ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PMF.mem_support_iff
@[simp]
theorem mem_support_iff (p : PMF α) (a : α) : a ∈ p.support ↔ p a ≠ 0 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: a ∈ p.support ↔ p a ≠ 0
-- PMF.support_nonempty
@[simp]
theorem support_nonempty (p : PMF α) : p.support.Nonempty 
Arguments: {α : Type u_1} (p : PMF α)
Goal: p.support.Nonempty
-- PMF.support_countable
@[simp]
theorem support_countable (p : PMF α) : p.support.Countable 
Arguments: {α : Type u_1} (p : PMF α)
Goal: p.support.Countable
-- PMF.apply_eq_zero_iff
theorem apply_eq_zero_iff (p : PMF α) (a : α) : p a = 0 ↔ a ∉ p.support 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: p a = 0 ↔ a ∉ p.support
-- PMF.apply_pos_iff
theorem apply_pos_iff (p : PMF α) (a : α) : 0 < p a ↔ a ∈ p.support 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: 0 < p a ↔ a ∈ p.support
-- PMF.apply_eq_one_iff
theorem apply_eq_one_iff (p : PMF α) (a : α) : p a = 1 ↔ p.support = {a} 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: p a = 1 ↔ p.support = {a}
-- PMF.coe_le_one
theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: p a ≤ 1
-- PMF.apply_ne_top
theorem apply_ne_top (p : PMF α) (a : α) : p a ≠ ∞ 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: p a ≠ ⊤
-- PMF.apply_lt_top
theorem apply_lt_top (p : PMF α) (a : α) : p a < ∞ 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: p a < ⊤
-- PMF.toOuterMeasure_apply
theorem toOuterMeasure_apply : p.toOuterMeasure s = ∑' x, s.indicator p x 
Arguments: {α : Type u_1} (p : PMF α) (s : Set α)
Goal: p.toOuterMeasure s = ∑' (x : α), s.indicator (⇑p) x
-- PMF.toOuterMeasure_caratheodory
@[simp]
theorem toOuterMeasure_caratheodory : p.toOuterMeasure.caratheodory = ⊤ 
Arguments: {α : Type u_1} (p : PMF α)
Goal: p.toOuterMeasure.caratheodory = ⊤
-- PMF.toOuterMeasure_apply_finset
@[simp]
theorem toOuterMeasure_apply_finset (s : Finset α) : p.toOuterMeasure s = ∑ x ∈ s, p x 
Arguments: {α : Type u_1} (p : PMF α) (s : Finset α)
Goal: p.toOuterMeasure ↑s = ∑ x ∈ s, p x
-- PMF.toOuterMeasure_apply_singleton
theorem toOuterMeasure_apply_singleton (a : α) : p.toOuterMeasure {a} = p a 
Arguments: {α : Type u_1} (p : PMF α) (a : α)
Goal: p.toOuterMeasure {a} = p a
-- PMF.toOuterMeasure_injective
theorem toOuterMeasure_injective : (toOuterMeasure : PMF α → OuterMeasure α).Injective 
Arguments: {α : Type u_1}
Goal: Function.Injective PMF.toOuterMeasure
-- PMF.toOuterMeasure_inj
@[simp]
theorem toOuterMeasure_inj {p q : PMF α} : p.toOuterMeasure = q.toOuterMeasure ↔ p = q 
Arguments: {α : Type u_1} {p q : PMF α}
Goal: p.toOuterMeasure = q.toOuterMeasure ↔ p = q
-- PMF.toOuterMeasure_apply_eq_zero_iff
theorem toOuterMeasure_apply_eq_zero_iff : p.toOuterMeasure s = 0 ↔ Disjoint p.support s 
Arguments: {α : Type u_1} (p : PMF α) (s : Set α)
Goal: p.toOuterMeasure s = 0 ↔ Disjoint p.support s
-- PMF.toOuterMeasure_apply_eq_one_iff
theorem toOuterMeasure_apply_eq_one_iff : p.toOuterMeasure s = 1 ↔ p.support ⊆ s 
Arguments: {α : Type u_1} (p : PMF α) (s : Set α)
Goal: p.toOuterMeasure s = 1 ↔ p.support ⊆ s
-- PMF.toOuterMeasure_apply_inter_support
@[simp]
theorem toOuterMeasure_apply_inter_support :
    p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s 
Arguments: {α : Type u_1} (p : PMF α) (s : Set α)
Goal: p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s
-- PMF.toOuterMeasure_mono
theorem toOuterMeasure_mono {s t : Set α} (h : s ∩ p.support ⊆ t) :
    p.toOuterMeasure s ≤ p.toOuterMeasure t 
