ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MvPowerSeries.C_inv
@[simp]
theorem C_inv (r : k) : (C σ k r)⁻¹ = C σ k r⁻¹ 
Arguments: {σ : Type u_1} {k : Type u_3} [Field k] (r : k)
Goal: ((MvPowerSeries.C σ k) r)⁻¹ = (MvPowerSeries.C σ k) r⁻¹
-- MvPowerSeries.X_inv
@[simp]
theorem X_inv (s : σ) : (X s : MvPowerSeries σ k)⁻¹ = 0 
Arguments: {σ : Type u_1} {k : Type u_3} [Field k] (s : σ)
Goal: (MvPowerSeries.X s)⁻¹ = 0
-- MvPowerSeries.smul_inv
@[simp]
theorem smul_inv (r : k) (φ : MvPowerSeries σ k) : (r • φ)⁻¹ = r⁻¹ • φ⁻¹ 
Arguments: {σ : Type u_1} {k : Type u_3} [Field k] (r : k) (φ : MvPowerSeries σ k)
Goal: (r • φ)⁻¹ = r⁻¹ • φ⁻¹
-- WittVector.frobenius_verschiebung
theorem frobenius_verschiebung (x : 𝕎 R) : frobenius (verschiebung x) = x * p 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R)
Goal: WittVector.frobenius (WittVector.verschiebung x) = x * ↑p
-- WittVector.verschiebung_zmod
theorem verschiebung_zmod (x : 𝕎 (ZMod p)) : verschiebung x = x * p 
Arguments: {p : ℕ} [hp : Fact (Nat.Prime p)] (x : WittVector p (ZMod p))
Goal: WittVector.verschiebung x = x * ↑p
-- WittVector.coeff_p_pow
theorem coeff_p_pow [CharP R p] (i : ℕ) : ((p : 𝕎 R) ^ i).coeff i = 1 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (i : ℕ)
Goal: (↑p ^ i).coeff i = 1
-- WittVector.coeff_p_pow_eq_zero
theorem coeff_p_pow_eq_zero [CharP R p] {i j : ℕ} (hj : j ≠ i) : ((p : 𝕎 R) ^ i).coeff j = 0 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] {i j : ℕ} (hj : j ≠ i)
Goal: (↑p ^ i).coeff j = 0
-- WittVector.coeff_p
theorem coeff_p [CharP R p] (i : ℕ) : (p : 𝕎 R).coeff i = if i = 1 then 1 else 0 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (i : ℕ)
Goal: (↑p).coeff i = if i = 1 then 1 else 0
-- WittVector.coeff_p_zero
@[simp]
theorem coeff_p_zero [CharP R p] : (p : 𝕎 R).coeff 0 = 0 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p]
Goal: (↑p).coeff 0 = 0
-- WittVector.coeff_p_one
@[simp]
theorem coeff_p_one [CharP R p] : (p : 𝕎 R).coeff 1 = 1 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p]
Goal: (↑p).coeff 1 = 1
-- WittVector.p_nonzero
theorem p_nonzero [Nontrivial R] [CharP R p] : (p : 𝕎 R) ≠ 0 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [Nontrivial R] [CharP R p]
Goal: ↑p ≠ 0
-- WittVector.FractionRing.p_nonzero
theorem FractionRing.p_nonzero [Nontrivial R] [CharP R p] : (p : FractionRing (𝕎 R)) ≠ 0 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [Nontrivial R] [CharP R p]
Goal: ↑p ≠ 0
-- WittVector.verschiebung_mul_frobenius
theorem verschiebung_mul_frobenius (x y : 𝕎 R) :
    verschiebung (x * frobenius y) = verschiebung x * y 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x y : WittVector p R)
Goal: WittVector.verschiebung (x * WittVector.frobenius y) = WittVector.verschiebung x * y
-- WittVector.mul_charP_coeff_zero
theorem mul_charP_coeff_zero [CharP R p] (x : 𝕎 R) : (x * p).coeff 0 = 0 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R)
Goal: (x * ↑p).coeff 0 = 0
-- WittVector.mul_charP_coeff_succ
theorem mul_charP_coeff_succ [CharP R p] (x : 𝕎 R) (i : ℕ) :
    (x * p).coeff (i + 1) = x.coeff i ^ p 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) (i : ℕ)
Goal: (x * ↑p).coeff (i + 1) = x.coeff i ^ p
-- WittVector.verschiebung_frobenius
theorem verschiebung_frobenius [CharP R p] (x : 𝕎 R) : verschiebung (frobenius x) = x * p 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R)
Goal: WittVector.verschiebung (WittVector.frobenius x) = x * ↑p
-- WittVector.verschiebung_frobenius_comm
theorem verschiebung_frobenius_comm [CharP R p] :
    Function.Commute (verschiebung : 𝕎 R → 𝕎 R) frobenius 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p]
Goal: Function.Commute ⇑WittVector.verschiebung ⇑WittVector.frobenius
-- WittVector.iterate_verschiebung_coeff
