ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsCoprime.add_mul_left_right_iff
theorem add_mul_left_right_iff {x y z : R} : IsCoprime x (y + x * z) ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime x (y + x * z) ↔ IsCoprime x y
-- IsCoprime.add_mul_right_right_iff
theorem add_mul_right_right_iff {x y z : R} : IsCoprime x (y + z * x) ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime x (y + z * x) ↔ IsCoprime x y
-- IsCoprime.mul_add_left_left_iff
theorem mul_add_left_left_iff {x y z : R} : IsCoprime (y * z + x) y ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime (y * z + x) y ↔ IsCoprime x y
-- IsCoprime.mul_add_right_left_iff
theorem mul_add_right_left_iff {x y z : R} : IsCoprime (z * y + x) y ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime (z * y + x) y ↔ IsCoprime x y
-- IsCoprime.mul_add_left_right_iff
theorem mul_add_left_right_iff {x y z : R} : IsCoprime x (x * z + y) ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime x (x * z + y) ↔ IsCoprime x y
-- IsCoprime.mul_add_right_right_iff
theorem mul_add_right_right_iff {x y z : R} : IsCoprime x (z * x + y) ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime x (z * x + y) ↔ IsCoprime x y
-- IsCoprime.neg_left
theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y)
Goal: IsCoprime (-x) y
-- IsCoprime.neg_left_iff
theorem neg_left_iff (x y : R) : IsCoprime (-x) y ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] (x y : R)
Goal: IsCoprime (-x) y ↔ IsCoprime x y
-- IsCoprime.neg_right
theorem neg_right {x y : R} (h : IsCoprime x y) : IsCoprime x (-y) 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y)
Goal: IsCoprime x (-y)
-- IsCoprime.neg_right_iff
theorem neg_right_iff (x y : R) : IsCoprime x (-y) ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] (x y : R)
Goal: IsCoprime x (-y) ↔ IsCoprime x y
-- IsCoprime.neg_neg
theorem neg_neg {x y : R} (h : IsCoprime x y) : IsCoprime (-x) (-y) 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y)
Goal: IsCoprime (-x) (-y)
-- IsCoprime.neg_neg_iff
theorem neg_neg_iff (x y : R) : IsCoprime (-x) (-y) ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] (x y : R)
Goal: IsCoprime (-x) (-y) ↔ IsCoprime x y
-- IsCoprime.sq_add_sq_ne_zero
theorem sq_add_sq_ne_zero {R : Type*} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b) :
    a ^ 2 + b ^ 2 ≠ 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {a b : R} (h : IsCoprime a b)
Goal: a ^ 2 + b ^ 2 ≠ 0
-- IsRelPrime.add_mul_left_left
theorem add_mul_left_left (h : IsRelPrime x y) : IsRelPrime (x + y * z) y 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (z : R) (h : IsRelPrime x y)
Goal: IsRelPrime (x + y * z) y
-- IsRelPrime.add_mul_right_left
theorem add_mul_right_left : IsRelPrime (x + z * y) y 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime (x + z * y) y
-- IsRelPrime.add_mul_left_right
theorem add_mul_left_right : IsRelPrime x (y + x * z) 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime x (y + x * z)
-- IsRelPrime.add_mul_right_right
theorem add_mul_right_right : IsRelPrime x (y + z * x) 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime x (y + z * x)
-- IsRelPrime.mul_add_left_left
theorem mul_add_left_left : IsRelPrime (y * z + x) y 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime (y * z + x) y
-- IsRelPrime.mul_add_right_left
theorem mul_add_right_left : IsRelPrime (z * y + x) y 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime (z * y + x) y
