ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.monotone_cdf
lemma monotone_cdf : Monotone (cdf μ) 
Arguments: (μ : MeasureTheory.Measure ℝ)
Goal: Monotone ↑(ProbabilityTheory.cdf μ)
-- ProbabilityTheory.tendsto_cdf_atBot
lemma tendsto_cdf_atBot : Tendsto (cdf μ) atBot (𝓝 0) 
Arguments: (μ : MeasureTheory.Measure ℝ)
Goal: Filter.Tendsto (↑(ProbabilityTheory.cdf μ)) Filter.atBot (𝓝 0)
-- ProbabilityTheory.tendsto_cdf_atTop
lemma tendsto_cdf_atTop : Tendsto (cdf μ) atTop (𝓝 1) 
Arguments: (μ : MeasureTheory.Measure ℝ)
Goal: Filter.Tendsto (↑(ProbabilityTheory.cdf μ)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.ofReal_cdf
lemma ofReal_cdf [IsProbabilityMeasure μ] (x : ℝ) : ENNReal.ofReal (cdf μ x) = μ (Iic x) 
Arguments: (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] (x : ℝ)
Goal: ENNReal.ofReal (↑(ProbabilityTheory.cdf μ) x) = μ (Set.Iic x)
-- ProbabilityTheory.cdf_eq_toReal
lemma cdf_eq_toReal [IsProbabilityMeasure μ] (x : ℝ) : cdf μ x = (μ (Iic x)).toReal 
Arguments: (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] (x : ℝ)
Goal: ↑(ProbabilityTheory.cdf μ) x = (μ (Set.Iic x)).toReal
-- ProbabilityTheory.measure_cdf
lemma measure_cdf [IsProbabilityMeasure μ] : (cdf μ).measure = μ 
Arguments: (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ]
Goal: (ProbabilityTheory.cdf μ).measure = μ
-- ProbabilityTheory.cdf_measure_stieltjesFunction
lemma cdf_measure_stieltjesFunction (f : StieltjesFunction) (hf0 : Tendsto f atBot (𝓝 0))
    (hf1 : Tendsto f atTop (𝓝 1)) :
    cdf f.measure = f 
Arguments: (f : StieltjesFunction) (hf0 : Filter.Tendsto (↑f) Filter.atBot (𝓝 0)) (hf1 : Filter.Tendsto (↑f) Filter.atTop (𝓝 1))
Goal: ProbabilityTheory.cdf f.measure = f
-- MeasureTheory.Measure.eq_of_cdf
lemma MeasureTheory.Measure.eq_of_cdf (μ ν : Measure ℝ) [IsProbabilityMeasure μ]
    [IsProbabilityMeasure ν] (h : cdf μ = cdf ν) : μ = ν 
Arguments: (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (h : ProbabilityTheory.cdf μ = ProbabilityTheory.cdf ν)
Goal: μ = ν
-- MeasureTheory.Measure.cdf_eq_iff
@[simp] lemma MeasureTheory.Measure.cdf_eq_iff (μ ν : Measure ℝ) [IsProbabilityMeasure μ]
    [IsProbabilityMeasure ν] :
    cdf μ = cdf ν ↔ μ = ν 
Arguments: (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν]
Goal: ProbabilityTheory.cdf μ = ProbabilityTheory.cdf ν ↔ μ = ν
-- MeasureTheory.Adapted.mul
@[to_additive]
protected theorem mul [Mul β] [ContinuousMul β] (hu : Adapted f u) (hv : Adapted f v) :
    Adapted f (u * v) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [Mul β] [ContinuousMul β] (hu : MeasureTheory.Adapted f u) (hv : MeasureTheory.Adapted f v)
Goal: MeasureTheory.Adapted f (u * v)
-- MeasureTheory.Adapted.add
@[to_additive]
protected theorem mul [Mul β] [ContinuousMul β] (hu : Adapted f u) (hv : Adapted f v) :
    Adapted f (u * v) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [Add β] [ContinuousAdd β] (hu : MeasureTheory.Adapted f u) (hv : MeasureTheory.Adapted f v)
Goal: MeasureTheory.Adapted f (u + v)
-- MeasureTheory.Adapted.div
@[to_additive]
protected theorem div [Div β] [ContinuousDiv β] (hu : Adapted f u) (hv : Adapted f v) :