theorem iterate_verschiebung_coeff (x : 𝕎 R) (n k : ℕ) :
    (verschiebung^[n] x).coeff (k + n) = x.coeff k 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n k : ℕ)
Goal: ((⇑WittVector.verschiebung)^[n] x).coeff (k + n) = x.coeff k
-- WittVector.iterate_verschiebung_mul_left
theorem iterate_verschiebung_mul_left (x y : 𝕎 R) (i : ℕ) :
    verschiebung^[i] x * y = verschiebung^[i] (x * frobenius^[i] y) 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x y : WittVector p R) (i : ℕ)
Goal: (⇑WittVector.verschiebung)^[i] x * y = (⇑WittVector.verschiebung)^[i] (x * (⇑WittVector.frobenius)^[i] y)
-- WittVector.iterate_verschiebung_mul
theorem iterate_verschiebung_mul (x y : 𝕎 R) (i j : ℕ) :
    verschiebung^[i] x * verschiebung^[j] y =
      verschiebung^[i + j] (frobenius^[j] x * frobenius^[i] y) 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x y : WittVector p R) (i j : ℕ)
Goal: (⇑WittVector.verschiebung)^[i] x * (⇑WittVector.verschiebung)^[j] y =
  (⇑WittVector.verschiebung)^[i + j] ((⇑WittVector.frobenius)^[j] x * (⇑WittVector.frobenius)^[i] y)
-- WittVector.iterate_frobenius_coeff
theorem iterate_frobenius_coeff (x : 𝕎 R) (i k : ℕ) :
    (frobenius^[i] x).coeff k = x.coeff k ^ p ^ i 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) (i k : ℕ)
Goal: ((⇑WittVector.frobenius)^[i] x).coeff k = x.coeff k ^ p ^ i
-- WittVector.iterate_verschiebung_mul_coeff
theorem iterate_verschiebung_mul_coeff (x y : 𝕎 R) (i j : ℕ) :
    (verschiebung^[i] x * verschiebung^[j] y).coeff (i + j) =
      x.coeff 0 ^ p ^ j * y.coeff 0 ^ p ^ i 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x y : WittVector p R) (i j : ℕ)
Goal: ((⇑WittVector.verschiebung)^[i] x * (⇑WittVector.verschiebung)^[j] y).coeff (i + j) =
  x.coeff 0 ^ p ^ j * y.coeff 0 ^ p ^ i
-- ProbabilityTheory.Kernel.iIndepSets.meas_biInter
lemma iIndepSets.meas_biInter (h : iIndepSets π κ μ) (s : Finset ι)
    {f : ι → Set Ω} (hf : ∀ i, i ∈ s → f i ∈ π i) :
    ∀ᵐ a ∂μ, κ a (⋂ i ∈ s, f i) = ∏ i ∈ s, κ a (f i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {π : ι → Set (Set Ω)} (h : ProbabilityTheory.Kernel.iIndepSets π κ μ) (s : Finset ι) {f : ι → Set Ω} (hf : ∀ i ∈ s, f i ∈ π i)
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (κ a) (f i)
-- ProbabilityTheory.Kernel.iIndepSets.meas_iInter
lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π κ μ) (hs : ∀ i, s i ∈ π i) :
    ∀ᵐ a ∂μ, κ a (⋂ i, s i) = ∏ i, κ a (s i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {π : ι → Set (Set Ω)} {s : ι → Set Ω} [Fintype ι] (h : ProbabilityTheory.Kernel.iIndepSets π κ μ) (hs : ∀ (i : ι), s i ∈ π i)
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i, s i) = ∏ i : ι, (κ a) (s i)
-- ProbabilityTheory.Kernel.iIndep.iIndepSets'
lemma iIndep.iIndepSets' (hμ : iIndep m κ μ) :
    iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (hμ : ProbabilityTheory.Kernel.iIndep m κ μ)
Goal: ProbabilityTheory.Kernel.iIndepSets (fun x => {s | MeasurableSet s}) κ μ
-- ProbabilityTheory.Kernel.iIndep.meas_biInter
lemma iIndep.meas_biInter (hμ : iIndep m κ μ) (hs : ∀ i, i ∈ S → MeasurableSet[m i] (s i)) :
    ∀ᵐ a ∂μ, κ a (⋂ i ∈ S, s i) = ∏ i ∈ S, κ a (s i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {s : ι → Set Ω} {S : Finset ι} (hμ : ProbabilityTheory.Kernel.iIndep m κ μ) (hs : ∀ i ∈ S, MeasurableSet (s i))
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
-- ProbabilityTheory.Kernel.iIndep.meas_iInter
lemma iIndep.meas_iInter [Fintype ι] (h : iIndep m κ μ) (hs : ∀ i, MeasurableSet[m i] (s i)) :
    ∀ᵐ a ∂μ, κ a (⋂ i, s i) = ∏ i, κ a (s i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {s : ι → Set Ω} [Fintype ι] (h : ProbabilityTheory.Kernel.iIndep m κ μ) (hs : ∀ (i : ι), MeasurableSet (s i))
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i, s i) = ∏ i : ι, (κ a) (s i)