Arguments: {α : Type u_1} (p : PMF α) {s t : Set α} (h : s ∩ p.support ⊆ t)
Goal: p.toOuterMeasure s ≤ p.toOuterMeasure t
-- PMF.toOuterMeasure_apply_eq_of_inter_support_eq
theorem toOuterMeasure_apply_eq_of_inter_support_eq {s t : Set α}
    (h : s ∩ p.support = t ∩ p.support) : p.toOuterMeasure s = p.toOuterMeasure t 
Arguments: {α : Type u_1} (p : PMF α) {s t : Set α} (h : s ∩ p.support = t ∩ p.support)
Goal: p.toOuterMeasure s = p.toOuterMeasure t
-- PMF.toOuterMeasure_apply_fintype
@[simp]
theorem toOuterMeasure_apply_fintype [Fintype α] : p.toOuterMeasure s = ∑ x, s.indicator p x 
Arguments: {α : Type u_1} (p : PMF α) (s : Set α) [Fintype α]
Goal: p.toOuterMeasure s = ∑ x : α, s.indicator (⇑p) x
-- PMF.toOuterMeasure_apply_le_toMeasure_apply
theorem toOuterMeasure_apply_le_toMeasure_apply : p.toOuterMeasure s ≤ p.toMeasure s 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α)
Goal: p.toOuterMeasure s ≤ p.toMeasure s
-- PMF.toMeasure_apply_eq_toOuterMeasure_apply
theorem toMeasure_apply_eq_toOuterMeasure_apply (hs : MeasurableSet s) :
    p.toMeasure s = p.toOuterMeasure s 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s)
Goal: p.toMeasure s = p.toOuterMeasure s
-- PMF.toMeasure_apply
theorem toMeasure_apply (hs : MeasurableSet s) : p.toMeasure s = ∑' x, s.indicator p x 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s)
Goal: p.toMeasure s = ∑' (x : α), s.indicator (⇑p) x
-- PMF.toMeasure_apply_singleton
theorem toMeasure_apply_singleton (a : α) (h : MeasurableSet ({a} : Set α)) :
    p.toMeasure {a} = p a 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (a : α) (h : MeasurableSet {a})
Goal: p.toMeasure {a} = p a
-- PMF.toMeasure_apply_eq_zero_iff
theorem toMeasure_apply_eq_zero_iff (hs : MeasurableSet s) :
    p.toMeasure s = 0 ↔ Disjoint p.support s 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s)
Goal: p.toMeasure s = 0 ↔ Disjoint p.support s
-- PMF.toMeasure_apply_eq_one_iff
theorem toMeasure_apply_eq_one_iff (hs : MeasurableSet s) : p.toMeasure s = 1 ↔ p.support ⊆ s 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s)
Goal: p.toMeasure s = 1 ↔ p.support ⊆ s
-- PMF.toMeasure_apply_inter_support
@[simp]
theorem toMeasure_apply_inter_support (hs : MeasurableSet s) (hp : MeasurableSet p.support) :
    p.toMeasure (s ∩ p.support) = p.toMeasure s 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s) (hp : MeasurableSet p.support)
Goal: p.toMeasure (s ∩ p.support) = p.toMeasure s
-- PMF.restrict_toMeasure_support
@[simp]
theorem restrict_toMeasure_support [MeasurableSingletonClass α] (p : PMF α) :
    Measure.restrict (toMeasure p) (support p) = toMeasure p 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α)
Goal: p.toMeasure.restrict p.support = p.toMeasure
-- PMF.toMeasure_mono
theorem toMeasure_mono {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t)
    (h : s ∩ p.support ⊆ t) : p.toMeasure s ≤ p.toMeasure t 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (h : s ∩ p.support ⊆ t)
Goal: p.toMeasure s ≤ p.toMeasure t
-- PMF.toMeasure_apply_eq_of_inter_support_eq
theorem toMeasure_apply_eq_of_inter_support_eq {s t : Set α} (hs : MeasurableSet s)
    (ht : MeasurableSet t) (h : s ∩ p.support = t ∩ p.support) : p.toMeasure s = p.toMeasure t 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (h : s ∩ p.support = t ∩ p.support)
Goal: p.toMeasure s = p.toMeasure t
-- PMF.toMeasure_injective
theorem toMeasure_injective : (toMeasure : PMF α → Measure α).Injective 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α]