    Adapted f (u / v) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [Div β] [ContinuousDiv β] (hu : MeasureTheory.Adapted f u) (hv : MeasureTheory.Adapted f v)
Goal: MeasureTheory.Adapted f (u / v)
-- MeasureTheory.Adapted.sub
@[to_additive]
protected theorem div [Div β] [ContinuousDiv β] (hu : Adapted f u) (hv : Adapted f v) :
    Adapted f (u / v) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [Sub β] [ContinuousSub β] (hu : MeasureTheory.Adapted f u) (hv : MeasureTheory.Adapted f v)
Goal: MeasureTheory.Adapted f (u - v)
-- MeasureTheory.Adapted.inv
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hu : Adapted f u) :
    Adapted f u⁻¹ 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [Group β] [TopologicalGroup β] (hu : MeasureTheory.Adapted f u)
Goal: MeasureTheory.Adapted f u⁻¹
-- MeasureTheory.Adapted.neg
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hu : Adapted f u) :
    Adapted f u⁻¹ 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [AddGroup β] [TopologicalAddGroup β] (hu : MeasureTheory.Adapted f u)
Goal: MeasureTheory.Adapted f (-u)
-- MeasureTheory.Adapted.smul
protected theorem smul [SMul ℝ β] [ContinuousSMul ℝ β] (c : ℝ) (hu : Adapted f u) :
    Adapted f (c • u) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [SMul ℝ β] [ContinuousSMul ℝ β] (c : ℝ) (hu : MeasureTheory.Adapted f u)
Goal: MeasureTheory.Adapted f (c • u)
-- MeasureTheory.Adapted.stronglyMeasurable
protected theorem stronglyMeasurable {i : ι} (hf : Adapted f u) : StronglyMeasurable[m] (u i) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} {i : ι} (hf : MeasureTheory.Adapted f u)
Goal: MeasureTheory.StronglyMeasurable (u i)
-- MeasureTheory.Adapted.stronglyMeasurable_le
theorem stronglyMeasurable_le {i j : ι} (hf : Adapted f u) (hij : i ≤ j) :
    StronglyMeasurable[f j] (u i) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} {i j : ι} (hf : MeasureTheory.Adapted f u) (hij : i ≤ j)
Goal: MeasureTheory.StronglyMeasurable (u i)
-- MeasureTheory.adapted_const
theorem adapted_const (f : Filtration ι m) (x : β) : Adapted f fun _ _ => x 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] (f : MeasureTheory.Filtration ι m) (x : β)
Goal: MeasureTheory.Adapted f fun x_1 x_2 => x
-- MeasureTheory.adapted_zero
theorem adapted_zero [Zero β] (f : Filtration ι m) : Adapted f (0 : ι → Ω → β) 
Arguments: {Ω : Type u_1} (β : Type u_2) {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] [Zero β] (f : MeasureTheory.Filtration ι m)
Goal: MeasureTheory.Adapted f 0
-- MeasureTheory.Filtration.adapted_natural
theorem Filtration.adapted_natural [MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β]
    {u : ι → Ω → β} (hum : ∀ i, StronglyMeasurable[m] (u i)) :
    Adapted (Filtration.natural u hum) u 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] [TopologicalSpace.MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β] {u : ι → Ω → β} (hum : ∀ (i : ι), MeasureTheory.StronglyMeasurable (u i))
Goal: MeasureTheory.Adapted (MeasureTheory.Filtration.natural u hum) u
-- MeasureTheory.progMeasurable_const
theorem progMeasurable_const [MeasurableSpace ι] (f : Filtration ι m) (b : β) :
