ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Valuation.Integers.mem_of_integral
theorem mem_of_integral {x : R} (hx : IsIntegral O x) : x ∈ v.integer 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (hv : v.Integers O) {x : R} (hx : IsIntegral O x)
Goal: x ∈ v.integer
-- Valuation.Integers.integralClosure
protected theorem integralClosure : integralClosure O R = ⊥ 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (hv : v.Integers O)
Goal: integralClosure O R = ⊥
-- Valuation.Integers.integrallyClosed
theorem integrallyClosed : IsIntegrallyClosed O 
Arguments: {K : Type u} {Γ₀ : Type v} [Field K] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation K Γ₀} {O : Type w} [CommRing O] [Algebra O K] [IsFractionRing O K] (hv : v.Integers O)
Goal: IsIntegrallyClosed O
-- multiplicity.addValuation_apply
@[simp]
theorem multiplicity.addValuation_apply {hp : Prime p} {r : R} :
    addValuation hp r = multiplicity p r 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] {p : R} [DecidableRel Dvd.dvd] {hp : Prime p} {r : R}
Goal: (multiplicity.addValuation hp) r = multiplicity p r
-- HopfAlgebra.mul_antipode_rTensor_comul
class HopfAlgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    Bialgebra R A where
    antipode : A →ₗ[R] A
    mul_antipode_rTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.rTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit
    mul_antipode_lTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.lTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : HopfAlgebra R A]
Goal: LinearMap.mul' R A ∘ₗ LinearMap.rTensor A HopfAlgebra.antipode ∘ₗ Coalgebra.comul =
  Algebra.linearMap R A ∘ₗ Coalgebra.counit
-- HopfAlgebra.mul_antipode_lTensor_comul
class HopfAlgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    Bialgebra R A where
    antipode : A →ₗ[R] A
    mul_antipode_rTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.rTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit
    mul_antipode_lTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.lTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : HopfAlgebra R A]
Goal: LinearMap.mul' R A ∘ₗ LinearMap.lTensor A HopfAlgebra.antipode ∘ₗ Coalgebra.comul =
  Algebra.linearMap R A ∘ₗ Coalgebra.counit
-- HopfAlgebra.mul_antipode_rTensor_comul_apply
@[simp]
theorem mul_antipode_rTensor_comul_apply (a : A) :
    LinearMap.mul' R A (antipode.rTensor A (Coalgebra.comul a)) =
    algebraMap R A (Coalgebra.counit a) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [HopfAlgebra R A] (a : A)
Goal: (LinearMap.mul' R A) ((LinearMap.rTensor A HopfAlgebra.antipode) (Coalgebra.comul a)) =
  (algebraMap R A) (Coalgebra.counit a)
-- HopfAlgebra.mul_antipode_lTensor_comul_apply
@[simp]
theorem mul_antipode_lTensor_comul_apply (a : A) :
    LinearMap.mul' R A (antipode.lTensor A (Coalgebra.comul a)) =
    algebraMap R A (Coalgebra.counit a) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [HopfAlgebra R A] (a : A)
Goal: (LinearMap.mul' R A) ((LinearMap.lTensor A HopfAlgebra.antipode) (Coalgebra.comul a)) =
  (algebraMap R A) (Coalgebra.counit a)
-- CommSemiring.antipode_eq_id
@[simp]
theorem antipode_eq_id : antipode (R := R) (A := R) = .id 
Arguments: (R : Type u) [CommSemiring R]
Goal: HopfAlgebra.antipode = LinearMap.id
-- QuotSMulTop.map_apply_mk
@[simp]
lemma map_apply_mk (f : M →ₗ[R] M') (x : M) :
    map r f (Submodule.Quotient.mk x) =
      (Submodule.Quotient.mk (f x) : QuotSMulTop r M') 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M)
Goal: ((QuotSMulTop.map r) f) (Submodule.Quotient.mk x) = Submodule.Quotient.mk (f x)
