ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Multiset.prod_X_sub_X_eq_sum_esymm
theorem prod_X_sub_X_eq_sum_esymm (s : Multiset R) :
    (s.map fun t => X - C t).prod =
      ∑ j ∈ Finset.range (Multiset.card s + 1),
        (-1) ^ j * (C (s.esymm j) * X ^ (Multiset.card s - j)) 
Arguments: {R : Type u_1} [CommRing R] (s : Multiset R)
Goal: (Multiset.map (fun t => Polynomial.X - Polynomial.C t) s).prod =
  ∑ j ∈ Finset.range (Multiset.card s + 1), (-1) ^ j * (Polynomial.C (s.esymm j) * Polynomial.X ^ (Multiset.card s - j))
-- Multiset.prod_X_sub_C_coeff
theorem prod_X_sub_C_coeff (s : Multiset R) {k : ℕ} (h : k ≤ Multiset.card s) :
    (s.map fun t => X - C t).prod.coeff k =
    (-1) ^ (Multiset.card s - k) * s.esymm (Multiset.card s - k) 
Arguments: {R : Type u_1} [CommRing R] (s : Multiset R) {k : ℕ} (h : k ≤ Multiset.card s)
Goal: (Multiset.map (fun t => Polynomial.X - Polynomial.C t) s).prod.coeff k =
  (-1) ^ (Multiset.card s - k) * s.esymm (Multiset.card s - k)
-- Polynomial.coeff_eq_esymm_roots_of_card
theorem _root_.Polynomial.coeff_eq_esymm_roots_of_card [IsDomain R] {p : R[X]}
    (hroots : Multiset.card p.roots = p.natDegree) {k : ℕ} (h : k ≤ p.natDegree) :
    p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] {p : R[X]} (hroots : Multiset.card p.roots = p.natDegree) {k : ℕ} (h : k ≤ p.natDegree)
Goal: p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k)
-- Polynomial.coeff_eq_esymm_roots_of_splits
theorem _root_.Polynomial.coeff_eq_esymm_roots_of_splits {F} [Field F] {p : F[X]}
    (hsplit : p.Splits (RingHom.id F)) {k : ℕ} (h : k ≤ p.natDegree) :
    p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k) 
Arguments: {F : Type u_2} [Field F] {p : F[X]} (hsplit : Polynomial.Splits (RingHom.id F) p) {k : ℕ} (h : k ≤ p.natDegree)
Goal: p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k)
-- MvPolynomial.prod_C_add_X_eq_sum_esymm
theorem MvPolynomial.prod_C_add_X_eq_sum_esymm :
    (∏ i : σ, (Polynomial.X + Polynomial.C (MvPolynomial.X i))) =
      ∑ j ∈ range (card σ + 1), Polynomial.C
        (MvPolynomial.esymm σ R j) * Polynomial.X ^ (card σ - j) 
Arguments: (R : Type u_1) (σ : Type u_2) [CommSemiring R] [Fintype σ]
Goal: ∏ i : σ, (Polynomial.X + Polynomial.C (MvPolynomial.X i)) =
  ∑ j ∈ Finset.range (Fintype.card σ + 1), Polynomial.C (MvPolynomial.esymm σ R j) * Polynomial.X ^ (Fintype.card σ - j)
-- MvPolynomial.prod_X_add_C_coeff
theorem MvPolynomial.prod_X_add_C_coeff (k : ℕ) (h : k ≤ card σ) :
    (∏ i : σ, (Polynomial.X + Polynomial.C (MvPolynomial.X i)) : Polynomial _).coeff k =
    MvPolynomial.esymm σ R (card σ - k) 
Arguments: (R : Type u_1) (σ : Type u_2) [CommSemiring R] [Fintype σ] (k : ℕ) (h : k ≤ Fintype.card σ)
Goal: (∏ i : σ, (Polynomial.X + Polynomial.C (MvPolynomial.X i))).coeff k = MvPolynomial.esymm σ R (Fintype.card σ - k)
-- Polynomial.coeff_scaleRoots
@[simp]
theorem coeff_scaleRoots (p : R[X]) (s : R) (i : ℕ) :
    (scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) (i : ℕ)
Goal: (p.scaleRoots s).coeff i = p.coeff i * s ^ (p.natDegree - i)
-- Polynomial.coeff_scaleRoots_natDegree
theorem coeff_scaleRoots_natDegree (p : R[X]) (s : R) :
    (scaleRoots p s).coeff p.natDegree = p.leadingCoeff 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R)
Goal: (p.scaleRoots s).coeff p.natDegree = p.leadingCoeff
-- Polynomial.zero_scaleRoots
@[simp]
theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 
Arguments: {R : Type u_1} [Semiring R] (s : R)
Goal: Polynomial.scaleRoots 0 s = 0
-- Polynomial.scaleRoots_ne_zero
theorem scaleRoots_ne_zero {p : R[X]} (hp : p ≠ 0) (s : R) : scaleRoots p s ≠ 0 
Arguments: {R : Type u_1} [Semiring R] {p : Polynomial R} (hp : p ≠ 0) (s : R)
Goal: p.scaleRoots s ≠ 0
-- Polynomial.support_scaleRoots_le
theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support ≤ p.support 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R)
Goal: (p.scaleRoots s).support ≤ p.support
-- Polynomial.support_scaleRoots_eq
theorem support_scaleRoots_eq (p : R[X]) {s : R} (hs : s ∈ nonZeroDivisors R) :
    (scaleRoots p s).support = p.support 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) {s : R} (hs : s ∈ nonZeroDivisors R)
Goal: (p.scaleRoots s).support = p.support
-- Polynomial.degree_scaleRoots
@[simp]
theorem degree_scaleRoots (p : R[X]) {s : R} : degree (scaleRoots p s) = degree p 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) {s : R}
Goal: (p.scaleRoots s).degree = p.degree
-- Polynomial.natDegree_scaleRoots
@[simp]
theorem natDegree_scaleRoots (p : R[X]) (s : R) : natDegree (scaleRoots p s) = natDegree p 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R)
Goal: (p.scaleRoots s).natDegree = p.natDegree
-- Polynomial.monic_scaleRoots_iff
theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) ↔ Monic p 
Arguments: {R : Type u_1} [Semiring R] {p : Polynomial R} (s : R)
Goal: (p.scaleRoots s).Monic ↔ p.Monic
-- Polynomial.map_scaleRoots
theorem map_scaleRoots (p : R[X]) (x : R) (f : R →+* S) (h : f p.leadingCoeff ≠ 0) :
    (p.scaleRoots x).map f = (p.map f).scaleRoots (f x) 
Arguments: {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (p : Polynomial R) (x : R) (f : R →+* S) (h : f p.leadingCoeff ≠ 0)
Goal: Polynomial.map f (p.scaleRoots x) = (Polynomial.map f p).scaleRoots (f x)
-- Polynomial.scaleRoots_C
@[simp]
lemma scaleRoots_C (r c : R) : (C c).scaleRoots r = C c 
Arguments: {R : Type u_1} [Semiring R] (r c : R)
Goal: (Polynomial.C c).scaleRoots r = Polynomial.C c
-- Polynomial.scaleRoots_one
@[simp]
lemma scaleRoots_one (p : R[X]) :
    p.scaleRoots 1 = p 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R)
Goal: p.scaleRoots 1 = p
-- Polynomial.scaleRoots_zero
@[simp]
lemma scaleRoots_zero (p : R[X]) :
    p.scaleRoots 0 = p.leadingCoeff • X ^ p.natDegree 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R)
Goal: p.scaleRoots 0 = p.leadingCoeff • Polynomial.X ^ p.natDegree
-- Polynomial.one_scaleRoots
@[simp]
lemma one_scaleRoots (r : R) :
    (1 : R[X]).scaleRoots r = 1 
Arguments: {R : Type u_1} [Semiring R] (r : R)
Goal: Polynomial.scaleRoots 1 r = 1
-- Polynomial.scaleRoots_eval₂_mul_of_commute
theorem scaleRoots_eval₂_mul_of_commute {p : S[X]} (f : S →+* A) (a : A) (s : S)
    (hsa : Commute (f s) a) (hf : ∀ s₁ s₂, Commute (f s₁) (f s₂)) :
    eval₂ f (f s * a) (scaleRoots p s) = f s ^ p.natDegree * eval₂ f a p 
Arguments: {S : Type u_2} {A : Type u_3} [Semiring S] [Semiring A] {p : Polynomial S} (f : S →+* A) (a : A) (s : S) (hsa : Commute (f s) a) (hf : ∀ (s₁ s₂ : S), Commute (f s₁) (f s₂))
Goal: Polynomial.eval₂ f (f s * a) (p.scaleRoots s) = f s ^ p.natDegree * Polynomial.eval₂ f a p
-- Polynomial.scaleRoots_eval₂_mul
theorem scaleRoots_eval₂_mul {p : S[X]} (f : S →+* R) (r : R) (s : S) :
    eval₂ f (f s * r) (scaleRoots p s) = f s ^ p.natDegree * eval₂ f r p 
Arguments: {R : Type u_1} {S : Type u_2} [Semiring S] [CommSemiring R] {p : Polynomial S} (f : S →+* R) (r : R) (s : S)
Goal: Polynomial.eval₂ f (f s * r) (p.scaleRoots s) = f s ^ p.natDegree * Polynomial.eval₂ f r p
-- Polynomial.scaleRoots_eval₂_eq_zero
theorem scaleRoots_eval₂_eq_zero {p : S[X]} (f : S →+* R) {r : R} {s : S} (hr : eval₂ f r p = 0) :
    eval₂ f (f s * r) (scaleRoots p s) = 0 
Arguments: {R : Type u_1} {S : Type u_2} [Semiring S] [CommSemiring R] {p : Polynomial S} (f : S →+* R) {r : R} {s : S} (hr : Polynomial.eval₂ f r p = 0)
Goal: Polynomial.eval₂ f (f s * r) (p.scaleRoots s) = 0
-- Polynomial.scaleRoots_aeval_eq_zero
theorem scaleRoots_aeval_eq_zero [Algebra R A] {p : R[X]} {a : A} {r : R} (ha : aeval a p = 0) :
    aeval (algebraMap R A r * a) (scaleRoots p r) = 0 