    ProgMeasurable f (fun _ _ => b : ι → Ω → β) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] [MeasurableSpace ι] (f : MeasureTheory.Filtration ι m) (b : β)
Goal: MeasureTheory.ProgMeasurable f fun x x => b
-- MeasureTheory.ProgMeasurable.adapted
protected theorem adapted (h : ProgMeasurable f u) : Adapted f u 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] (h : MeasureTheory.ProgMeasurable f u)
Goal: MeasureTheory.Adapted f u
-- MeasureTheory.ProgMeasurable.comp
protected theorem comp {t : ι → Ω → ι} [TopologicalSpace ι] [BorelSpace ι] [MetrizableSpace ι]
    (h : ProgMeasurable f u) (ht : ProgMeasurable f t) (ht_le : ∀ i ω, t i ω ≤ i) :
    ProgMeasurable f fun i ω => u (t i ω) ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] {t : ι → Ω → ι} [TopologicalSpace ι] [BorelSpace ι] [TopologicalSpace.MetrizableSpace ι] (h : MeasureTheory.ProgMeasurable f u) (ht : MeasureTheory.ProgMeasurable f t) (ht_le : ∀ (i : ι) (ω : Ω), t i ω ≤ i)
Goal: MeasureTheory.ProgMeasurable f fun i ω => u (t i ω) ω
-- MeasureTheory.ProgMeasurable.add
@[to_additive]
protected theorem mul [Mul β] [ContinuousMul β] (hu : ProgMeasurable f u)
    (hv : ProgMeasurable f v) : ProgMeasurable f fun i ω => u i ω * v i ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [Add β] [ContinuousAdd β] (hu : MeasureTheory.ProgMeasurable f u) (hv : MeasureTheory.ProgMeasurable f v)
Goal: MeasureTheory.ProgMeasurable f fun i ω => u i ω + v i ω
-- MeasureTheory.ProgMeasurable.mul
@[to_additive]
protected theorem mul [Mul β] [ContinuousMul β] (hu : ProgMeasurable f u)
    (hv : ProgMeasurable f v) : ProgMeasurable f fun i ω => u i ω * v i ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [Mul β] [ContinuousMul β] (hu : MeasureTheory.ProgMeasurable f u) (hv : MeasureTheory.ProgMeasurable f v)
Goal: MeasureTheory.ProgMeasurable f fun i ω => u i ω * v i ω
-- MeasureTheory.ProgMeasurable.finset_prod'
@[to_additive]
protected theorem finset_prod' {γ} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β}
    {s : Finset γ} (h : ∀ c ∈ s, ProgMeasurable f (U c)) : ProgMeasurable f (∏ c ∈ s, U c) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] {γ : Type u_4} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β} {s : Finset γ} (h : ∀ c ∈ s, MeasureTheory.ProgMeasurable f (U c))
Goal: MeasureTheory.ProgMeasurable f (∏ c ∈ s, U c)
-- MeasureTheory.ProgMeasurable.finset_sum'
@[to_additive]
protected theorem finset_prod' {γ} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β}
    {s : Finset γ} (h : ∀ c ∈ s, ProgMeasurable f (U c)) : ProgMeasurable f (∏ c ∈ s, U c) 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] {γ : Type u_4} [AddCommMonoid β] [ContinuousAdd β] {U : γ → ι → Ω → β} {s : Finset γ} (h : ∀ c ∈ s, MeasureTheory.ProgMeasurable f (U c))
Goal: MeasureTheory.ProgMeasurable f (∑ c ∈ s, U c)
-- MeasureTheory.ProgMeasurable.finset_prod
@[to_additive]
protected theorem finset_prod {γ} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β}
    {s : Finset γ} (h : ∀ c ∈ s, ProgMeasurable f (U c)) :
    ProgMeasurable f fun i a => ∏ c ∈ s, U c i a 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] {γ : Type u_4} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β} {s : Finset γ} (h : ∀ c ∈ s, MeasureTheory.ProgMeasurable f (U c))
Goal: MeasureTheory.ProgMeasurable f fun i a => ∏ c ∈ s, U c i a
-- MeasureTheory.ProgMeasurable.finset_sum
@[to_additive]
protected theorem finset_prod {γ} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β}
    {s : Finset γ} (h : ∀ c ∈ s, ProgMeasurable f (U c)) :
    ProgMeasurable f fun i a => ∏ c ∈ s, U c i a 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] {γ : Type u_4} [AddCommMonoid β] [ContinuousAdd β] {U : γ → ι → Ω → β} {s : Finset γ} (h : ∀ c ∈ s, MeasureTheory.ProgMeasurable f (U c))
Goal: MeasureTheory.ProgMeasurable f fun i a => ∑ c ∈ s, U c i a
-- MeasureTheory.ProgMeasurable.neg
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hu : ProgMeasurable f u) :
    ProgMeasurable f fun i ω => (u i ω)⁻¹ 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [AddGroup β] [TopologicalAddGroup β] (hu : MeasureTheory.ProgMeasurable f u)
Goal: MeasureTheory.ProgMeasurable f fun i ω => -u i ω
-- MeasureTheory.ProgMeasurable.inv
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hu : ProgMeasurable f u) :
    ProgMeasurable f fun i ω => (u i ω)⁻¹ 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [Group β] [TopologicalGroup β] (hu : MeasureTheory.ProgMeasurable f u)
Goal: MeasureTheory.ProgMeasurable f fun i ω => (u i ω)⁻¹
-- MeasureTheory.ProgMeasurable.sub
@[to_additive]
protected theorem div [Group β] [TopologicalGroup β] (hu : ProgMeasurable f u)
    (hv : ProgMeasurable f v) : ProgMeasurable f fun i ω => u i ω / v i ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [AddGroup β] [TopologicalAddGroup β] (hu : MeasureTheory.ProgMeasurable f u) (hv : MeasureTheory.ProgMeasurable f v)
Goal: MeasureTheory.ProgMeasurable f fun i ω => u i ω - v i ω
-- MeasureTheory.ProgMeasurable.div
@[to_additive]
protected theorem div [Group β] [TopologicalGroup β] (hu : ProgMeasurable f u)
    (hv : ProgMeasurable f v) : ProgMeasurable f fun i ω => u i ω / v i ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u v : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [Group β] [TopologicalGroup β] (hu : MeasureTheory.ProgMeasurable f u) (hv : MeasureTheory.ProgMeasurable f v)
Goal: MeasureTheory.ProgMeasurable f fun i ω => u i ω / v i ω
-- MeasureTheory.progMeasurable_of_tendsto'
theorem progMeasurable_of_tendsto' {γ} [MeasurableSpace ι] [PseudoMetrizableSpace β]
    (fltr : Filter γ) [fltr.NeBot] [fltr.IsCountablyGenerated] {U : γ → ι → Ω → β}
    (h : ∀ l, ProgMeasurable f (U l)) (h_tendsto : Tendsto U fltr (𝓝 u)) : ProgMeasurable f u 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} {γ : Type u_4} [MeasurableSpace ι] [TopologicalSpace.PseudoMetrizableSpace β] (fltr : Filter γ) [fltr.NeBot] [fltr.IsCountablyGenerated] {U : γ → ι → Ω → β} (h : ∀ (l : γ), MeasureTheory.ProgMeasurable f (U l)) (h_tendsto : Filter.Tendsto U fltr (𝓝 u))
Goal: MeasureTheory.ProgMeasurable f u
-- MeasureTheory.progMeasurable_of_tendsto
theorem progMeasurable_of_tendsto [MeasurableSpace ι] [PseudoMetrizableSpace β] {U : ℕ → ι → Ω → β}
    (h : ∀ l, ProgMeasurable f (U l)) (h_tendsto : Tendsto U atTop (𝓝 u)) : ProgMeasurable f u 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [MeasurableSpace ι] [TopologicalSpace.PseudoMetrizableSpace β] {U : ℕ → ι → Ω → β} (h : ∀ (l : ℕ), MeasureTheory.ProgMeasurable f (U l)) (h_tendsto : Filter.Tendsto U Filter.atTop (𝓝 u))
Goal: MeasureTheory.ProgMeasurable f u
-- MeasureTheory.Adapted.progMeasurable_of_continuous
theorem Adapted.progMeasurable_of_continuous [TopologicalSpace ι] [MetrizableSpace ι]
    [SecondCountableTopology ι] [MeasurableSpace ι] [OpensMeasurableSpace ι]
    [PseudoMetrizableSpace β] (h : Adapted f u) (hu_cont : ∀ ω, Continuous fun i => u i ω) :
    ProgMeasurable f u 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [TopologicalSpace ι] [TopologicalSpace.MetrizableSpace ι] [SecondCountableTopology ι] [MeasurableSpace ι] [OpensMeasurableSpace ι] [TopologicalSpace.PseudoMetrizableSpace β] (h : MeasureTheory.Adapted f u) (hu_cont : ∀ (ω : Ω), Continuous fun i => u i ω)
Goal: MeasureTheory.ProgMeasurable f u
-- MeasureTheory.Adapted.progMeasurable_of_discrete
theorem Adapted.progMeasurable_of_discrete [TopologicalSpace ι] [DiscreteTopology ι]
    [SecondCountableTopology ι] [MeasurableSpace ι] [OpensMeasurableSpace ι]
    [PseudoMetrizableSpace β] (h : Adapted f u) : ProgMeasurable f u 
Arguments: {Ω : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace Ω} [TopologicalSpace β] [Preorder ι] {u : ι → Ω → β} {f : MeasureTheory.Filtration ι m} [TopologicalSpace ι] [DiscreteTopology ι] [SecondCountableTopology ι] [MeasurableSpace ι] [OpensMeasurableSpace ι] [TopologicalSpace.PseudoMetrizableSpace β] (h : MeasureTheory.Adapted f u)
Goal: MeasureTheory.ProgMeasurable f u
-- MeasureTheory.Predictable.adapted
theorem Predictable.adapted {f : Filtration ℕ m} {u : ℕ → Ω → β} (hu : Adapted f fun n => u (n + 1))
    (hu0 : StronglyMeasurable[f 0] (u 0)) : Adapted f u 
Arguments: {Ω : Type u_1} {β : Type u_2} {m : MeasurableSpace Ω} [TopologicalSpace β] {f : MeasureTheory.Filtration ℕ m} {u : ℕ → Ω → β} (hu : MeasureTheory.Adapted f fun n => u (n + 1)) (hu0 : MeasureTheory.StronglyMeasurable (u 0))
Goal: MeasureTheory.Adapted f u
-- ProbabilityTheory.Kernel.rnDerivAux_nonneg
lemma rnDerivAux_nonneg (hκη : κ ≤ η) {a : α} {x : γ} : 0 ≤ rnDerivAux κ η a x 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} {κ η : ProbabilityTheory.Kernel α γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (hκη : κ ≤ η) {a : α} {x : γ}
Goal: 0 ≤ κ.rnDerivAux η a x
-- ProbabilityTheory.Kernel.rnDerivAux_le_one
lemma rnDerivAux_le_one [IsFiniteKernel η] (hκη : κ ≤ η) {a : α} :
    rnDerivAux κ η a ≤ᵐ[η a] 1 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} {κ η : ProbabilityTheory.Kernel α γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] [ProbabilityTheory.IsFiniteKernel η] (hκη : κ ≤ η) {a : α}
Goal: κ.rnDerivAux η a ≤ᵐ[η a] 1
-- ProbabilityTheory.Kernel.measurable_rnDerivAux
lemma measurable_rnDerivAux (κ η : Kernel α γ) :
    Measurable (fun p : α × γ ↦ Kernel.rnDerivAux κ η p.1 p.2) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ)
Goal: Measurable fun p => κ.rnDerivAux η p.1 p.2
-- ProbabilityTheory.Kernel.measurable_rnDerivAux_right
lemma measurable_rnDerivAux_right (κ η : Kernel α γ) (a : α) :
    Measurable (fun x : γ ↦ rnDerivAux κ η a x) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α)
Goal: Measurable fun x => κ.rnDerivAux η a x
-- ProbabilityTheory.Kernel.setLIntegral_rnDerivAux
lemma setLIntegral_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η]
    (a : α) {s : Set γ} (hs : MeasurableSet s) :
    ∫⁻ x in s, ENNReal.ofReal (rnDerivAux κ (κ + η) a x) ∂(κ + η) a = κ a s 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α) {s : Set γ} (hs : MeasurableSet s)
Goal: ∫⁻ (x : γ) in s, ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) ∂(κ + η) a = (κ a) s
-- ProbabilityTheory.Kernel.withDensity_rnDerivAux
lemma withDensity_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
    withDensity (κ + η) (fun a x ↦ Real.toNNReal (rnDerivAux κ (κ + η) a x)) = κ 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η]
Goal: ((κ + η).withDensity fun a x => ↑(κ.rnDerivAux (κ + η) a x).toNNReal) = κ
-- ProbabilityTheory.Kernel.withDensity_one_sub_rnDerivAux
lemma withDensity_one_sub_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
    withDensity (κ + η) (fun a x ↦ Real.toNNReal (1 - rnDerivAux κ (κ + η) a x)) = η 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η]
Goal: ((κ + η).withDensity fun a x => ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal) = η
-- ProbabilityTheory.Kernel.mem_mutuallySingularSetSlice
lemma mem_mutuallySingularSetSlice (κ η : Kernel α γ) (a : α) (x : γ) :
    x ∈ mutuallySingularSetSlice κ η a ↔ 1 ≤ rnDerivAux κ (κ + η) a x 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α) (x : γ)
Goal: x ∈ κ.mutuallySingularSetSlice η a ↔ 1 ≤ κ.rnDerivAux (κ + η) a x
-- ProbabilityTheory.Kernel.not_mem_mutuallySingularSetSlice
lemma not_mem_mutuallySingularSetSlice (κ η : Kernel α γ) (a : α) (x : γ) :
    x ∉ mutuallySingularSetSlice κ η a ↔ rnDerivAux κ (κ + η) a x < 1 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α) (x : γ)
Goal: x ∉ κ.mutuallySingularSetSlice η a ↔ κ.rnDerivAux (κ + η) a x < 1
-- ProbabilityTheory.Kernel.measurableSet_mutuallySingularSet
lemma measurableSet_mutuallySingularSet (κ η : Kernel α γ) :
    MeasurableSet (mutuallySingularSet κ η) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ)
Goal: MeasurableSet (κ.mutuallySingularSet η)
-- ProbabilityTheory.Kernel.measurableSet_mutuallySingularSetSlice
lemma measurableSet_mutuallySingularSetSlice (κ η : Kernel α γ) (a : α) :
    MeasurableSet (mutuallySingularSetSlice κ η a) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α)
Goal: MeasurableSet (κ.mutuallySingularSetSlice η a)
-- ProbabilityTheory.Kernel.measure_mutuallySingularSetSlice
lemma measure_mutuallySingularSetSlice (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η]
    (a : α) :
    η a (mutuallySingularSetSlice κ η a) = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (η a) (κ.mutuallySingularSetSlice η a) = 0
-- ProbabilityTheory.Kernel.rnDeriv_def
noncomputable
irreducible_def rnDeriv (κ η : Kernel α γ) (a : α) (x : γ) : ℝ≥0∞ 
Arguments: {α : Type u_3} {γ : Type u_4} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α) (x : γ)
Goal: κ.rnDeriv η a x = ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) / ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x)