ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.disintegrate
lemma disintegrate : ρ.fst ⊗ₘ ρCond = ρ 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} (ρ : MeasureTheory.Measure (α × Ω)) (ρCond : ProbabilityTheory.Kernel α Ω) [ρ.IsCondKernel ρCond]
Goal: ρ.fst ⊗ₘ ρCond = ρ
-- MeasureTheory.Measure.IsCondKernel.isSFiniteKernel
lemma IsCondKernel.isSFiniteKernel (hρ : ρ ≠ 0) : IsSFiniteKernel ρCond 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} (ρ : MeasureTheory.Measure (α × Ω)) (ρCond : ProbabilityTheory.Kernel α Ω) [ρ.IsCondKernel ρCond] (hρ : ρ ≠ 0)
Goal: ProbabilityTheory.IsSFiniteKernel ρCond
-- _private.Mathlib.Probability.Kernel.Disintegration.Basic.0.MeasureTheory.Measure.IsCondKernel.apply_of_ne_zero_of_measurableSet
private lemma IsCondKernel.apply_of_ne_zero_of_measurableSet [MeasurableSingletonClass α] {x : α}
    (hx : ρ.fst {x} ≠ 0) {s : Set Ω} (hs : MeasurableSet s) :
    ρCond x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ] (ρCond : ProbabilityTheory.Kernel α Ω) [ρ.IsCondKernel ρCond] [MeasurableSingletonClass α] {x : α} (hx : ρ.fst {x} ≠ 0) {s : Set Ω} (hs : MeasurableSet s)
Goal: (ρCond x) s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s)
-- MeasureTheory.Measure.IsCondKernel.apply_of_ne_zero
lemma IsCondKernel.apply_of_ne_zero [MeasurableSingletonClass α] {x : α}
    (hx : ρ.fst {x} ≠ 0) (s : Set Ω) : ρCond x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ] (ρCond : ProbabilityTheory.Kernel α Ω) [ρ.IsCondKernel ρCond] [MeasurableSingletonClass α] {x : α} (hx : ρ.fst {x} ≠ 0) (s : Set Ω)
Goal: (ρCond x) s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s)
-- MeasureTheory.Measure.IsCondKernel.isProbabilityMeasure
lemma IsCondKernel.isProbabilityMeasure [MeasurableSingletonClass α] {a : α} (ha : ρ.fst {a} ≠ 0) :
    IsProbabilityMeasure (ρCond a) 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ] (ρCond : ProbabilityTheory.Kernel α Ω) [ρ.IsCondKernel ρCond] [MeasurableSingletonClass α] {a : α} (ha : ρ.fst {a} ≠ 0)
Goal: MeasureTheory.IsProbabilityMeasure (ρCond a)
-- MeasureTheory.Measure.IsCondKernel.isMarkovKernel
lemma IsCondKernel.isMarkovKernel [MeasurableSingletonClass α] (hρ : ∀ a, ρ.fst {a} ≠ 0) :
    IsMarkovKernel ρCond 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ] (ρCond : ProbabilityTheory.Kernel α Ω) [ρ.IsCondKernel ρCond] [MeasurableSingletonClass α] (hρ : ∀ (a : α), ρ.fst {a} ≠ 0)
Goal: ProbabilityTheory.IsMarkovKernel ρCond
-- ProbabilityTheory.Kernel.IsCondKernel.disintegrate
class IsCondKernel : Prop where
  protected disintegrate : κ.fst ⊗ₖ κCond = κ
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α (β × Ω)} {κCond : ProbabilityTheory.Kernel (α × β) Ω} [self : κ.IsCondKernel κCond]
Goal: κ.fst ⊗ₖ κCond = κ
-- ProbabilityTheory.Kernel.disintegrate
lemma disintegrate : κ.fst ⊗ₖ κCond = κ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mΩ : MeasurableSpace Ω} (κ : ProbabilityTheory.Kernel α (β × Ω)) (κCond : ProbabilityTheory.Kernel (α × β) Ω) [κ.IsCondKernel κCond]
Goal: κ.fst ⊗ₖ κCond = κ
-- ProbabilityTheory.Kernel.condKernelCountable_apply
lemma condKernelCountable_apply (h_atom : ∀ x y, x ∈ measurableAtom y → κCond x = κCond y)
    (p : α × β) : condKernelCountable κCond h_atom p = κCond p.1 p.2 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mΩ : MeasurableSpace Ω} [Countable α] (κCond : α → ProbabilityTheory.Kernel β Ω) (h_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y) (p : α × β)
Goal: (ProbabilityTheory.Kernel.condKernelCountable κCond h_atom) p = (κCond p.1) p.2
-- ProbabilityTheory.IdentDistrib.map_eq
structure IdentDistrib (f : α → γ) (g : β → γ)
    (μ : Measure α := by volume_tac)
    (ν : Measure β := by volume_tac) : Prop where
  aemeasurable_fst : AEMeasurable f μ
  aemeasurable_snd : AEMeasurable g ν
  map_eq : Measure.map f μ = Measure.map g ν
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : α → γ} {g : β → γ} {μ : autoParam (MeasureTheory.Measure α) _auto✝} {ν : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map g ν
-- ProbabilityTheory.IdentDistrib.aemeasurable_fst
structure IdentDistrib (f : α → γ) (g : β → γ)
    (μ : Measure α := by volume_tac)
    (ν : Measure β := by volume_tac) : Prop where
  aemeasurable_fst : AEMeasurable f μ
  aemeasurable_snd : AEMeasurable g ν
  map_eq : Measure.map f μ = Measure.map g ν
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : α → γ} {g : β → γ} {μ : autoParam (MeasureTheory.Measure α) _auto✝} {ν : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: AEMeasurable f μ
-- ProbabilityTheory.IdentDistrib.aemeasurable_snd
structure IdentDistrib (f : α → γ) (g : β → γ)
    (μ : Measure α := by volume_tac)
    (ν : Measure β := by volume_tac) : Prop where
  aemeasurable_fst : AEMeasurable f μ
  aemeasurable_snd : AEMeasurable g ν
  map_eq : Measure.map f μ = Measure.map g ν
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : α → γ} {g : β → γ} {μ : autoParam (MeasureTheory.Measure α) _auto✝} {ν : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: AEMeasurable g ν
-- ProbabilityTheory.IdentDistrib.refl
protected theorem refl (hf : AEMeasurable f μ) : IdentDistrib f f μ μ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → γ} (hf : AEMeasurable f μ)
Goal: ProbabilityTheory.IdentDistrib f f μ μ
-- ProbabilityTheory.IdentDistrib.symm
protected theorem symm (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib g f ν μ
-- ProbabilityTheory.IdentDistrib.trans
protected theorem trans {ρ : Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν)
    (h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {ρ : MeasureTheory.Measure δ} {h : δ → γ} (h₁ : ProbabilityTheory.IdentDistrib f g μ ν) (h₂ : ProbabilityTheory.IdentDistrib g h ν ρ)
Goal: ProbabilityTheory.IdentDistrib f h μ ρ
-- ProbabilityTheory.IdentDistrib.comp_of_aemeasurable
protected theorem comp_of_aemeasurable {u : γ → δ} (h : IdentDistrib f g μ ν)
    (hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {u : γ → δ} (h : ProbabilityTheory.IdentDistrib f g μ ν) (hu : AEMeasurable u (MeasureTheory.Measure.map f μ))
Goal: ProbabilityTheory.IdentDistrib (u ∘ f) (u ∘ g) μ ν
-- ProbabilityTheory.IdentDistrib.comp
protected theorem comp {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) :
    IdentDistrib (u ∘ f) (u ∘ g) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {u : γ → δ} (h : ProbabilityTheory.IdentDistrib f g μ ν) (hu : Measurable u)
Goal: ProbabilityTheory.IdentDistrib (u ∘ f) (u ∘ g) μ ν
-- ProbabilityTheory.IdentDistrib.of_ae_eq
protected theorem of_ae_eq {g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) :
    IdentDistrib f g μ μ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g)
Goal: ProbabilityTheory.IdentDistrib f g μ μ
-- MeasureTheory.AEMeasurable.identDistrib_mk
lemma _root_.MeasureTheory.AEMeasurable.identDistrib_mk
    (hf : AEMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → γ} (hf : AEMeasurable f μ)
Goal: ProbabilityTheory.IdentDistrib f (AEMeasurable.mk f hf) μ μ
-- MeasureTheory.AEStronglyMeasurable.identDistrib_mk
lemma _root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk
    [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]
    (hf : AEStronglyMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → γ} [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ProbabilityTheory.IdentDistrib f (MeasureTheory.AEStronglyMeasurable.mk f hf) μ μ
-- ProbabilityTheory.IdentDistrib.measure_mem_eq
theorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :
    μ (f ⁻¹' s) = ν (g ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : ProbabilityTheory.IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s)
Goal: μ (f ⁻¹' s) = ν (g ⁻¹' s)
-- ProbabilityTheory.IdentDistrib.ae_snd
theorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x})
    (hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : ProbabilityTheory.IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x}) (hp : ∀ᵐ (x : α) ∂μ, p (f x))
Goal: ∀ᵐ (x : β) ∂ν, p (g x)
-- ProbabilityTheory.IdentDistrib.ae_mem_snd
theorem ae_mem_snd (h : IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t)
    (ht : ∀ᵐ x ∂μ, f x ∈ t) : ∀ᵐ x ∂ν, g x ∈ t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : ProbabilityTheory.IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t) (ht : ∀ᵐ (x : α) ∂μ, f x ∈ t)
Goal: ∀ᵐ (x : β) ∂ν, g x ∈ t
-- ProbabilityTheory.IdentDistrib.aestronglyMeasurable_fst
theorem aestronglyMeasurable_fst [TopologicalSpace γ] [MetrizableSpace γ] [OpensMeasurableSpace γ]
    [SecondCountableTopology γ] (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [TopologicalSpace γ] [TopologicalSpace.MetrizableSpace γ] [OpensMeasurableSpace γ] [SecondCountableTopology γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- ProbabilityTheory.IdentDistrib.aestronglyMeasurable_snd
theorem aestronglyMeasurable_snd [TopologicalSpace γ] [MetrizableSpace γ] [BorelSpace γ]
    (h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [TopologicalSpace γ] [TopologicalSpace.MetrizableSpace γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable g ν
-- ProbabilityTheory.IdentDistrib.aestronglyMeasurable_iff
theorem aestronglyMeasurable_iff [TopologicalSpace γ] [MetrizableSpace γ] [BorelSpace γ]
    (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [TopologicalSpace γ] [TopologicalSpace.MetrizableSpace γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: MeasureTheory.AEStronglyMeasurable f μ ↔ MeasureTheory.AEStronglyMeasurable g ν
-- ProbabilityTheory.IdentDistrib.essSup_eq
theorem essSup_eq [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ]
    [OrderClosedTopology γ] (h : IdentDistrib f g μ ν) : essSup f μ = essSup g ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ] [OrderClosedTopology γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: essSup f μ = essSup g ν
-- ProbabilityTheory.IdentDistrib.lintegral_eq
theorem lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) :
    ∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : β), g x ∂ν
-- ProbabilityTheory.IdentDistrib.integral_eq
theorem integral_eq [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ]
    (h : IdentDistrib f g μ ν) : ∫ x, f x ∂μ = ∫ x, g x ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ∫ (x : α), f x ∂μ = ∫ (x : β), g x ∂ν
-- ProbabilityTheory.IdentDistrib.eLpNorm_eq
theorem eLpNorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν)
    (p : ℝ≥0∞) : eLpNorm f p μ = eLpNorm g p ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f p μ = MeasureTheory.eLpNorm g p ν
-- ProbabilityTheory.IdentDistrib.memℒp_snd
theorem memℒp_snd [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : IdentDistrib f g μ ν)
    (hf : Memℒp f p μ) : Memℒp g p ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : ProbabilityTheory.IdentDistrib f g μ ν) (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp g p ν
-- ProbabilityTheory.IdentDistrib.memℒp_iff
theorem memℒp_iff [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : IdentDistrib f g μ ν) :
    Memℒp f p μ ↔ Memℒp g p ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: MeasureTheory.Memℒp f p μ ↔ MeasureTheory.Memℒp g p ν
-- ProbabilityTheory.IdentDistrib.integrable_snd
theorem integrable_snd [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν)
    (hf : Integrable f μ) : Integrable g ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable g ν
-- ProbabilityTheory.IdentDistrib.integrable_iff
theorem integrable_iff [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) :
    Integrable f μ ↔ Integrable g ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.Integrable g ν
-- ProbabilityTheory.IdentDistrib.norm
protected theorem norm [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) :
    IdentDistrib (fun x => ‖f x‖) (fun x => ‖g x‖) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib (fun x => ‖f x‖) (fun x => ‖g x‖) μ ν
-- ProbabilityTheory.IdentDistrib.nnnorm
protected theorem nnnorm [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) :
    IdentDistrib (fun x => ‖f x‖₊) (fun x => ‖g x‖₊) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib (fun x => ‖f x‖₊) (fun x => ‖g x‖₊) μ ν
-- ProbabilityTheory.IdentDistrib.pow
protected theorem pow [Pow γ ℕ] [MeasurablePow γ ℕ] (h : IdentDistrib f g μ ν) {n : ℕ} :
    IdentDistrib (fun x => f x ^ n) (fun x => g x ^ n) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Pow γ ℕ] [MeasurablePow γ ℕ] (h : ProbabilityTheory.IdentDistrib f g μ ν) {n : ℕ}
Goal: ProbabilityTheory.IdentDistrib (fun x => f x ^ n) (fun x => g x ^ n) μ ν
-- ProbabilityTheory.IdentDistrib.sq
protected theorem sq [Pow γ ℕ] [MeasurablePow γ ℕ] (h : IdentDistrib f g μ ν) :
    IdentDistrib (fun x => f x ^ 2) (fun x => g x ^ 2) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Pow γ ℕ] [MeasurablePow γ ℕ] (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib (fun x => f x ^ 2) (fun x => g x ^ 2) μ ν
-- ProbabilityTheory.IdentDistrib.coe_nnreal_ennreal
protected theorem coe_nnreal_ennreal {f : α → ℝ≥0} {g : β → ℝ≥0} (h : IdentDistrib f g μ ν) :
    IdentDistrib (fun x => (f x : ℝ≥0∞)) (fun x => (g x : ℝ≥0∞)) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ℝ≥0} {g : β → ℝ≥0} (h : ProbabilityTheory.IdentDistrib f g μ ν)
Goal: ProbabilityTheory.IdentDistrib (fun x => ↑(f x)) (fun x => ↑(g x)) μ ν
-- ProbabilityTheory.IdentDistrib.add_const
@[to_additive]
theorem mul_const [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => f x * c) (fun x => g x * c) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Add γ] [MeasurableAdd γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => f x + c) (fun x => g x + c) μ ν
-- ProbabilityTheory.IdentDistrib.mul_const
@[to_additive]
theorem mul_const [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => f x * c) (fun x => g x * c) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Mul γ] [MeasurableMul γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => f x * c) (fun x => g x * c) μ ν
-- ProbabilityTheory.IdentDistrib.const_add
@[to_additive]
theorem const_mul [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => c * f x) (fun x => c * g x) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Add γ] [MeasurableAdd γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => c + f x) (fun x => c + g x) μ ν
-- ProbabilityTheory.IdentDistrib.const_mul
@[to_additive]
theorem const_mul [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => c * f x) (fun x => c * g x) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Mul γ] [MeasurableMul γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => c * f x) (fun x => c * g x) μ ν
-- ProbabilityTheory.IdentDistrib.div_const
@[to_additive]
theorem div_const [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => f x / c) (fun x => g x / c) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Div γ] [MeasurableDiv γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => f x / c) (fun x => g x / c) μ ν
-- ProbabilityTheory.IdentDistrib.sub_const
@[to_additive]
theorem div_const [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => f x / c) (fun x => g x / c) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Sub γ] [MeasurableSub γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => f x - c) (fun x => g x - c) μ ν
-- ProbabilityTheory.IdentDistrib.const_div
@[to_additive]
theorem const_div [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) :
    IdentDistrib (fun x => c / f x) (fun x => c / g x) μ ν 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Div γ] [MeasurableDiv γ] (h : ProbabilityTheory.IdentDistrib f g μ ν) (c : γ)
Goal: ProbabilityTheory.IdentDistrib (fun x => c / f x) (fun x => c / g x) μ ν