Arguments: {R : Type u_1} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {p : Polynomial R} {a : A} {r : R} (ha : (Polynomial.aeval a) p = 0)
Goal: (Polynomial.aeval ((algebraMap R A) r * a)) (p.scaleRoots r) = 0
-- Polynomial.scaleRoots_eval₂_eq_zero_of_eval₂_div_eq_zero
theorem scaleRoots_eval₂_eq_zero_of_eval₂_div_eq_zero {p : S[X]} {f : S →+* K}
    (hf : Function.Injective f) {r s : S} (hr : eval₂ f (f r / f s) p = 0)
    (hs : s ∈ nonZeroDivisors S) : eval₂ f (f r) (scaleRoots p s) = 0 
Arguments: {S : Type u_2} {K : Type u_4} [Semiring S] [Field K] {p : Polynomial S} {f : S →+* K} (hf : Function.Injective ⇑f) {r s : S} (hr : Polynomial.eval₂ f (f r / f s) p = 0) (hs : s ∈ nonZeroDivisors S)
Goal: Polynomial.eval₂ f (f r) (p.scaleRoots s) = 0
-- Polynomial.scaleRoots_aeval_eq_zero_of_aeval_div_eq_zero
theorem scaleRoots_aeval_eq_zero_of_aeval_div_eq_zero [Algebra R K]
    (inj : Function.Injective (algebraMap R K)) {p : R[X]} {r s : R}
    (hr : aeval (algebraMap R K r / algebraMap R K s) p = 0) (hs : s ∈ nonZeroDivisors R) :
    aeval (algebraMap R K r) (scaleRoots p s) = 0 
Arguments: {R : Type u_1} {K : Type u_4} [CommSemiring R] [Field K] [Algebra R K] (inj : Function.Injective ⇑(algebraMap R K)) {p : Polynomial R} {r s : R} (hr : (Polynomial.aeval ((algebraMap R K) r / (algebraMap R K) s)) p = 0) (hs : s ∈ nonZeroDivisors R)
Goal: (Polynomial.aeval ((algebraMap R K) r)) (p.scaleRoots s) = 0
-- Polynomial.scaleRoots_mul
@[simp]
lemma scaleRoots_mul (p : R[X]) (r s) :
    p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s 
Arguments: {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r s : R)
Goal: p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s
-- Polynomial.mul_scaleRoots
lemma mul_scaleRoots (p q : R[X]) (r : R) :
    r ^ (natDegree p + natDegree q - natDegree (p * q)) • (p * q).scaleRoots r =
      p.scaleRoots r * q.scaleRoots r 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R)
Goal: r ^ (p.natDegree + q.natDegree - (p * q).natDegree) • (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r
-- Polynomial.mul_scaleRoots'
lemma mul_scaleRoots' (p q : R[X]) (r : R) (h : leadingCoeff p * leadingCoeff q ≠ 0) :
    (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (h : p.leadingCoeff * q.leadingCoeff ≠ 0)
Goal: (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r
-- Polynomial.mul_scaleRoots_of_noZeroDivisors
lemma mul_scaleRoots_of_noZeroDivisors (p q : R[X]) (r : R) [NoZeroDivisors R] :
    (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) [NoZeroDivisors R]
Goal: (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r
-- Polynomial.add_scaleRoots_of_natDegree_eq
lemma add_scaleRoots_of_natDegree_eq (p q : R[X]) (r : R) (h : natDegree p = natDegree q) :
    r ^ (natDegree p - natDegree (p + q)) • (p + q).scaleRoots r =
      p.scaleRoots r + q.scaleRoots r 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (h : p.natDegree = q.natDegree)
Goal: r ^ (p.natDegree - (p + q).natDegree) • (p + q).scaleRoots r = p.scaleRoots r + q.scaleRoots r
-- Polynomial.scaleRoots_dvd'
lemma scaleRoots_dvd' (p q : R[X]) {r : R} (hr : IsUnit r)
    (hpq : p ∣ q) : p.scaleRoots r ∣ q.scaleRoots r 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r) (hpq : p ∣ q)
Goal: p.scaleRoots r ∣ q.scaleRoots r
-- Polynomial.scaleRoots_dvd
lemma scaleRoots_dvd (p q : R[X]) {r : R} [NoZeroDivisors R] (hpq : p ∣ q) :
    p.scaleRoots r ∣ q.scaleRoots r 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} [NoZeroDivisors R] (hpq : p ∣ q)
Goal: p.scaleRoots r ∣ q.scaleRoots r
-- Polynomial.scaleRoots_dvd_iff
lemma scaleRoots_dvd_iff (p q : R[X]) {r : R} (hr : IsUnit r) :
    p.scaleRoots r ∣ q.scaleRoots r ↔ p ∣ q 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r)
Goal: p.scaleRoots r ∣ q.scaleRoots r ↔ p ∣ q
-- Polynomial.isCoprime_scaleRoots
lemma isCoprime_scaleRoots (p q : R[X]) (r : R) (hr : IsUnit r) (h : IsCoprime p q) :
    IsCoprime (p.scaleRoots r) (q.scaleRoots r) 
Arguments: {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (hr : IsUnit r) (h : IsCoprime p q)
Goal: IsCoprime (p.scaleRoots r) (q.scaleRoots r)
-- ProbabilityTheory.Kernel.densityProcess_def
lemma densityProcess_def (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ) (a : α) (s : Set β) :
    (fun t ↦ densityProcess κ ν n a t s)
      = fun t ↦ (κ a (countablePartitionSet n t ×ˢ s) / ν a (countablePartitionSet n t)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (a : α) (s : Set β)
Goal: (fun t => κ.densityProcess ν n a t s) = fun t =>
  ((κ a) (MeasurableSpace.countablePartitionSet n t ×ˢ s) / (ν a) (MeasurableSpace.countablePartitionSet n t)).toReal
-- ProbabilityTheory.Kernel.measurable_densityProcess_countableFiltration_aux
lemma measurable_densityProcess_countableFiltration_aux (κ : Kernel α (γ × β)) (ν : Kernel α γ)
    (n : ℕ) {s : Set β} (hs : MeasurableSet s) :
    Measurable[mα.prod (countableFiltration γ n)] (fun (p : α × γ) ↦
      κ p.1 (countablePartitionSet n p.2 ×ˢ s) / ν p.1 (countablePartitionSet n p.2)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun p =>
  (κ p.1) (MeasurableSpace.countablePartitionSet n p.2 ×ˢ s) / (ν p.1) (MeasurableSpace.countablePartitionSet n p.2)
-- ProbabilityTheory.Kernel.measurable_densityProcess_aux
lemma measurable_densityProcess_aux (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
    {s : Set β} (hs : MeasurableSet s) :
    Measurable (fun (p : α × γ) ↦
      κ p.1 (countablePartitionSet n p.2 ×ˢ s) / ν p.1 (countablePartitionSet n p.2)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun p =>
  (κ p.1) (MeasurableSpace.countablePartitionSet n p.2 ×ˢ s) / (ν p.1) (MeasurableSpace.countablePartitionSet n p.2)
-- ProbabilityTheory.Kernel.measurable_densityProcess
lemma measurable_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
    {s : Set β} (hs : MeasurableSet s) :
    Measurable (fun (p : α × γ) ↦ densityProcess κ ν n p.1 p.2 s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun p => κ.densityProcess ν n p.1 p.2 s
-- ProbabilityTheory.Kernel.measurable_densityProcess_left
lemma measurable_densityProcess_left (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
    (x : γ) {s : Set β} (hs : MeasurableSet s) :
    Measurable (fun a ↦ densityProcess κ ν n a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (x : γ) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun a => κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.measurable_densityProcess_right
lemma measurable_densityProcess_right (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
    {s : Set β} (a : α) (hs : MeasurableSet s) :
    Measurable (fun x ↦ densityProcess κ ν n a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) {s : Set β} (a : α) (hs : MeasurableSet s)
Goal: Measurable fun x => κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.measurable_countableFiltration_densityProcess
lemma measurable_countableFiltration_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    Measurable[countableFiltration γ n] (fun x ↦ densityProcess κ ν n a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun x => κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.stronglyMeasurable_countableFiltration_densityProcess
lemma stronglyMeasurable_countableFiltration_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ)
    (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) :
    StronglyMeasurable[countableFiltration γ n] (fun x ↦ densityProcess κ ν n a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.StronglyMeasurable fun x => κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.adapted_densityProcess
lemma adapted_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (a : α)
    {s : Set β} (hs : MeasurableSet s) :
    Adapted (countableFiltration γ) (fun n x ↦ densityProcess κ ν n a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Adapted (ProbabilityTheory.countableFiltration γ) fun n x => κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.densityProcess_nonneg
lemma densityProcess_nonneg (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
    (a : α) (x : γ) (s : Set β) :
    0 ≤ densityProcess κ ν n a x s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (a : α) (x : γ) (s : Set β)
Goal: 0 ≤ κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.meas_countablePartitionSet_le_of_fst_le
lemma meas_countablePartitionSet_le_of_fst_le (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ)
    (s : Set β) :
    κ a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β)
Goal: (κ a) (MeasurableSpace.countablePartitionSet n x ×ˢ s) ≤ (ν a) (MeasurableSpace.countablePartitionSet n x)
-- ProbabilityTheory.Kernel.densityProcess_le_one
lemma densityProcess_le_one (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β) :
    densityProcess κ ν n a x s ≤ 1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β)
Goal: κ.densityProcess ν n a x s ≤ 1
-- ProbabilityTheory.Kernel.eLpNorm_densityProcess_le
lemma eLpNorm_densityProcess_le (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (s : Set β) :
    eLpNorm (fun x ↦ densityProcess κ ν n a x s) 1 (ν a) ≤ ν a univ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (n : ℕ) (a : α) (s : Set β)
Goal: MeasureTheory.eLpNorm (fun x => κ.densityProcess ν n a x s) 1 (ν a) ≤ (ν a) Set.univ
-- ProbabilityTheory.Kernel.integrable_densityProcess
lemma integrable_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (n : ℕ)
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    Integrable (fun x ↦ densityProcess κ ν n a x s) (ν a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (fun x => κ.densityProcess ν n a x s) (ν a)
-- ProbabilityTheory.Kernel.setIntegral_densityProcess_of_mem
lemma setIntegral_densityProcess_of_mem (hκν : fst κ ≤ ν) [hν : IsFiniteKernel ν]
    (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {u : Set γ}
    (hu : u ∈ countablePartition γ n) :
    ∫ x in u, densityProcess κ ν n a x s ∂(ν a) = (κ a (u ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [hν : ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {u : Set γ} (hu : u ∈ MeasurableSpace.countablePartition γ n)
Goal: ∫ (x : γ) in u, κ.densityProcess ν n a x s ∂ν a = ((κ a) (u ×ˢ s)).toReal
-- ProbabilityTheory.Kernel.setIntegral_densityProcess
lemma setIntegral_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
    (hA : MeasurableSet[countableFiltration γ n] A) :
    ∫ x in A, densityProcess κ ν n a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A)
Goal: ∫ (x : γ) in A, κ.densityProcess ν n a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
-- ProbabilityTheory.Kernel.integral_densityProcess
lemma integral_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) :
    ∫ x, densityProcess κ ν n a x s ∂(ν a) = (κ a (univ ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ∫ (x : γ), κ.densityProcess ν n a x s ∂ν a = ((κ a) (Set.univ ×ˢ s)).toReal
-- ProbabilityTheory.Kernel.setIntegral_densityProcess_of_le
lemma setIntegral_densityProcess_of_le (hκν : fst κ ≤ ν)
    [IsFiniteKernel ν] {n m : ℕ} (hnm : n ≤ m) (a : α) {s : Set β} (hs : MeasurableSet s)
    {A : Set γ} (hA : MeasurableSet[countableFiltration γ n] A) :
    ∫ x in A, densityProcess κ ν m a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] {n m : ℕ} (hnm : n ≤ m) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A)
Goal: ∫ (x : γ) in A, κ.densityProcess ν m a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal