ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.CondIndepFun.comp
theorem CondIndepFun.comp {γ γ' : Type*} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'}
    {_mγ : MeasurableSpace γ} {_mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'}
    (hfg : CondIndepFun m' hm' f g μ) (hφ : Measurable φ) (hψ : Measurable ψ) :
    CondIndepFun m' hm' (φ ∘ f) (ψ ∘ g) μ 
Arguments: {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'} {γ : Type u_5} {γ' : Type u_6} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} {_mγ : MeasurableSpace γ} {_mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'} (hfg : ProbabilityTheory.CondIndepFun m' hm' f g μ) (hφ : Measurable φ) (hψ : Measurable ψ)
Goal: ProbabilityTheory.CondIndepFun m' hm' (φ ∘ f) (ψ ∘ g) μ
-- ProbabilityTheory.CondIndepFun.neg_right
theorem CondIndepFun.neg_right {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β']
    [MeasurableNeg β'] (hfg : CondIndepFun m' hm' f g μ)  :
    CondIndepFun m' hm' f (-g) μ 
Arguments: {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β'] [MeasurableNeg β'] (hfg : ProbabilityTheory.CondIndepFun m' hm' f g μ)
Goal: ProbabilityTheory.CondIndepFun m' hm' f (-g) μ
-- ProbabilityTheory.CondIndepFun.neg_left
theorem CondIndepFun.neg_left {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β]
    [MeasurableNeg β] (hfg : CondIndepFun m' hm' f g μ) :
    CondIndepFun m' hm' (-f) g μ 
Arguments: {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β] [MeasurableNeg β] (hfg : ProbabilityTheory.CondIndepFun m' hm' f g μ)
Goal: ProbabilityTheory.CondIndepFun m' hm' (-f) g μ
-- ProbabilityTheory.iCondIndepFun.of_subsingleton
@[nontriviality]
lemma iCondIndepFun.of_subsingleton [Subsingleton ι] : iCondIndepFun m' hm' m f μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_5} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} [Subsingleton ι]
Goal: ProbabilityTheory.iCondIndepFun m' hm' m f μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_finset
theorem iCondIndepFun.condIndepFun_finset {β : ι → Type*}
    {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (S T : Finset ι) (hST : Disjoint S T)
    (hf_Indep : iCondIndepFun m' hm' m f μ) (hf_meas : ∀ i, Measurable (f i)) :
    CondIndepFun m' hm' (fun a (i : S) => f i a) (fun a (i : T) => f i a) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_6} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (S T : Finset ι) (hST : Disjoint S T) (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' m f μ) (hf_meas : ∀ (i : ι), Measurable (f i))
Goal: ProbabilityTheory.CondIndepFun m' hm' (fun a i => f (↑i) a) (fun a i => f (↑i) a) μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_prod_mk
theorem iCondIndepFun.condIndepFun_prod_mk {β : ι → Type*}
    {m : ∀ i, MeasurableSpace (β i)} {f : ∀ i, Ω → β i} (hf_Indep : iCondIndepFun m' hm' m f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    CondIndepFun m' hm' (fun a => (f i a, f j a)) (f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_6} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' m f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (fun a => (f i a, f j a)) (f k) μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_prod_mk_prod_mk
lemma iCondIndepFun.condIndepFun_prod_mk_prod_mk (h_indep : iCondIndepFun m' hm' m f μ)
    (hf : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    CondIndepFun m' hm' (fun a ↦ (f i a, f j a)) (fun a ↦ (f k a, f l a)) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_5} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (h_indep : ProbabilityTheory.iCondIndepFun m' hm' m f μ) (hf : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.CondIndepFun m' hm' (fun a => (f i a, f j a)) (fun a => (f k a, f l a)) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_add_left
@[to_additive]
lemma iCondIndepFun.indepFun_mul_left (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    CondIndepFun m' hm' (f i * f j) (f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Add β] [MeasurableAdd₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i + f j) (f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_mul_left
@[to_additive]
lemma iCondIndepFun.indepFun_mul_left (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    CondIndepFun m' hm' (f i * f j) (f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i * f j) (f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_add_right
@[to_additive]
lemma iCondIndepFun.indepFun_mul_right (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    CondIndepFun m' hm' (f i) (f j * f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Add β] [MeasurableAdd₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i) (f j + f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_mul_right
@[to_additive]
lemma iCondIndepFun.indepFun_mul_right (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    CondIndepFun m' hm' (f i) (f j * f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i) (f j * f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_mul_mul
@[to_additive]
lemma iCondIndepFun.indepFun_mul_mul (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    CondIndepFun m' hm' (f i * f j) (f k * f l) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i * f j) (f k * f l) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_add_add
@[to_additive]
lemma iCondIndepFun.indepFun_mul_mul (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    CondIndepFun m' hm' (f i * f j) (f k * f l) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Add β] [MeasurableAdd₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i + f j) (f k + f l) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_sub_left
@[to_additive]
lemma iCondIndepFun.indepFun_div_left (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    CondIndepFun m' hm' (f i / f j) (f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Sub β] [MeasurableSub₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i - f j) (f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_div_left
@[to_additive]
lemma iCondIndepFun.indepFun_div_left (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    CondIndepFun m' hm' (f i / f j) (f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i / f j) (f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_div_right
@[to_additive]
lemma iCondIndepFun.indepFun_div_right (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    CondIndepFun m' hm' (f i) (f j / f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i) (f j / f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_sub_right
@[to_additive]
lemma iCondIndepFun.indepFun_div_right (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    CondIndepFun m' hm' (f i) (f j / f k) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Sub β] [MeasurableSub₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i) (f j - f k) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_sub_sub
@[to_additive]
lemma iCondIndepFun.indepFun_div_div (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    CondIndepFun m' hm' (f i / f j) (f k / f l) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Sub β] [MeasurableSub₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i - f j) (f k - f l) μ
-- ProbabilityTheory.iCondIndepFun.indepFun_div_div
@[to_additive]
lemma iCondIndepFun.indepFun_div_div (hf_indep : iCondIndepFun m' hm' (fun _ ↦ m) f μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    CondIndepFun m' hm' (f i / f j) (f k / f l) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β} (hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i / f j) (f k / f l) μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_finset_prod_of_not_mem
@[to_additive]
theorem iCondIndepFun.condIndepFun_finset_prod_of_not_mem
    (hf_Indep : iCondIndepFun m' hm' (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i))
    {s : Finset ι} {i : ι} (hi : i ∉ s) :
    CondIndepFun m' hm' (∏ j ∈ s, f j) (f i) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β} (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s)
Goal: ProbabilityTheory.CondIndepFun m' hm' (∏ j ∈ s, f j) (f i) μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_finset_sum_of_not_mem
@[to_additive]
theorem iCondIndepFun.condIndepFun_finset_prod_of_not_mem
    (hf_Indep : iCondIndepFun m' hm' (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i))
    {s : Finset ι} {i : ι} (hi : i ∉ s) :
    CondIndepFun m' hm' (∏ j ∈ s, f j) (f i) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [AddCommMonoid β] [MeasurableAdd₂ β] {f : ι → Ω → β} (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s)
Goal: ProbabilityTheory.CondIndepFun m' hm' (∑ j ∈ s, f j) (f i) μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_sum_range_succ
@[to_additive]
theorem iCondIndepFun.condIndepFun_prod_range_succ {f : ℕ → Ω → β}
    (hf_Indep : iCondIndepFun m' hm' (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
    CondIndepFun m' hm' (∏ j ∈ Finset.range n, f j) (f n) μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [AddCommMonoid β] [MeasurableAdd₂ β] {f : ℕ → Ω → β} (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ℕ), Measurable (f i)) (n : ℕ)
Goal: ProbabilityTheory.CondIndepFun m' hm' (∑ j ∈ Finset.range n, f j) (f n) μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun_prod_range_succ
@[to_additive]
theorem iCondIndepFun.condIndepFun_prod_range_succ {f : ℕ → Ω → β}
    (hf_Indep : iCondIndepFun m' hm' (fun _ => m) f μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
    CondIndepFun m' hm' (∏ j ∈ Finset.range n, f j) (f n) μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : Type u_5} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ℕ → Ω → β} (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ) (hf_meas : ∀ (i : ℕ), Measurable (f i)) (n : ℕ)
Goal: ProbabilityTheory.CondIndepFun m' hm' (∏ j ∈ Finset.range n, f j) (f n) μ
-- ProbabilityTheory.iCondIndepSet.iCondIndepFun_indicator
theorem iCondIndepSet.iCondIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β}
    {s : ι → Set Ω} (hs : iCondIndepSet m' hm' s μ) :
    iCondIndepFun m' hm' (fun _n => m) (fun n => (s n).indicator fun _ω => 1) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {β : Type u_3} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω} (hs : ProbabilityTheory.iCondIndepSet m' hm' s μ)
Goal: ProbabilityTheory.iCondIndepFun m' hm' (fun _n => m) (fun n => (s n).indicator fun _ω => 1) μ
-- ProbabilityTheory.measurableSet_partitionFiltration_of_mem
lemma measurableSet_partitionFiltration_of_mem (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) {s : Set α}
    (hs : s ∈ memPartition t n) :
    MeasurableSet[partitionFiltration ht n] s 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ) {s : Set α} (hs : s ∈ memPartition t n)
Goal: MeasurableSet s
-- ProbabilityTheory.measurableSet_partitionFiltration_memPartitionSet
lemma measurableSet_partitionFiltration_memPartitionSet (ht : ∀ n, MeasurableSet (t n))
    (n : ℕ) (a : α) :
    MeasurableSet[partitionFiltration ht n] (memPartitionSet t n a) 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ) (a : α)
Goal: MeasurableSet (memPartitionSet t n a)
-- ProbabilityTheory.measurable_memPartitionSet_subtype
lemma measurable_memPartitionSet_subtype (ht : ∀ n, MeasurableSet (t n)) (n : ℕ)
    (m : MeasurableSpace (memPartition t n)) :
    @Measurable α (memPartition t n) (partitionFiltration ht n) m
      (fun a ↦ ⟨memPartitionSet t n a, memPartitionSet_mem t n a⟩) 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ)
Goal: ∀ (m_1 : MeasurableSpace ↑(memPartition t n)), Measurable fun a => ⟨memPartitionSet t n a, ⋯⟩
-- ProbabilityTheory.measurable_partitionFiltration_memPartitionSet
lemma measurable_partitionFiltration_memPartitionSet (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) :
    Measurable[partitionFiltration ht n] (memPartitionSet t n) 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ)
Goal: Measurable (memPartitionSet t n)
-- ProbabilityTheory.measurable_memPartitionSet
lemma measurable_memPartitionSet (ht : ∀ n, MeasurableSet (t n)) (n : ℕ) :
    Measurable (memPartitionSet t n) 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (n : ℕ)
Goal: Measurable (memPartitionSet t n)
-- ProbabilityTheory.iSup_partitionFiltration_eq_generateFrom_range
lemma iSup_partitionFiltration_eq_generateFrom_range (ht : ∀ n, MeasurableSet (t n)) :
    ⨆ n, partitionFiltration ht n = generateFrom (Set.range t) 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n))
Goal: ⨆ n, ↑(ProbabilityTheory.partitionFiltration ht) n = MeasurableSpace.generateFrom (Set.range t)
-- ProbabilityTheory.iSup_partitionFiltration
lemma iSup_partitionFiltration (ht : ∀ n, MeasurableSet (t n))
    (ht_range : generateFrom (Set.range t) = m) :
    ⨆ n, partitionFiltration ht n = m 
Arguments: {α : Type u_1} [m : MeasurableSpace α] {t : ℕ → Set α} (ht : ∀ (n : ℕ), MeasurableSet (t n)) (ht_range : MeasurableSpace.generateFrom (Set.range t) = m)
Goal: ⨆ n, ↑(ProbabilityTheory.partitionFiltration ht) n = m
-- ProbabilityTheory.measurableSet_countableFiltration_of_mem
lemma measurableSet_countableFiltration_of_mem (n : ℕ) {s : Set α}
    (hs : s ∈ countablePartition α n) :
    MeasurableSet[countableFiltration α n] s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ) {s : Set α} (hs : s ∈ MeasurableSpace.countablePartition α n)
Goal: MeasurableSet s
-- ProbabilityTheory.measurableSet_countableFiltration_countablePartitionSet
lemma measurableSet_countableFiltration_countablePartitionSet (n : ℕ) (t : α) :
    MeasurableSet[countableFiltration α n] (countablePartitionSet n t) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ) (t : α)
Goal: MeasurableSet (MeasurableSpace.countablePartitionSet n t)
-- ProbabilityTheory.measurable_countablePartitionSet_subtype
lemma measurable_countablePartitionSet_subtype (n : ℕ)
    (m : MeasurableSpace (countablePartition α n)) :
    @Measurable α (countablePartition α n) (countableFiltration α n) m
      (fun a ↦ ⟨countablePartitionSet n a, countablePartitionSet_mem n a⟩) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ) (m : MeasurableSpace ↑(MeasurableSpace.countablePartition α n))
Goal: Measurable fun a => ⟨MeasurableSpace.countablePartitionSet n a, ⋯⟩
-- ProbabilityTheory.measurable_countableFiltration_countablePartitionSet
lemma measurable_countableFiltration_countablePartitionSet (α : Type*)
    [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
    Measurable[countableFiltration α n] (countablePartitionSet n) 
Arguments: (α : Type u_2) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: Measurable (MeasurableSpace.countablePartitionSet n)
-- ProbabilityTheory.measurable_countablePartitionSet
lemma measurable_countablePartitionSet (α : Type*) [MeasurableSpace α] [CountablyGenerated α]
    (n : ℕ) :
    Measurable (countablePartitionSet (α := α) n) 
Arguments: (α : Type u_2) [MeasurableSpace α] [MeasurableSpace.CountablyGenerated α] (n : ℕ)
Goal: Measurable (MeasurableSpace.countablePartitionSet n)
-- ProbabilityTheory.iSup_countableFiltration
lemma iSup_countableFiltration (α : Type*) [m : MeasurableSpace α] [CountablyGenerated α] :
    ⨆ n, countableFiltration α n = m 
Arguments: (α : Type u_2) [m : MeasurableSpace α] [MeasurableSpace.CountablyGenerated α]
Goal: ⨆ n, ↑(ProbabilityTheory.countableFiltration α) n = m
-- ProbabilityTheory.condDistrib_def
noncomputable irreducible_def condDistrib {_ : MeasurableSpace α} [MeasurableSpace β] (Y : α → Ω)
    (X : α → β) (μ : Measure α) [IsFiniteMeasure μ] : Kernel β Ω 
Arguments: {α : Type u_5} {β : Type u_6} {Ω : Type u_7} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω]
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSpace β] (Y : α → Ω) (X : α → β) (μ : MeasureTheory.Measure α)
  [inst_1 : MeasureTheory.IsFiniteMeasure μ],
  ProbabilityTheory.condDistrib Y X μ = (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).condKernel