Goal: Function.Injective PMF.toMeasure
-- PMF.toMeasure_inj
@[simp]
theorem toMeasure_inj {p q : PMF α} : p.toMeasure = q.toMeasure ↔ p = q 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {p q : PMF α}
Goal: p.toMeasure = q.toMeasure ↔ p = q
-- PMF.toMeasure_apply_finset
@[simp]
theorem toMeasure_apply_finset (s : Finset α) : p.toMeasure s = ∑ x ∈ s, p x 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) [MeasurableSingletonClass α] (s : Finset α)
Goal: p.toMeasure ↑s = ∑ x ∈ s, p x
-- PMF.toMeasure_apply_of_finite
theorem toMeasure_apply_of_finite (hs : s.Finite) : p.toMeasure s = ∑' x, s.indicator p x 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) [MeasurableSingletonClass α] (hs : s.Finite)
Goal: p.toMeasure s = ∑' (x : α), s.indicator (⇑p) x
-- PMF.toMeasure_apply_fintype
@[simp]
theorem toMeasure_apply_fintype [Fintype α] : p.toMeasure s = ∑ x, s.indicator p x 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) [MeasurableSingletonClass α] [Fintype α]
Goal: p.toMeasure s = ∑ x : α, s.indicator (⇑p) x
-- MeasureTheory.Measure.toPMF_apply
theorem toPMF_apply (x : α) : μ.toPMF x = μ {x} 
Arguments: {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (x : α)
Goal: μ.toPMF x = μ {x}
-- MeasureTheory.Measure.toPMF_toMeasure
@[simp]
theorem toPMF_toMeasure : μ.toPMF.toMeasure = μ 
Arguments: {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ]
Goal: μ.toPMF.toMeasure = μ
-- PMF.toMeasure_toPMF
@[simp]
theorem toMeasure_toPMF : p.toMeasure.toPMF = p 
Arguments: {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α)
Goal: p.toMeasure.toPMF = p
-- PMF.toMeasure_eq_iff_eq_toPMF
theorem toMeasure_eq_iff_eq_toPMF (μ : Measure α) [IsProbabilityMeasure μ] :
    p.toMeasure = μ ↔ p = μ.toPMF 
Arguments: {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ]
Goal: p.toMeasure = μ ↔ p = μ.toPMF
-- PMF.toPMF_eq_iff_toMeasure_eq
theorem toPMF_eq_iff_toMeasure_eq (μ : Measure α) [IsProbabilityMeasure μ] :
    μ.toPMF = p ↔ μ = p.toMeasure 
Arguments: {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ]
Goal: μ.toPMF = p ↔ μ = p.toMeasure
-- ProbabilityTheory.moment_zero
@[simp]
theorem moment_zero (hp : p ≠ 0) : moment 0 p μ = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {p : ℕ} {μ : MeasureTheory.Measure Ω} (hp : p ≠ 0)
Goal: ProbabilityTheory.moment 0 p μ = 0
-- ProbabilityTheory.centralMoment_zero
@[simp]
theorem centralMoment_zero (hp : p ≠ 0) : centralMoment 0 p μ = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {p : ℕ} {μ : MeasureTheory.Measure Ω} (hp : p ≠ 0)
Goal: ProbabilityTheory.centralMoment 0 p μ = 0
-- ProbabilityTheory.centralMoment_one'
theorem centralMoment_one' [IsFiniteMeasure μ] (h_int : Integrable X μ) :
    centralMoment X 1 μ = (1 - (μ Set.univ).toReal) * μ[X] 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h_int : MeasureTheory.Integrable X μ)
Goal: ProbabilityTheory.centralMoment X 1 μ = (1 - (μ Set.univ).toReal) * ∫ (x : Ω), X x ∂μ
-- ProbabilityTheory.centralMoment_one
@[simp]
theorem centralMoment_one [IsProbabilityMeasure μ] : centralMoment X 1 μ = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
Goal: ProbabilityTheory.centralMoment X 1 μ = 0
-- ProbabilityTheory.centralMoment_two_eq_variance
theorem centralMoment_two_eq_variance [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
    centralMoment X 2 μ = variance 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.centralMoment X 2 μ = ProbabilityTheory.variance X μ