-- QuotSMulTop.map_comp_mkQ
lemma map_comp_mkQ (f : M →ₗ[R] M') :
    map r f ∘ₗ mkQ (r • ⊤) = mkQ (r • ⊤) ∘ₗ f 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M')
Goal: (QuotSMulTop.map r) f ∘ₗ (r • ⊤).mkQ = (r • ⊤).mkQ ∘ₗ f
-- QuotSMulTop.map_id
@[simp]
lemma map_id : map r (LinearMap.id : M →ₗ[R] M) = .id 
Arguments: {R : Type u_2} [CommRing R] (r : R) (M : Type u_1) [AddCommGroup M] [Module R M]
Goal: (QuotSMulTop.map r) LinearMap.id = LinearMap.id
-- QuotSMulTop.map_comp
@[simp]
lemma map_comp (g : M' →ₗ[R] M'') (f : M →ₗ[R] M') :
    map r (g ∘ₗ f) = map r g ∘ₗ map r f 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} {M'' : Type u_4} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M''] (g : M' →ₗ[R] M'') (f : M →ₗ[R] M')
Goal: (QuotSMulTop.map r) (g ∘ₗ f) = (QuotSMulTop.map r) g ∘ₗ (QuotSMulTop.map r) f
-- QuotSMulTop.equivQuotTensor_naturality_mk
lemma equivQuotTensor_naturality_mk (f : M →ₗ[R] M') (x : M) :
    equivQuotTensor r M' (map r f (Submodule.Quotient.mk x)) =
      f.lTensor (R ⧸ Ideal.span {r})
        (equivQuotTensor r M (Submodule.Quotient.mk x)) 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M)
Goal: (QuotSMulTop.equivQuotTensor r M') (((QuotSMulTop.map r) f) (Submodule.Quotient.mk x)) =
  (LinearMap.lTensor (R ⧸ Ideal.span {r}) f) ((QuotSMulTop.equivQuotTensor r M) (Submodule.Quotient.mk x))
-- QuotSMulTop.equivQuotTensor_naturality
lemma equivQuotTensor_naturality (f : M →ₗ[R] M') :
    equivQuotTensor r M' ∘ₗ map r f =
      f.lTensor (R ⧸ Ideal.span {r}) ∘ₗ equivQuotTensor r M 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M')
Goal: ↑(QuotSMulTop.equivQuotTensor r M') ∘ₗ (QuotSMulTop.map r) f =
  LinearMap.lTensor (R ⧸ Ideal.span {r}) f ∘ₗ ↑(QuotSMulTop.equivQuotTensor r M)
-- QuotSMulTop.equivTensorQuot_naturality_mk
lemma equivTensorQuot_naturality_mk (f : M →ₗ[R] M') (x : M) :
    equivTensorQuot r M' (map r f (Submodule.Quotient.mk x)) =
      f.rTensor (R ⧸ Ideal.span {r})
        (equivTensorQuot r M (Submodule.Quotient.mk x)) 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M)
Goal: (QuotSMulTop.equivTensorQuot r M') (((QuotSMulTop.map r) f) (Submodule.Quotient.mk x)) =
  (LinearMap.rTensor (R ⧸ Ideal.span {r}) f) ((QuotSMulTop.equivTensorQuot r M) (Submodule.Quotient.mk x))
-- QuotSMulTop.equivTensorQuot_naturality
lemma equivTensorQuot_naturality (f : M →ₗ[R] M') :
    equivTensorQuot r M' ∘ₗ map r f =
      f.rTensor (R ⧸ Ideal.span {r}) ∘ₗ equivTensorQuot r M 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M')
Goal: ↑(QuotSMulTop.equivTensorQuot r M') ∘ₗ (QuotSMulTop.map r) f =
  LinearMap.rTensor (R ⧸ Ideal.span {r}) f ∘ₗ ↑(QuotSMulTop.equivTensorQuot r M)
-- QuotSMulTop.map_surjective
lemma map_surjective {f : M →ₗ[R] M'} (hf : Surjective f) : Surjective (map r f) 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] {f : M →ₗ[R] M'} (hf : Function.Surjective ⇑f)
Goal: Function.Surjective ⇑((QuotSMulTop.map r) f)
-- QuotSMulTop.map_exact
lemma map_exact {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''}
    (hfg : Exact f g) (hg : Surjective g) : Exact (map r f) (map r g) 
Arguments: {R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} {M'' : Type u_4} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M''] {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''} (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g)
Goal: Function.Exact ⇑((QuotSMulTop.map r) f) ⇑((QuotSMulTop.map r) g)
-- MvPowerSeries.coeff_truncFun
theorem coeff_truncFun (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
    (truncFun n φ).coeff m = if m < n then coeff R m φ else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (n m : σ →₀ ℕ) (φ : MvPowerSeries σ R)
Goal: MvPolynomial.coeff m (MvPowerSeries.truncFun n φ) = if m < n then (MvPowerSeries.coeff R m) φ else 0
-- MvPowerSeries.coeff_trunc
theorem coeff_trunc (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
    (trunc R n φ).coeff m = if m < n then coeff R m φ else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (n m : σ →₀ ℕ) (φ : MvPowerSeries σ R)
Goal: MvPolynomial.coeff m ((MvPowerSeries.trunc R n) φ) = if m < n then (MvPowerSeries.coeff R m) φ else 0
-- MvPowerSeries.trunc_one
@[simp]
theorem trunc_one (n : σ →₀ ℕ) (hnn : n ≠ 0) : trunc R n 1 = 1 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (n : σ →₀ ℕ) (hnn : n ≠ 0)
Goal: (MvPowerSeries.trunc R n) 1 = 1
-- MvPowerSeries.trunc_c
@[simp]
theorem trunc_c (n : σ →₀ ℕ) (hnn : n ≠ 0) (a : R) : trunc R n (C σ R a) = MvPolynomial.C a 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (n : σ →₀ ℕ) (hnn : n ≠ 0) (a : R)
Goal: (MvPowerSeries.trunc R n) ((MvPowerSeries.C σ R) a) = MvPolynomial.C a
-- Int.gcd_eq_one_iff_coprime
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b = 1 ↔ IsCoprime a b 
Arguments: {a b : ℤ}
Goal: a.gcd b = 1 ↔ IsCoprime a b
-- Int.coprime_iff_nat_coprime
theorem coprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs 
Arguments: {a b : ℤ}
Goal: IsCoprime a b ↔ a.natAbs.Coprime b.natAbs
-- Int.gcd_ne_one_iff_gcd_mul_right_ne_one
theorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :
    a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 
Arguments: {a : ℤ} {m n : ℕ}
Goal: a.gcd (↑m * ↑n) ≠ 1 ↔ a.gcd ↑m ≠ 1 ∨ a.gcd ↑n ≠ 1
-- Int.sq_of_gcd_eq_one
theorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :
    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 
Arguments: {a b c : ℤ} (h : a.gcd b = 1) (heq : a * b = c ^ 2)
Goal: ∃ a0, a = a0 ^ 2 ∨ a = -a0 ^ 2
-- Int.sq_of_coprime
theorem sq_of_coprime {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2) :
    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 
Arguments: {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2)
Goal: ∃ a0, a = a0 ^ 2 ∨ a = -a0 ^ 2
-- Int.natAbs_euclideanDomain_gcd
theorem natAbs_euclideanDomain_gcd (a b : ℤ) :
    Int.natAbs (EuclideanDomain.gcd a b) = Int.gcd a b 
Arguments: (a b : ℤ)
Goal: (EuclideanDomain.gcd a b).natAbs = a.gcd b
-- Int.Prime.dvd_mul
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :
    p ∣ m.natAbs ∨ p ∣ n.natAbs 
Arguments: {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : ↑p ∣ m * n)
Goal: p ∣ m.natAbs ∨ p ∣ n.natAbs
-- Int.Prime.dvd_mul'
theorem Int.Prime.dvd_mul' {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :
    (p : ℤ) ∣ m ∨ (p : ℤ) ∣ n 
Arguments: {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : ↑p ∣ m * n)
Goal: ↑p ∣ m ∨ ↑p ∣ n
-- Int.Prime.dvd_pow
theorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :
    p ∣ n.natAbs 
Arguments: {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : ↑p ∣ n ^ k)
Goal: p ∣ n.natAbs
-- Int.Prime.dvd_pow'
theorem Int.Prime.dvd_pow' {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :
    (p : ℤ) ∣ n 
Arguments: {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : ↑p ∣ n ^ k)
Goal: ↑p ∣ n
-- prime_two_or_dvd_of_dvd_two_mul_pow_self_two
theorem prime_two_or_dvd_of_dvd_two_mul_pow_self_two {m : ℤ} {p : ℕ} (hp : Nat.Prime p)
    (h : (p : ℤ) ∣ 2 * m ^ 2) : p = 2 ∨ p ∣ Int.natAbs m 
Arguments: {m : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : ↑p ∣ 2 * m ^ 2)
Goal: p = 2 ∨ p ∣ m.natAbs
-- Int.exists_prime_and_dvd
theorem Int.exists_prime_and_dvd {n : ℤ} (hn : n.natAbs ≠ 1) : ∃ p, Prime p ∧ p ∣ n 
Arguments: {n : ℤ} (hn : n.natAbs ≠ 1)
Goal: ∃ p, Prime p ∧ p ∣ n
-- Int.prime_iff_natAbs_prime
theorem Int.prime_iff_natAbs_prime {k : ℤ} : Prime k ↔ Nat.Prime k.natAbs 
Arguments: {k : ℤ}
Goal: Prime k ↔ Nat.Prime k.natAbs
-- Int.zmultiples_natAbs
theorem zmultiples_natAbs (a : ℤ) :
    AddSubgroup.zmultiples (a.natAbs : ℤ) = AddSubgroup.zmultiples a 
Arguments: (a : ℤ)
Goal: AddSubgroup.zmultiples ↑a.natAbs = AddSubgroup.zmultiples a
-- Int.span_natAbs
theorem span_natAbs (a : ℤ) : Ideal.span ({(a.natAbs : ℤ)} : Set ℤ) = Ideal.span {a} 
Arguments: (a : ℤ)
Goal: Ideal.span {↑a.natAbs} = Ideal.span {a}
-- Int.eq_pow_of_mul_eq_pow_odd_left
theorem eq_pow_of_mul_eq_pow_odd_left {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k)
    (h : a * b = c ^ k) : ∃ d, a = d ^ k 
Arguments: {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k) (h : a * b = c ^ k)
Goal: ∃ d, a = d ^ k
-- Int.eq_pow_of_mul_eq_pow_odd_right
theorem eq_pow_of_mul_eq_pow_odd_right {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k)
    (h : a * b = c ^ k) : ∃ d, b = d ^ k 
Arguments: {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k) (h : a * b = c ^ k)
Goal: ∃ d, b = d ^ k
-- Int.eq_pow_of_mul_eq_pow_odd
theorem eq_pow_of_mul_eq_pow_odd {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k)
    (h : a * b = c ^ k) : (∃ d, a = d ^ k) ∧ ∃ e, b = e ^ k 
Arguments: {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k) (h : a * b = c ^ k)
Goal: (∃ d, a = d ^ k) ∧ ∃ e, b = e ^ k
-- MeasureTheory.AEStronglyMeasurable.truncation
theorem _root_.MeasureTheory.AEStronglyMeasurable.truncation (hf : AEStronglyMeasurable f μ)
    {A : ℝ} : AEStronglyMeasurable (truncation f A) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ}
Goal: MeasureTheory.AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ
-- ProbabilityTheory.abs_truncation_le_bound
theorem abs_truncation_le_bound (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |A| 
Arguments: {α : Type u_1} (f : α → ℝ) (A : ℝ) (x : α)
Goal: |ProbabilityTheory.truncation f A x| ≤ |A|
-- ProbabilityTheory.truncation_zero
@[simp]
theorem truncation_zero (f : α → ℝ) : truncation f 0 = 0 
Arguments: {α : Type u_1} (f : α → ℝ)
Goal: ProbabilityTheory.truncation f 0 = 0
-- ProbabilityTheory.abs_truncation_le_abs_self
theorem abs_truncation_le_abs_self (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |f x| 
Arguments: {α : Type u_1} (f : α → ℝ) (A : ℝ) (x : α)
Goal: |ProbabilityTheory.truncation f A x| ≤ |f x|
-- ProbabilityTheory.truncation_eq_self
theorem truncation_eq_self {f : α → ℝ} {A : ℝ} {x : α} (h : |f x| < A) :
    truncation f A x = f x 
Arguments: {α : Type u_1} {f : α → ℝ} {A : ℝ} {x : α} (h : |f x| < A)
Goal: ProbabilityTheory.truncation f A x = f x
-- ProbabilityTheory.truncation_eq_of_nonneg
theorem truncation_eq_of_nonneg {f : α → ℝ} {A : ℝ} (h : ∀ x, 0 ≤ f x) :
    truncation f A = indicator (Set.Ioc 0 A) id ∘ f 
Arguments: {α : Type u_1} {f : α → ℝ} {A : ℝ} (h : ∀ (x : α), 0 ≤ f x)
Goal: ProbabilityTheory.truncation f A = (Set.Ioc 0 A).indicator id ∘ f
-- ProbabilityTheory.truncation_nonneg
theorem truncation_nonneg {f : α → ℝ} (A : ℝ) {x : α} (h : 0 ≤ f x) : 0 ≤ truncation f A x 
Arguments: {α : Type u_1} {f : α → ℝ} (A : ℝ) {x : α} (h : 0 ≤ f x)
Goal: 0 ≤ ProbabilityTheory.truncation f A x
-- MeasureTheory.AEStronglyMeasurable.memℒp_truncation
theorem _root_.MeasureTheory.AEStronglyMeasurable.memℒp_truncation [IsFiniteMeasure μ]
    (hf : AEStronglyMeasurable f μ) {A : ℝ} {p : ℝ≥0∞} : Memℒp (truncation f A) p μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} {p : ℝ≥0∞}
Goal: MeasureTheory.Memℒp (ProbabilityTheory.truncation f A) p μ
-- MeasureTheory.AEStronglyMeasurable.integrable_truncation
theorem _root_.MeasureTheory.AEStronglyMeasurable.integrable_truncation [IsFiniteMeasure μ]
    (hf : AEStronglyMeasurable f μ) {A : ℝ} : Integrable (truncation f A) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ}
Goal: MeasureTheory.Integrable (ProbabilityTheory.truncation f A) μ
-- ProbabilityTheory.moment_truncation_eq_intervalIntegral
theorem moment_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A)
    {n : ℕ} (hn : n ≠ 0) : ∫ x, truncation f A x ^ n ∂μ = ∫ y in -A..A, y ^ n ∂Measure.map f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A) {n : ℕ} (hn : n ≠ 0)
Goal: ∫ (x : α), ProbabilityTheory.truncation f A x ^ n ∂μ = ∫ (y : ℝ) in -A..A, y ^ n ∂MeasureTheory.Measure.map f μ
-- ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg
theorem moment_truncation_eq_intervalIntegral_of_nonneg (hf : AEStronglyMeasurable f μ) {A : ℝ}
    {n : ℕ} (hn : n ≠ 0) (h'f : 0 ≤ f) :
    ∫ x, truncation f A x ^ n ∂μ = ∫ y in (0)..A, y ^ n ∂Measure.map f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} {n : ℕ} (hn : n ≠ 0) (h'f : 0 ≤ f)
Goal: ∫ (x : α), ProbabilityTheory.truncation f A x ^ n ∂μ = ∫ (y : ℝ) in 0 ..A, y ^ n ∂MeasureTheory.Measure.map f μ
-- ProbabilityTheory.integral_truncation_eq_intervalIntegral
theorem integral_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f μ) {A : ℝ}
    (hA : 0 ≤ A) : ∫ x, truncation f A x ∂μ = ∫ y in -A..A, y ∂Measure.map f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A)
Goal: ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂MeasureTheory.Measure.map f μ
-- ProbabilityTheory.integral_truncation_eq_intervalIntegral_of_nonneg
theorem integral_truncation_eq_intervalIntegral_of_nonneg (hf : AEStronglyMeasurable f μ) {A : ℝ}
    (h'f : 0 ≤ f) : ∫ x, truncation f A x ∂μ = ∫ y in (0)..A, y ∂Measure.map f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} (h'f : 0 ≤ f)
Goal: ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ = ∫ (y : ℝ) in 0 ..A, y ∂MeasureTheory.Measure.map f μ
-- ProbabilityTheory.integral_truncation_le_integral_of_nonneg
theorem integral_truncation_le_integral_of_nonneg (hf : Integrable f μ) (h'f : 0 ≤ f) {A : ℝ} :
    ∫ x, truncation f A x ∂μ ≤ ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) (h'f : 0 ≤ f) {A : ℝ}
Goal: ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ ≤ ∫ (x : α), f x ∂μ
-- ProbabilityTheory.tendsto_integral_truncation
theorem tendsto_integral_truncation {f : α → ℝ} (hf : Integrable f μ) :
    Tendsto (fun A => ∫ x, truncation f A x ∂μ) atTop (𝓝 (∫ x, f x ∂μ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: Filter.Tendsto (fun A => ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ) Filter.atTop (𝓝 (∫ (x : α), f x ∂μ))
-- ProbabilityTheory.IdentDistrib.truncation
theorem IdentDistrib.truncation {β : Type*} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ}
    {g : β → ℝ} (h : IdentDistrib f g μ ν) {A : ℝ} :
    IdentDistrib (truncation f A) (truncation g A) μ ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_2} [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → ℝ} {g : β → ℝ} (h : ProbabilityTheory.IdentDistrib f g μ ν) {A : ℝ}
Goal: ProbabilityTheory.IdentDistrib (ProbabilityTheory.truncation f A) (ProbabilityTheory.truncation g A) μ ν
-- ProbabilityTheory.sum_prob_mem_Ioc_le
theorem sum_prob_mem_Ioc_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) {K : ℕ} {N : ℕ}
    (hKN : K ≤ N) :
    ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc (j : ℝ) N} ≤ ENNReal.ofReal (𝔼[X] + 1) 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X : Ω → ℝ} (hint : MeasureTheory.Integrable X ℙ) (hnonneg : 0 ≤ X) {K N : ℕ} (hKN : K ≤ N)
Goal: ∑ j ∈ Finset.range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)
-- ProbabilityTheory.tsum_prob_mem_Ioi_lt_top
theorem tsum_prob_mem_Ioi_lt_top {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) :
    (∑' j : ℕ, ℙ {ω | X ω ∈ Set.Ioi (j : ℝ)}) < ∞ 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X : Ω → ℝ} (hint : MeasureTheory.Integrable X ℙ) (hnonneg : 0 ≤ X)
Goal: ∑' (j : ℕ), ℙ {ω | X ω ∈ Set.Ioi ↑j} < ⊤
-- ProbabilityTheory.sum_variance_truncation_le
theorem sum_variance_truncation_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) :
    ∑ j ∈ range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[truncation X j ^ 2] ≤ 2 * 𝔼[X] 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X : Ω → ℝ} (hint : MeasureTheory.Integrable X ℙ) (hnonneg : 0 ≤ X) (K : ℕ)
Goal: ∑ j ∈ Finset.range K, (↑j ^ 2)⁻¹ * ∫ (a : Ω), (ProbabilityTheory.truncation X ↑j ^ 2) a ≤ 2 * ∫ (a : Ω), X a
-- ProbabilityTheory.strong_law_aux1
theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) : ∀ᵐ ω, ∀ᶠ n : ℕ in atTop,
    |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω - 𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]| <
    ε * ⌊c ^ n⌋₊ 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) ℙ) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j) ℙ) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) ℙ ℙ) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε)
Goal: ∀ᵐ (ω : Ω),
  ∀ᶠ (n : ℕ) in Filter.atTop,
    |∑ i ∈ Finset.range ⌊c ^ n⌋₊, ProbabilityTheory.truncation (X i) (↑i) ω -
          ∫ (a : Ω), (∑ i ∈ Finset.range ⌊c ^ n⌋₊, ProbabilityTheory.truncation (X i) ↑i) a| <
      ε * ↑⌊c ^ n⌋₊
-- ProbabilityTheory.strong_law_aux2
theorem strong_law_aux2 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω -
      𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]) =o[atTop] fun n : ℕ => (⌊c ^ n⌋₊ : ℝ) 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) ℙ) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j) ℙ) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) ℙ ℙ) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) {c : ℝ} (c_one : 1 < c)
Goal: ∀ᵐ (ω : Ω),
  (fun n =>
      ∑ i ∈ Finset.range ⌊c ^ n⌋₊, ProbabilityTheory.truncation (X i) (↑i) ω -
        ∫ (a : Ω), (∑ i ∈ Finset.range ⌊c ^ n⌋₊, ProbabilityTheory.truncation (X i) ↑i) a) =o[Filter.atTop]
    fun n => ↑⌊c ^ n⌋₊
-- ProbabilityTheory.strong_law_aux3
theorem strong_law_aux3 :
    (fun n => 𝔼[∑ i ∈ range n, truncation (X i) i] - n * 𝔼[X 0]) =o[atTop] ((↑) : ℕ → ℝ) 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) ℙ) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) ℙ ℙ)
Goal: (fun n =>
    (∫ (a : Ω), (∑ i ∈ Finset.range n, ProbabilityTheory.truncation (X i) ↑i) a) -
      ↑n * ∫ (a : Ω), X 0 a) =o[Filter.atTop]
  Nat.cast
-- ProbabilityTheory.strong_law_aux4
theorem strong_law_aux4 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω - ⌊c ^ n⌋₊ * 𝔼[X 0]) =o[atTop]
    fun n : ℕ => (⌊c ^ n⌋₊ : ℝ) 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) ℙ) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j) ℙ) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) ℙ ℙ) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) {c : ℝ} (c_one : 1 < c)
Goal: ∀ᵐ (ω : Ω),
  (fun n =>
      ∑ i ∈ Finset.range ⌊c ^ n⌋₊, ProbabilityTheory.truncation (X i) (↑i) ω -
        ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[Filter.atTop]
    fun n => ↑⌊c ^ n⌋₊