-- IsRelPrime.mul_add_left_right
theorem mul_add_left_right : IsRelPrime x (x * z + y) 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime x (x * z + y)
-- IsRelPrime.mul_add_right_right
theorem mul_add_right_right : IsRelPrime x (z * x + y) 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y) (z : R)
Goal: IsRelPrime x (z * x + y)
-- IsRelPrime.add_mul_left_left_iff
theorem add_mul_left_left_iff : IsRelPrime (x + y * z) y ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime (x + y * z) y ↔ IsRelPrime x y
-- IsRelPrime.add_mul_right_left_iff
theorem add_mul_right_left_iff : IsRelPrime (x + z * y) y ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime (x + z * y) y ↔ IsRelPrime x y
-- IsRelPrime.add_mul_left_right_iff
theorem add_mul_left_right_iff : IsRelPrime x (y + x * z) ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime x (y + x * z) ↔ IsRelPrime x y
-- IsRelPrime.add_mul_right_right_iff
theorem add_mul_right_right_iff : IsRelPrime x (y + z * x) ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime x (y + z * x) ↔ IsRelPrime x y
-- IsRelPrime.mul_add_left_left_iff
theorem mul_add_left_left_iff {x y z : R} : IsRelPrime (y * z + x) y ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime (y * z + x) y ↔ IsRelPrime x y
-- IsRelPrime.mul_add_right_left_iff
theorem mul_add_right_left_iff {x y z : R} : IsRelPrime (z * y + x) y ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime (z * y + x) y ↔ IsRelPrime x y
-- IsRelPrime.mul_add_left_right_iff
theorem mul_add_left_right_iff {x y z : R} : IsRelPrime x (x * z + y) ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime x (x * z + y) ↔ IsRelPrime x y
-- IsRelPrime.mul_add_right_right_iff
theorem mul_add_right_right_iff {x y z : R} : IsRelPrime x (z * x + y) ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] {x y z : R}
Goal: IsRelPrime x (z * x + y) ↔ IsRelPrime x y
-- IsRelPrime.neg_left
theorem neg_left : IsRelPrime (-x) y 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y)
Goal: IsRelPrime (-x) y
-- IsRelPrime.neg_right
theorem neg_right : IsRelPrime x (-y) 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y)
Goal: IsRelPrime x (-y)
-- IsRelPrime.neg_neg
protected theorem neg_neg : IsRelPrime (-x) (-y) 
Arguments: {R : Type u_1} [CommRing R] {x y : R} (h : IsRelPrime x y)
Goal: IsRelPrime (-x) (-y)
-- IsRelPrime.neg_left_iff
theorem neg_left_iff (x y : R) : IsRelPrime (-x) y ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] (x y : R)
Goal: IsRelPrime (-x) y ↔ IsRelPrime x y
-- IsRelPrime.neg_right_iff
theorem neg_right_iff (x y : R) : IsRelPrime x (-y) ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] (x y : R)
Goal: IsRelPrime x (-y) ↔ IsRelPrime x y
-- IsRelPrime.neg_neg_iff
theorem neg_neg_iff (x y : R) : IsRelPrime (-x) (-y) ↔ IsRelPrime x y 
Arguments: {R : Type u_1} [CommRing R] (x y : R)
Goal: IsRelPrime (-x) (-y) ↔ IsRelPrime x y
-- MeasureTheory.Submartingale.expected_stoppedValue_mono
theorem Submartingale.expected_stoppedValue_mono [SigmaFiniteFiltration μ 𝒢]
    (hf : Submartingale f 𝒢 μ) (hτ : IsStoppingTime 𝒢 τ) (hπ : IsStoppingTime 𝒢 π) (hle : τ ≤ π)
    {N : ℕ} (hbdd : ∀ ω, π ω ≤ N) : μ[stoppedValue f τ] ≤ μ[stoppedValue f π] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {τ π : Ω → ℕ} [MeasureTheory.SigmaFiniteFiltration μ 𝒢] (hf : MeasureTheory.Submartingale f 𝒢 μ) (hτ : MeasureTheory.IsStoppingTime 𝒢 τ) (hπ : MeasureTheory.IsStoppingTime 𝒢 π) (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ (ω : Ω), π ω ≤ N)