-- ProbabilityTheory.mgf_zero_fun
@[simp]
theorem mgf_zero_fun : mgf 0 μ t = (μ Set.univ).toReal 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ}
Goal: ProbabilityTheory.mgf 0 μ t = (μ Set.univ).toReal
-- ProbabilityTheory.cgf_zero_fun
@[simp]
theorem cgf_zero_fun : cgf 0 μ t = log (μ Set.univ).toReal 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ}
Goal: ProbabilityTheory.cgf 0 μ t = Real.log (μ Set.univ).toReal
-- ProbabilityTheory.mgf_zero_measure
@[simp]
theorem mgf_zero_measure : mgf X (0 : Measure Ω) t = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {t : ℝ}
Goal: ProbabilityTheory.mgf X 0 t = 0
-- ProbabilityTheory.cgf_zero_measure
@[simp]
theorem cgf_zero_measure : cgf X (0 : Measure Ω) t = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {t : ℝ}
Goal: ProbabilityTheory.cgf X 0 t = 0
-- ProbabilityTheory.mgf_const'
@[simp]
theorem mgf_const' (c : ℝ) : mgf (fun _ => c) μ t = (μ Set.univ).toReal * exp (t * c) 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} (c : ℝ)
Goal: ProbabilityTheory.mgf (fun x => c) μ t = (μ Set.univ).toReal * rexp (t * c)
-- ProbabilityTheory.mgf_const
theorem mgf_const (c : ℝ) [IsProbabilityMeasure μ] : mgf (fun _ => c) μ t = exp (t * c) 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} (c : ℝ) [MeasureTheory.IsProbabilityMeasure μ]
Goal: ProbabilityTheory.mgf (fun x => c) μ t = rexp (t * c)
-- ProbabilityTheory.cgf_const'
@[simp]
theorem cgf_const' [IsFiniteMeasure μ] (hμ : μ ≠ 0) (c : ℝ) :
    cgf (fun _ => c) μ t = log (μ Set.univ).toReal + t * c 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0) (c : ℝ)
Goal: ProbabilityTheory.cgf (fun x => c) μ t = Real.log (μ Set.univ).toReal + t * c
-- ProbabilityTheory.cgf_const
@[simp]
theorem cgf_const [IsProbabilityMeasure μ] (c : ℝ) : cgf (fun _ => c) μ t = t * c 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsProbabilityMeasure μ] (c : ℝ)
Goal: ProbabilityTheory.cgf (fun x => c) μ t = t * c
-- ProbabilityTheory.mgf_zero'
@[simp]
theorem mgf_zero' : mgf X μ 0 = (μ Set.univ).toReal 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω}
Goal: ProbabilityTheory.mgf X μ 0 = (μ Set.univ).toReal
-- ProbabilityTheory.mgf_zero
theorem mgf_zero [IsProbabilityMeasure μ] : mgf X μ 0 = 1 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
Goal: ProbabilityTheory.mgf X μ 0 = 1
-- ProbabilityTheory.cgf_zero'
@[simp]
theorem cgf_zero' : cgf X μ 0 = log (μ Set.univ).toReal 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω}
Goal: ProbabilityTheory.cgf X μ 0 = Real.log (μ Set.univ).toReal
-- ProbabilityTheory.cgf_zero
theorem cgf_zero [IsProbabilityMeasure μ] : cgf X μ 0 = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
Goal: ProbabilityTheory.cgf X μ 0 = 0
-- ProbabilityTheory.mgf_undef
theorem mgf_undef (hX : ¬Integrable (fun ω => exp (t * X ω)) μ) : mgf X μ t = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} (hX : ¬MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ)
Goal: ProbabilityTheory.mgf X μ t = 0
-- ProbabilityTheory.cgf_undef
theorem cgf_undef (hX : ¬Integrable (fun ω => exp (t * X ω)) μ) : cgf X μ t = 0 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} (hX : ¬MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ)
Goal: ProbabilityTheory.cgf X μ t = 0
-- ProbabilityTheory.mgf_nonneg
theorem mgf_nonneg : 0 ≤ mgf X μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ}
Goal: 0 ≤ ProbabilityTheory.mgf X μ t
-- ProbabilityTheory.mgf_pos'
theorem mgf_pos' (hμ : μ ≠ 0) (h_int_X : Integrable (fun ω => exp (t * X ω)) μ) :
    0 < mgf X μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} (hμ : μ ≠ 0) (h_int_X : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ)
Goal: 0 < ProbabilityTheory.mgf X μ t
-- ProbabilityTheory.mgf_pos
theorem mgf_pos [IsProbabilityMeasure μ] (h_int_X : Integrable (fun ω => exp (t * X ω)) μ) :
    0 < mgf X μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsProbabilityMeasure μ] (h_int_X : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ)
Goal: 0 < ProbabilityTheory.mgf X μ t
-- ProbabilityTheory.mgf_neg
theorem mgf_neg : mgf (-X) μ t = mgf X μ (-t) 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ}
Goal: ProbabilityTheory.mgf (-X) μ t = ProbabilityTheory.mgf X μ (-t)
-- ProbabilityTheory.cgf_neg
theorem cgf_neg : cgf (-X) μ t = cgf X μ (-t) 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ}
Goal: ProbabilityTheory.cgf (-X) μ t = ProbabilityTheory.cgf X μ (-t)
-- ProbabilityTheory.IndepFun.exp_mul
theorem IndepFun.exp_mul {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ) (s t : ℝ) :
    IndepFun (fun ω => exp (s * X ω)) (fun ω => exp (t * Y ω)) μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y : Ω → ℝ} (h_indep : ProbabilityTheory.IndepFun X Y μ) (s t : ℝ)
Goal: ProbabilityTheory.IndepFun (fun ω => rexp (s * X ω)) (fun ω => rexp (t * Y ω)) μ
-- ProbabilityTheory.IndepFun.mgf_add
theorem IndepFun.mgf_add {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ)
    (hX : AEStronglyMeasurable (fun ω => exp (t * X ω)) μ)
    (hY : AEStronglyMeasurable (fun ω => exp (t * Y ω)) μ) :
    mgf (X + Y) μ t = mgf X μ t * mgf Y μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} {X Y : Ω → ℝ} (h_indep : ProbabilityTheory.IndepFun X Y μ) (hX : MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * X ω)) μ) (hY : MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * Y ω)) μ)
Goal: ProbabilityTheory.mgf (X + Y) μ t = ProbabilityTheory.mgf X μ t * ProbabilityTheory.mgf Y μ t
-- ProbabilityTheory.IndepFun.mgf_add'
theorem IndepFun.mgf_add' {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
    (hY : AEStronglyMeasurable Y μ) : mgf (X + Y) μ t = mgf X μ t * mgf Y μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} {X Y : Ω → ℝ} (h_indep : ProbabilityTheory.IndepFun X Y μ) (hX : MeasureTheory.AEStronglyMeasurable X μ) (hY : MeasureTheory.AEStronglyMeasurable Y μ)
Goal: ProbabilityTheory.mgf (X + Y) μ t = ProbabilityTheory.mgf X μ t * ProbabilityTheory.mgf Y μ t
-- ProbabilityTheory.IndepFun.cgf_add
theorem IndepFun.cgf_add {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ)
    (h_int_X : Integrable (fun ω => exp (t * X ω)) μ)
    (h_int_Y : Integrable (fun ω => exp (t * Y ω)) μ) :
    cgf (X + Y) μ t = cgf X μ t + cgf Y μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} {X Y : Ω → ℝ} (h_indep : ProbabilityTheory.IndepFun X Y μ) (h_int_X : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ) (h_int_Y : MeasureTheory.Integrable (fun ω => rexp (t * Y ω)) μ)
Goal: ProbabilityTheory.cgf (X + Y) μ t = ProbabilityTheory.cgf X μ t + ProbabilityTheory.cgf Y μ t
-- ProbabilityTheory.aestronglyMeasurable_exp_mul_add
theorem aestronglyMeasurable_exp_mul_add {X Y : Ω → ℝ}
    (h_int_X : AEStronglyMeasurable (fun ω => exp (t * X ω)) μ)
    (h_int_Y : AEStronglyMeasurable (fun ω => exp (t * Y ω)) μ) :
    AEStronglyMeasurable (fun ω => exp (t * (X + Y) ω)) μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} {X Y : Ω → ℝ} (h_int_X : MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * X ω)) μ) (h_int_Y : MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * Y ω)) μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * (X + Y) ω)) μ
-- ProbabilityTheory.aestronglyMeasurable_exp_mul_sum
theorem aestronglyMeasurable_exp_mul_sum {X : ι → Ω → ℝ} {s : Finset ι}
    (h_int : ∀ i ∈ s, AEStronglyMeasurable (fun ω => exp (t * X i ω)) μ) :
    AEStronglyMeasurable (fun ω => exp (t * (∑ i ∈ s, X i) ω)) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} {X : ι → Ω → ℝ} {s : Finset ι} (h_int : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * X i ω)) μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
-- ProbabilityTheory.IndepFun.integrable_exp_mul_add
theorem IndepFun.integrable_exp_mul_add {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ)
    (h_int_X : Integrable (fun ω => exp (t * X ω)) μ)
    (h_int_Y : Integrable (fun ω => exp (t * Y ω)) μ) :
    Integrable (fun ω => exp (t * (X + Y) ω)) μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} {X Y : Ω → ℝ} (h_indep : ProbabilityTheory.IndepFun X Y μ) (h_int_X : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ) (h_int_Y : MeasureTheory.Integrable (fun ω => rexp (t * Y ω)) μ)
Goal: MeasureTheory.Integrable (fun ω => rexp (t * (X + Y) ω)) μ
-- ProbabilityTheory.iIndepFun.integrable_exp_mul_sum
theorem iIndepFun.integrable_exp_mul_sum [IsProbabilityMeasure μ] {X : ι → Ω → ℝ}
    (h_indep : iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ i, Measurable (X i))
    {s : Finset ι} (h_int : ∀ i ∈ s, Integrable (fun ω => exp (t * X i ω)) μ) :
    Integrable (fun ω => exp (t * (∑ i ∈ s, X i) ω)) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsProbabilityMeasure μ] {X : ι → Ω → ℝ} (h_indep : ProbabilityTheory.iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ (i : ι), Measurable (X i)) {s : Finset ι} (h_int : ∀ i ∈ s, MeasureTheory.Integrable (fun ω => rexp (t * X i ω)) μ)
Goal: MeasureTheory.Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
-- ProbabilityTheory.iIndepFun.mgf_sum
theorem iIndepFun.mgf_sum [IsProbabilityMeasure μ] {X : ι → Ω → ℝ}
    (h_indep : iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ i, Measurable (X i))
    (s : Finset ι) : mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsProbabilityMeasure μ] {X : ι → Ω → ℝ} (h_indep : ProbabilityTheory.iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ (i : ι), Measurable (X i)) (s : Finset ι)
Goal: ProbabilityTheory.mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, ProbabilityTheory.mgf (X i) μ t
-- ProbabilityTheory.iIndepFun.cgf_sum
theorem iIndepFun.cgf_sum [IsProbabilityMeasure μ] {X : ι → Ω → ℝ}
    (h_indep : iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ i, Measurable (X i))
    {s : Finset ι} (h_int : ∀ i ∈ s, Integrable (fun ω => exp (t * X i ω)) μ) :
    cgf (∑ i ∈ s, X i) μ t = ∑ i ∈ s, cgf (X i) μ t 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsProbabilityMeasure μ] {X : ι → Ω → ℝ} (h_indep : ProbabilityTheory.iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ (i : ι), Measurable (X i)) {s : Finset ι} (h_int : ∀ i ∈ s, MeasureTheory.Integrable (fun ω => rexp (t * X i ω)) μ)
Goal: ProbabilityTheory.cgf (∑ i ∈ s, X i) μ t = ∑ i ∈ s, ProbabilityTheory.cgf (X i) μ t
-- ProbabilityTheory.measure_ge_le_exp_mul_mgf
theorem measure_ge_le_exp_mul_mgf [IsFiniteMeasure μ] (ε : ℝ) (ht : 0 ≤ t)
    (h_int : Integrable (fun ω => exp (t * X ω)) μ) :
    (μ {ω | ε ≤ X ω}).toReal ