ[μ] γ) : (f * g).toGerm = f.toGerm * g.toGerm 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Mul γ] [ContinuousMul γ] (f g : α →ₘ[μ] γ)
Goal: (f * g).toGerm = f.toGerm * g.toGerm
-- MeasureTheory.AEEqFun.mk_pow
@[simp]
theorem mk_pow (f : α → γ) (hf) (n : ℕ) :
    (mk f hf : α →ₘ[μ] γ) ^ n =
      mk (f ^ n) ((_root_.continuous_pow n).comp_aestronglyMeasurable hf) 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (f : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (n : ℕ)
Goal: MeasureTheory.AEEqFun.mk f hf ^ n = MeasureTheory.AEEqFun.mk (f ^ n) ⋯
-- MeasureTheory.AEEqFun.coeFn_pow
theorem coeFn_pow (f : α →ₘ[μ] γ) (n : ℕ) : ⇑(f ^ n) =ᵐ[μ] (⇑f) ^ n 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (f : α →ₘ[μ] γ) (n : ℕ)
Goal: ↑(f ^ n) =ᵐ[μ] ↑f ^ n
-- MeasureTheory.AEEqFun.pow_toGerm
@[simp]
theorem pow_toGerm (f : α →ₘ[μ] γ) (n : ℕ) : (f ^ n).toGerm = f.toGerm ^ n 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (f : α →ₘ[μ] γ) (n : ℕ)
Goal: (f ^ n).toGerm = f.toGerm ^ n
-- MeasureTheory.AEEqFun.toGermMonoidHom_apply
@[to_additive (attr := simps) "`AEEqFun.toGerm` as an `AddMonoidHom`."]
def toGermMonoidHom : (α →ₘ[μ] γ) →* (ae μ).Germ γ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (f : α →ₘ[μ] γ)
Goal: MeasureTheory.AEEqFun.toGermMonoidHom f = f.toGerm
-- MeasureTheory.AEEqFun.toGermAddMonoidHom_apply
@[to_additive (attr := simps) "`AEEqFun.toGerm` as an `AddMonoidHom`."]
def toGermMonoidHom : (α →ₘ[μ] γ) →* (ae μ).Germ γ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddMonoid γ] [ContinuousAdd γ] (f : α →ₘ[μ] γ)
Goal: MeasureTheory.AEEqFun.toGermAddMonoidHom f = f.toGerm
-- MeasureTheory.AEEqFun.inv_mk
@[to_additive (attr := simp)]
theorem inv_mk (f : α → γ) (hf) : (mk f hf : α →ₘ[μ] γ)⁻¹ = mk f⁻¹ hf.inv 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: (MeasureTheory.AEEqFun.mk f hf)⁻¹ = MeasureTheory.AEEqFun.mk f⁻¹ ⋯
-- MeasureTheory.AEEqFun.neg_mk
@[to_additive (attr := simp)]
theorem inv_mk (f : α → γ) (hf) : (mk f hf : α →ₘ[μ] γ)⁻¹ = mk f⁻¹ hf.inv 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: -MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (-f) ⋯
-- MeasureTheory.AEEqFun.coeFn_neg
@[to_additive]
theorem coeFn_inv (f : α →ₘ[μ] γ) : ⇑f⁻¹ =ᵐ[μ] f⁻¹ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f : α →ₘ[μ] γ)
Goal: ↑(-f) =ᵐ[μ] -↑f
-- MeasureTheory.AEEqFun.coeFn_inv
@[to_additive]
theorem coeFn_inv (f : α →ₘ[μ] γ) : ⇑f⁻¹ =ᵐ[μ] f⁻¹ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : α →ₘ[μ] γ)
Goal: ↑f⁻¹ =ᵐ[μ] (↑f)⁻¹
-- MeasureTheory.AEEqFun.inv_toGerm
@[to_additive]
theorem inv_toGerm (f : α →ₘ[μ] γ) : f⁻¹.toGerm = f.toGerm⁻¹ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : α →ₘ[μ] γ)
Goal: f⁻¹.toGerm = f.toGerm⁻¹
-- MeasureTheory.AEEqFun.neg_toGerm
@[to_additive]
theorem inv_toGerm (f : α →ₘ[μ] γ) : f⁻¹.toGerm = f.toGerm⁻¹ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f : α →ₘ[μ] γ)
Goal: (-f).toGerm = -f.toGerm
-- MeasureTheory.AEEqFun.mk_div
@[to_additive (attr := simp, nolint simpNF)] -- Porting note: LHS does not simplify.
theorem mk_div (f g : α → γ) (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    mk (f / g) (hf.div hg) = (mk f hf : α →ₘ[μ] γ) / mk g hg 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f g : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEEqFun.mk (f / g) ⋯ = MeasureTheory.AEEqFun.mk f hf / MeasureTheory.AEEqFun.mk g hg
-- MeasureTheory.AEEqFun.mk_sub
@[to_additive (attr := simp, nolint simpNF)] -- Porting note: LHS does not simplify.
theorem mk_div (f g : α → γ) (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    mk (f / g) (hf.div hg) = (mk f hf : α →ₘ[μ] γ) / mk g hg 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f g : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEEqFun.mk (f - g) ⋯ = MeasureTheory.AEEqFun.mk f hf - MeasureTheory.AEEqFun.mk g hg
-- MeasureTheory.AEEqFun.coeFn_div
@[to_additive]
theorem coeFn_div (f g : α →ₘ[μ] γ) : ⇑(f / g) =ᵐ[μ] f / g 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f g : α →ₘ[μ] γ)
Goal: ↑(f / g) =ᵐ[μ] ↑f / ↑g
-- MeasureTheory.AEEqFun.coeFn_sub
@[to_additive]
theorem coeFn_div (f g : α →ₘ[μ] γ) : ⇑(f / g) =ᵐ[μ] f / g 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f g : α →ₘ[μ] γ)
Goal: ↑(f - g) =ᵐ[μ] ↑f - ↑g
-- MeasureTheory.AEEqFun.div_toGerm
@[to_additive]
theorem div_toGerm (f g : α →ₘ[μ] γ) : (f / g).toGerm = f.toGerm / g.toGerm 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f g : α →ₘ[μ] γ)
Goal: (f / g).toGerm = f.toGerm / g.toGerm
-- MeasureTheory.AEEqFun.sub_toGerm
@[to_additive]
theorem div_toGerm (f g : α →ₘ[μ] γ) : (f / g).toGerm = f.toGerm / g.toGerm 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f g : α →ₘ[μ] γ)
Goal: (f - g).toGerm = f.toGerm - g.toGerm
-- MeasureTheory.AEEqFun.mk_zpow
@[simp]
theorem mk_zpow (f : α → γ) (hf) (n : ℤ) :
    (mk f hf : α →ₘ[μ] γ) ^ n = mk (f ^ n) ((continuous_zpow n).comp_aestronglyMeasurable hf) 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (n : ℤ)
Goal: MeasureTheory.AEEqFun.mk f hf ^ n = MeasureTheory.AEEqFun.mk (f ^ n) ⋯
-- MeasureTheory.AEEqFun.coeFn_zpow
theorem coeFn_zpow (f : α →ₘ[μ] γ) (n : ℤ) : ⇑(f ^ n) =ᵐ[μ] (⇑f) ^ n 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : α →ₘ[μ] γ) (n : ℤ)
Goal: ↑(f ^ n) =ᵐ[μ] ↑f ^ n
-- MeasureTheory.AEEqFun.zpow_toGerm
@[simp]
theorem zpow_toGerm (f : α →ₘ[μ] γ) (n : ℤ) : (f ^ n).toGerm = f.toGerm ^ n 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : α →ₘ[μ] γ) (n : ℤ)
Goal: (f ^ n).toGerm = f.toGerm ^ n
-- MeasureTheory.AEEqFun.lintegral_mk
@[simp]
theorem lintegral_mk (f : α → ℝ≥0∞) (hf) : (mk f hf : α →ₘ[μ] ℝ≥0∞).lintegral = ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: (MeasureTheory.AEEqFun.mk f hf).lintegral = ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.AEEqFun.lintegral_coeFn
theorem lintegral_coeFn (f : α →ₘ[μ] ℝ≥0∞) : ∫⁻ a, f a ∂μ = f.lintegral 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (f : α →ₘ[μ] ℝ≥0∞)
Goal: ∫⁻ (a : α), ↑f a ∂μ = f.lintegral
-- MeasureTheory.AEEqFun.lintegral_zero
@[simp]
nonrec theorem lintegral_zero : lintegral (0 : α →ₘ[μ] ℝ≥0∞) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.AEEqFun.lintegral 0 = 0
-- MeasureTheory.AEEqFun.lintegral_eq_zero_iff
@[simp]
theorem lintegral_eq_zero_iff {f : α →ₘ[μ] ℝ≥0∞} : lintegral f = 0 ↔ f = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α →ₘ[μ] ℝ≥0∞}
Goal: f.lintegral = 0 ↔ f = 0