Goal: ∫ (x : Ω), MeasureTheory.stoppedValue f τ x ∂μ ≤ ∫ (x : Ω), MeasureTheory.stoppedValue f π x ∂μ
-- MeasureTheory.submartingale_of_expected_stoppedValue_mono
theorem submartingale_of_expected_stoppedValue_mono [IsFiniteMeasure μ] (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ τ π : Ω → ℕ, IsStoppingTime 𝒢 τ → IsStoppingTime 𝒢 π →
      τ ≤ π → (∃ N, ∀ ω, π ω ≤ N) → μ[stoppedValue f τ] ≤ μ[stoppedValue f π]) :
    Submartingale f 𝒢 μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) (hf :
  ∀ (τ π : Ω → ℕ),
    MeasureTheory.IsStoppingTime 𝒢 τ →
      MeasureTheory.IsStoppingTime 𝒢 π →
        τ ≤ π →
          (∃ N, ∀ (ω : Ω), π ω ≤ N) →
            ∫ (x : Ω), MeasureTheory.stoppedValue f τ x ∂μ ≤ ∫ (x : Ω), MeasureTheory.stoppedValue f π x ∂μ)
Goal: MeasureTheory.Submartingale f 𝒢 μ
-- MeasureTheory.submartingale_iff_expected_stoppedValue_mono
theorem submartingale_iff_expected_stoppedValue_mono [IsFiniteMeasure μ] (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ) :
    Submartingale f 𝒢 μ ↔ ∀ τ π : Ω → ℕ, IsStoppingTime 𝒢 τ → IsStoppingTime 𝒢 π →
      τ ≤ π → (∃ N, ∀ x, π x ≤ N) → μ[stoppedValue f τ] ≤ μ[stoppedValue f π] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ)
Goal: MeasureTheory.Submartingale f 𝒢 μ ↔
  ∀ (τ π : Ω → ℕ),
    MeasureTheory.IsStoppingTime 𝒢 τ →
      MeasureTheory.IsStoppingTime 𝒢 π →
        τ ≤ π →
          (∃ N, ∀ (x : Ω), π x ≤ N) →
            ∫ (x : Ω), MeasureTheory.stoppedValue f τ x ∂μ ≤ ∫ (x : Ω), MeasureTheory.stoppedValue f π x ∂μ
-- MeasureTheory.Submartingale.stoppedProcess
protected theorem Submartingale.stoppedProcess [IsFiniteMeasure μ] (h : Submartingale f 𝒢 μ)
    (hτ : IsStoppingTime 𝒢 τ) : Submartingale (stoppedProcess f τ) 𝒢 μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {τ : Ω → ℕ} [MeasureTheory.IsFiniteMeasure μ] (h : MeasureTheory.Submartingale f 𝒢 μ) (hτ : MeasureTheory.IsStoppingTime 𝒢 τ)
Goal: MeasureTheory.Submartingale (MeasureTheory.stoppedProcess f τ) 𝒢 μ
-- MeasureTheory.smul_le_stoppedValue_hitting
theorem smul_le_stoppedValue_hitting [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) {ε : ℝ≥0}
    (n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
    ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
      stoppedValue f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω ∂μ) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hsub : MeasureTheory.Submartingale f 𝒢 μ) {ε : ℝ≥0} (n : ℕ)
Goal: ε • μ {ω | ↑ε ≤ (Finset.range (n + 1)).sup' ⋯ fun k => f k ω} ≤
  ENNReal.ofReal
    (∫ (ω : Ω) in {ω | ↑ε ≤ (Finset.range (n + 1)).sup' ⋯ fun k => f k ω},
      MeasureTheory.stoppedValue f (MeasureTheory.hitting f {y | ↑ε ≤ y} 0 n) ω ∂μ)
-- MeasureTheory.maximal_ineq
theorem maximal_ineq [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0}
    (n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
    ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
      f n ω ∂μ) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hsub : MeasureTheory.Submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0} (n : ℕ)
Goal: ε • μ {ω | ↑ε ≤ (Finset.range (n + 1)).sup' ⋯ fun k => f k ω} ≤
  ENNReal.ofReal (∫ (ω : Ω) in {ω | ↑ε ≤ (Finset.range (n + 1)).sup' ⋯ fun k => f k ω}, f n ω ∂μ)
-- Polynomial.aeval_map_algebraMap
@[simp]
theorem aeval_map_algebraMap (x : B) (p : R[X]) : aeval x (map (algebraMap R A) p) = aeval x p 
Arguments: {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : B) (p : R[X])
Goal: (Polynomial.aeval x) (Polynomial.map (algebraMap R A) p) = (Polynomial.aeval x) p
-- Polynomial.eval_map_algebraMap
@[simp]
lemma eval_map_algebraMap (P : R[X]) (a : A) :
    (map (algebraMap R A) P).eval a = aeval a P 
Arguments: {R : Type u_1} (A : Type u_2) [CommSemiring R] [CommSemiring A] [Algebra R A] (P : R[X]) (a : A)
Goal: Polynomial.eval a (Polynomial.map (algebraMap R A) P) = (Polynomial.aeval a) P
-- Polynomial.aeval_algebraMap_apply
theorem aeval_algebraMap_apply (x : A) (p : R[X]) :
    aeval (algebraMap A B x) p = algebraMap A B (aeval x p) 
Arguments: {R : Type u_1} {A : Type u_2} (B : Type u_3) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : A) (p : R[X])
Goal: (Polynomial.aeval ((algebraMap A B) x)) p = (algebraMap A B) ((Polynomial.aeval x) p)
-- Polynomial.aeval_algebraMap_eq_zero_iff
@[simp]
theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X]) :
    aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 
Arguments: {R : Type u_1} {A : Type u_2} (B : Type u_3) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X])
Goal: (Polynomial.aeval ((algebraMap A B) x)) p = 0 ↔ (Polynomial.aeval x) p = 0
-- Polynomial.aeval_algebraMap_eq_zero_iff_of_injective
theorem aeval_algebraMap_eq_zero_iff_of_injective {x : A} {p : R[X]}
    (h : Function.Injective (algebraMap A B)) : aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : A} {p : R[X]} (h : Function.Injective ⇑(algebraMap A B))
Goal: (Polynomial.aeval ((algebraMap A B) x)) p = 0 ↔ (Polynomial.aeval x) p = 0
-- Subalgebra.aeval_coe
@[simp]
theorem aeval_coe (S : Subalgebra R A) (x : S) (p : R[X]) : aeval (x : A) p = aeval x p 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] (S : Subalgebra R A) (x : ↥S) (p : R[X])
Goal: (Polynomial.aeval ↑x) p = ↑((Polynomial.aeval x) p)
-- Polynomial.X_pow_sub_X_sub_one_irreducible_aux
theorem X_pow_sub_X_sub_one_irreducible_aux (z : ℂ) : ¬(z ^ n = z + 1 ∧ z ^ n + z ^ 2 = 0) 
Arguments: {n : ℕ} (z : ℂ)
Goal: ¬(z ^ n = z + 1 ∧ z ^ n + z ^ 2 = 0)
-- Polynomial.X_pow_sub_X_sub_one_irreducible
theorem X_pow_sub_X_sub_one_irreducible (hn1 : n ≠ 1) : Irreducible (X ^ n - X - 1 : ℤ[X]) 
Arguments: {n : ℕ} (hn1 : n ≠ 1)
Goal: Irreducible (Polynomial.X ^ n - Polynomial.X - 1)
-- Polynomial.X_pow_sub_X_sub_one_irreducible_rat
theorem X_pow_sub_X_sub_one_irreducible_rat (hn1 : n ≠ 1) : Irreducible (X ^ n - X - 1 : ℚ[X]) 
Arguments: {n : ℕ} (hn1 : n ≠ 1)
Goal: Irreducible (Polynomial.X ^ n - Polynomial.X - 1)
-- MeasureTheory.Memℒp.evariance_lt_top
theorem _root_.MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
    evariance X μ < ∞ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.Memℒp X 2 μ)
Goal: ProbabilityTheory.evariance X μ < ⊤
-- ProbabilityTheory.evariance_eq_top
theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
    evariance X μ = ∞ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hXm : MeasureTheory.AEStronglyMeasurable X μ) (hX : ¬MeasureTheory.Memℒp X 2 μ)
Goal: ProbabilityTheory.evariance X μ = ⊤
-- ProbabilityTheory.evariance_lt_top_iff_memℒp
theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
    evariance X μ < ∞ ↔ Memℒp X 2 μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.AEStronglyMeasurable X μ)
Goal: ProbabilityTheory.evariance X μ < ⊤ ↔ MeasureTheory.Memℒp X 2 μ
-- MeasureTheory.Memℒp.ofReal_variance_eq
theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
    ENNReal.ofReal (variance X μ) = evariance X μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.Memℒp X 2 μ)
Goal: ENNReal.ofReal (ProbabilityTheory.variance X μ) = ProbabilityTheory.evariance X μ
-- ProbabilityTheory.evariance_eq_lintegral_ofReal
theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
    evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.evariance X μ = ∫⁻ (ω : Ω), ENNReal.ofReal ((X ω - ∫ (x : Ω), X x ∂μ) ^ 2) ∂μ
-- MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero
theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
    (hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} (hX : MeasureTheory.Memℒp X 2 μ) (hXint : ∫ (x : Ω), X x ∂μ = 0)
Goal: ProbabilityTheory.variance X μ = ∫ (x : Ω), (X ^ 2) x ∂μ
-- MeasureTheory.Memℒp.variance_eq
theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
    variance X μ = μ[(X - fun _ => μ[X] :) ^ (2 : Nat)] 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hX : MeasureTheory.Memℒp X 2 μ)
Goal: ProbabilityTheory.variance X μ = ∫ (x : Ω), ((X - fun x => ∫ (x : Ω), X x ∂μ) ^ 2) x ∂μ
-- ProbabilityTheory.evariance_zero
@[simp]
theorem evariance_zero : evariance 0 μ = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω}
Goal: ProbabilityTheory.evariance 0 μ = 0
-- ProbabilityTheory.evariance_eq_zero_iff
theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) :
    evariance X μ = 0 ↔ X =ᵐ[μ] fun _ => μ[X] 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} (hX : AEMeasurable X μ)
Goal: ProbabilityTheory.evariance X μ = 0 ↔ X =ᵐ[μ] fun x => ∫ (x : Ω), X x ∂μ
-- ProbabilityTheory.evariance_mul
theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
    evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * evariance X μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} (c : ℝ) (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * ProbabilityTheory.evariance X μ
-- ProbabilityTheory.variance_zero
@[simp]
theorem variance_zero (μ : Measure Ω) : variance 0 μ = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.variance 0 μ = 0
-- ProbabilityTheory.variance_nonneg
theorem variance_nonneg (X : Ω → ℝ) (μ : Measure Ω) : 0 ≤ variance X μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω)
Goal: 0 ≤ ProbabilityTheory.variance X μ
-- ProbabilityTheory.variance_mul
theorem variance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
    variance (fun ω => c * X ω) μ = c ^ 2 * variance X μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} (c : ℝ) (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.variance (fun ω => c * X ω) μ = c ^ 2 * ProbabilityTheory.variance X μ
-- ProbabilityTheory.variance_smul
theorem variance_smul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
    variance (c • X) μ = c ^ 2 * variance X μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} (c : ℝ) (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.variance (c • X) μ = c ^ 2 * ProbabilityTheory.variance X μ
-- ProbabilityTheory.variance_smul'
theorem variance_smul' {A : Type*} [CommSemiring A] [Algebra A ℝ] (c : A) (X : Ω → ℝ)
    (μ : Measure Ω) : variance (c • X) μ = c ^ 2 • variance X μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {A : Type u_2} [CommSemiring A] [Algebra A ℝ] (c : A) (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.variance (c • X) μ = c ^ 2 • ProbabilityTheory.variance X μ
-- ProbabilityTheory.variance_def'
theorem variance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Memℒp X 2) :
    Var[X] = 𝔼[X ^ 2] - 𝔼[X] ^ 2 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X : Ω → ℝ} (hX : MeasureTheory.Memℒp X 2 ℙ)
Goal: ProbabilityTheory.variance X ℙ = (∫ (a : Ω), (X ^ 2) a) - (∫ (a : Ω), X a) ^ 2
-- ProbabilityTheory.variance_le_expectation_sq
theorem variance_le_expectation_sq [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ}
    (hm : AEStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X : Ω → ℝ} (hm : MeasureTheory.AEStronglyMeasurable X ℙ)
Goal: ProbabilityTheory.variance X ℙ ≤ ∫ (a : Ω), (X ^ 2) a
-- ProbabilityTheory.evariance_def'
theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) :
    eVar[X] = (∫⁻ ω, (‖X ω‖₊ ^ 2 :)) - ENNReal.ofReal (𝔼[X] ^ 2) 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X : Ω → ℝ} (hX : MeasureTheory.AEStronglyMeasurable X ℙ)
Goal: ProbabilityTheory.evariance X ℙ = (∫⁻ (ω : Ω), ↑(‖X ω‖₊ ^ 2)) - ENNReal.ofReal ((∫ (a : Ω), X a) ^ 2)
-- ProbabilityTheory.meas_ge_le_evariance_div_sq
theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) {c : ℝ≥0}
    (hc : c ≠ 0) : ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {X : Ω → ℝ} (hX : MeasureTheory.AEStronglyMeasurable X ℙ) {c : ℝ≥0} (hc : c ≠ 0)
Goal: ℙ {ω | ↑c ≤ |X ω - ∫ (a : Ω), X a|} ≤ ProbabilityTheory.evariance X ℙ / ↑c ^ 2
-- ProbabilityTheory.meas_ge_le_variance_div_sq
theorem meas_ge_le_variance_div_sq [@IsFiniteMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Memℒp X 2) {c : ℝ}
    (hc : 0 < c) : ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ENNReal.ofReal (Var[X] / c ^ 2) 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ} (hX : MeasureTheory.Memℒp X 2 ℙ) {c : ℝ} (hc : 0 < c)
Goal: ℙ {ω | c ≤ |X ω - ∫ (a : Ω), X a|} ≤ ENNReal.ofReal (ProbabilityTheory.variance X ℙ / c ^ 2)
-- ProbabilityTheory.IndepFun.variance_add
theorem IndepFun.variance_add [@IsProbabilityMeasure Ω _ ℙ] {X Y : Ω → ℝ} (hX : Memℒp X 2)
    (hY : Memℒp Y 2) (h : @IndepFun _ _ _ (_) _ _ X Y ℙ) : Var[X + Y] = Var[X] + Var[Y] 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {X Y : Ω → ℝ} (hX : MeasureTheory.Memℒp X 2 ℙ) (hY : MeasureTheory.Memℒp Y 2 ℙ) (h : ProbabilityTheory.IndepFun X Y ℙ)
Goal: ProbabilityTheory.variance (X + Y) ℙ = ProbabilityTheory.variance X ℙ + ProbabilityTheory.variance Y ℙ
-- ProbabilityTheory.IndepFun.variance_sum
theorem IndepFun.variance_sum [@IsProbabilityMeasure Ω _ ℙ] {ι : Type*} {X : ι → Ω → ℝ}
    {s : Finset ι} (hs : ∀ i ∈ s, @Memℒp _ _ _ (_) (X i) 2 ℙ)
    (h : Set.Pairwise ↑s fun i j => @IndepFun _ _ _ (_) _ _ (X i) (X j) ℙ) :
    Var[∑ i ∈ s, X i] = ∑ i ∈ s, Var[X i] 
Arguments: {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure ℙ] {ι : Type u_2} {X : ι → Ω → ℝ} {s : Finset ι} (hs : ∀ i ∈ s, MeasureTheory.Memℒp (X i) 2 ℙ) (h : (↑s).Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j) ℙ)
Goal: ProbabilityTheory.variance (∑ i ∈ s, X i) ℙ = ∑ i ∈ s, ProbabilityTheory.variance (X i) ℙ
-- MeasureTheory.not_frequently_of_upcrossings_lt_top
theorem not_frequently_of_upcrossings_lt_top (hab : a < b) (hω : upcrossings a b f ω ≠ ∞) :
    ¬((∃ᶠ n in atTop, f n ω < a) ∧ ∃ᶠ n in atTop, b < f n ω) 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {ω : Ω} (hab : a < b) (hω : MeasureTheory.upcrossings a b f ω ≠ ⊤)
Goal: ¬((∃ᶠ (n : ℕ) in Filter.atTop, f n ω < a) ∧ ∃ᶠ (n : ℕ) in Filter.atTop, b < f n ω)