-- ProbabilityTheory.Kernel.iIndepFun.iIndep
protected lemma iIndepFun.iIndep (hf : iIndepFun mβ f κ μ) :
    iIndep (fun x ↦ (mβ x).comap (f x)) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {β : ι → Type u_4} {mβ : (i : ι) → MeasurableSpace (β i)} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : (x : ι) → Ω → β x} (hf : ProbabilityTheory.Kernel.iIndepFun mβ f κ μ)
Goal: ProbabilityTheory.Kernel.iIndep (fun x => MeasurableSpace.comap (f x) (mβ x)) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.meas_biInter
lemma iIndepFun.meas_biInter (hf : iIndepFun mβ f κ μ)
    (hs : ∀ i, i ∈ S → MeasurableSet[(mβ i).comap (f i)] (s i)) :
    ∀ᵐ a ∂μ, κ a (⋂ i ∈ S, s i) = ∏ i ∈ S, κ a (s i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {β : ι → Type u_4} {mβ : (i : ι) → MeasurableSpace (β i)} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {s : ι → Set Ω} {S : Finset ι} {f : (x : ι) → Ω → β x} (hf : ProbabilityTheory.Kernel.iIndepFun mβ f κ μ) (hs : ∀ i ∈ S, MeasurableSet (s i))
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
-- ProbabilityTheory.Kernel.iIndepFun.meas_iInter
lemma iIndepFun.meas_iInter [Fintype ι] (hf : iIndepFun mβ f κ μ)
    (hs : ∀ i, MeasurableSet[(mβ i).comap (f i)] (s i)) :
    ∀ᵐ a ∂μ, κ a (⋂ i, s i) = ∏ i, κ a (s i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {β : ι → Type u_4} {mβ : (i : ι) → MeasurableSpace (β i)} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {s : ι → Set Ω} {f : (x : ι) → Ω → β x} [Fintype ι] (hf : ProbabilityTheory.Kernel.iIndepFun mβ f κ μ) (hs : ∀ (i : ι), MeasurableSet (s i))
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i, s i) = ∏ i : ι, (κ a) (s i)
-- ProbabilityTheory.Kernel.IndepFun.meas_inter
lemma IndepFun.meas_inter {β γ : Type*} [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ]
    {f : Ω → β} {g : Ω → γ} (hfg : IndepFun f g κ μ)
    {s t : Set Ω} (hs : MeasurableSet[mβ.comap f] s) (ht : MeasurableSet[mγ.comap g] t) :
    ∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_5} {γ : Type u_6} [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] {f : Ω → β} {g : Ω → γ} (hfg : ProbabilityTheory.Kernel.IndepFun f g κ μ) {s t : Set Ω} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (s ∩ t) = (κ a) s * (κ a) t
-- ProbabilityTheory.Kernel.IndepSets.symm
@[symm]
theorem IndepSets.symm {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α}
    {s₁ s₂ : Set (Set Ω)} (h : IndepSets s₁ s₂ κ μ) :
    IndepSets s₂ s₁ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set (Set Ω)} (h : ProbabilityTheory.Kernel.IndepSets s₁ s₂ κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets s₂ s₁ κ μ
-- ProbabilityTheory.Kernel.Indep.symm
@[symm]
theorem Indep.symm {m₁ m₂ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
    {μ : Measure α} (h : Indep m₁ m₂ κ μ) :
    Indep m₂ m₁ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m₁ m₂ _mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h : ProbabilityTheory.Kernel.Indep m₁ m₂ κ μ)
Goal: ProbabilityTheory.Kernel.Indep m₂ m₁ κ μ
-- ProbabilityTheory.Kernel.indep_bot_right
theorem indep_bot_right (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] :
    Indep m' ⊥ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ]
Goal: ProbabilityTheory.Kernel.Indep m' ⊥ κ μ
-- ProbabilityTheory.Kernel.indep_bot_left
theorem indep_bot_left (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] :
    Indep ⊥ m' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ]
Goal: ProbabilityTheory.Kernel.Indep ⊥ m' κ μ
-- ProbabilityTheory.Kernel.indepSet_empty_right
theorem indepSet_empty_right {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] (s : Set Ω) :
    IndepSet s ∅ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (s : Set Ω)
Goal: ProbabilityTheory.Kernel.IndepSet s ∅ κ μ
-- ProbabilityTheory.Kernel.indepSet_empty_left
theorem indepSet_empty_left {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
    {μ : Measure α} [IsMarkovKernel κ] (s : Set Ω) :
    IndepSet ∅ s κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (s : Set Ω)
Goal: ProbabilityTheory.Kernel.IndepSet ∅ s κ μ
-- ProbabilityTheory.Kernel.indepSets_of_indepSets_of_le_left
theorem indepSets_of_indepSets_of_le_left {s₁ s₂ s₃ : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h_indep : IndepSets s₁ s₂ κ μ) (h31 : s₃ ⊆ s₁) :
    IndepSets s₃ s₂ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s₁ s₂ s₃ : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.IndepSets s₁ s₂ κ μ) (h31 : s₃ ⊆ s₁)
Goal: ProbabilityTheory.Kernel.IndepSets s₃ s₂ κ μ
-- ProbabilityTheory.Kernel.indepSets_of_indepSets_of_le_right
theorem indepSets_of_indepSets_of_le_right {s₁ s₂ s₃ : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h_indep : IndepSets s₁ s₂ κ μ) (h32 : s₃ ⊆ s₂) :
    IndepSets s₁ s₃ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s₁ s₂ s₃ : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.IndepSets s₁ s₂ κ μ) (h32 : s₃ ⊆ s₂)
Goal: ProbabilityTheory.Kernel.IndepSets s₁ s₃ κ μ
-- ProbabilityTheory.Kernel.indep_of_indep_of_le_left
theorem indep_of_indep_of_le_left {m₁ m₂ m₃ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h_indep : Indep m₁ m₂ κ μ) (h31 : m₃ ≤ m₁) :
    Indep m₃ m₂ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m₁ m₂ m₃ _mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.Indep m₁ m₂ κ μ) (h31 : m₃ ≤ m₁)
Goal: ProbabilityTheory.Kernel.Indep m₃ m₂ κ μ
-- ProbabilityTheory.Kernel.indep_of_indep_of_le_right
theorem indep_of_indep_of_le_right {m₁ m₂ m₃ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h_indep : Indep m₁ m₂ κ μ) (h32 : m₃ ≤ m₂) :
    Indep m₁ m₃ κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m₁ m₂ m₃ _mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.Indep m₁ m₂ κ μ) (h32 : m₃ ≤ m₂)
Goal: ProbabilityTheory.Kernel.Indep m₁ m₃ κ μ
-- ProbabilityTheory.Kernel.IndepSets.union
theorem IndepSets.union {s₁ s₂ s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α}
    (h₁ : IndepSets s₁ s' κ μ) (h₂ : IndepSets s₂ s' κ μ) :
    IndepSets (s₁ ∪ s₂) s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s₁ s₂ s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h₁ : ProbabilityTheory.Kernel.IndepSets s₁ s' κ μ) (h₂ : ProbabilityTheory.Kernel.IndepSets s₂ s' κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets (s₁ ∪ s₂) s' κ μ
-- ProbabilityTheory.Kernel.IndepSets.union_iff
@[simp]
theorem IndepSets.union_iff {s₁ s₂ s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} :
    IndepSets (s₁ ∪ s₂) s' κ μ ↔ IndepSets s₁ s' κ μ ∧ IndepSets s₂ s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s₁ s₂ s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α}
Goal: ProbabilityTheory.Kernel.IndepSets (s₁ ∪ s₂) s' κ μ ↔
  ProbabilityTheory.Kernel.IndepSets s₁ s' κ μ ∧ ProbabilityTheory.Kernel.IndepSets s₂ s' κ μ
-- ProbabilityTheory.Kernel.IndepSets.iUnion
theorem IndepSets.iUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (hyp : ∀ n, IndepSets (s n) s' κ μ) :
    IndepSets (⋃ n, s n) s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (hyp : ∀ (n : ι), ProbabilityTheory.Kernel.IndepSets (s n) s' κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets (⋃ n, s n) s' κ μ
-- ProbabilityTheory.Kernel.IndepSets.bUnion
theorem IndepSets.bUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} {u : Set ι} (hyp : ∀ n ∈ u, IndepSets (s n) s' κ μ) :
    IndepSets (⋃ n ∈ u, s n) s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {u : Set ι} (hyp : ∀ n ∈ u, ProbabilityTheory.Kernel.IndepSets (s n) s' κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets (⋃ n ∈ u, s n) s' κ μ
-- ProbabilityTheory.Kernel.IndepSets.inter
theorem IndepSets.inter {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h₁ : IndepSets s₁ s' κ μ) :
    IndepSets (s₁ ∩ s₂) s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h₁ : ProbabilityTheory.Kernel.IndepSets s₁ s' κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets (s₁ ∩ s₂) s' κ μ
-- ProbabilityTheory.Kernel.IndepSets.iInter
theorem IndepSets.iInter {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h : ∃ n, IndepSets (s n) s' κ μ) :
    IndepSets (⋂ n, s n) s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h : ∃ n, ProbabilityTheory.Kernel.IndepSets (s n) s' κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets (⋂ n, s n) s' κ μ
-- ProbabilityTheory.Kernel.IndepSets.bInter
theorem IndepSets.bInter {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} {u : Set ι} (h : ∃ n ∈ u, IndepSets (s n) s' κ μ) :
    IndepSets (⋂ n ∈ u, s n) s' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {u : Set ι} (h : ∃ n ∈ u, ProbabilityTheory.Kernel.IndepSets (s n) s' κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets (⋂ n ∈ u, s n) s' κ μ
-- ProbabilityTheory.Kernel.iIndep_comap_mem_iff
theorem iIndep_comap_mem_iff {f : ι → Set Ω} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} :
    iIndep (fun i => MeasurableSpace.comap (· ∈ f i) ⊤) κ μ ↔ iIndepSet f κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {f : ι → Set Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α}
Goal: ProbabilityTheory.Kernel.iIndep (fun i => MeasurableSpace.comap (fun x => x ∈ f i) ⊤) κ μ ↔
  ProbabilityTheory.Kernel.iIndepSet f κ μ
-- ProbabilityTheory.Kernel.iIndepSets_singleton_iff
theorem iIndepSets_singleton_iff {s : ι → Set Ω} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} :
    iIndepSets (fun i ↦ {s i}) κ μ ↔
      ∀ S : Finset ι, ∀ᵐ a ∂μ, κ a (⋂ i ∈ S, s i) = ∏ i ∈ S, κ a (s i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → Set Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α}
Goal: ProbabilityTheory.Kernel.iIndepSets (fun i => {s i}) κ μ ↔
  ∀ (S : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
-- ProbabilityTheory.Kernel.indepSets_singleton_iff
theorem indepSets_singleton_iff {s t : Set Ω} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} :
    IndepSets {s} {t} κ μ ↔ ∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s t : Set Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α}
Goal: ProbabilityTheory.Kernel.IndepSets {s} {t} κ μ ↔ ∀ᵐ (a : α) ∂μ, (κ a) (s ∩ t) = (κ a) s * (κ a) t
-- ProbabilityTheory.Kernel.iIndepSets.indepSets
theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} (h_indep : iIndepSets s κ μ) {i j : ι} (hij : i ≠ j) :
    IndepSets (s i) (s j) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → Set (Set Ω)} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.iIndepSets s κ μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.Kernel.IndepSets (s i) (s j) κ μ
-- ProbabilityTheory.Kernel.iIndep.indep
theorem iIndep.indep {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α}
    (h_indep : iIndep m κ μ) {i j : ι} (hij : i ≠ j) : Indep (m i) (m j) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.iIndep m κ μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.Kernel.Indep (m i) (m j) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun
theorem iIndepFun.indepFun {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} {β : ι → Type*}
    {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f κ μ) {i j : ι}
    (hij : i ≠ j) : IndepFun (f i) (f j) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_4} {m : (x : ι) → MeasurableSpace (β x)} {f : (i : ι) → Ω → β i} (hf_Indep : ProbabilityTheory.Kernel.iIndepFun m f κ μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.Kernel.IndepFun (f i) (f j) κ μ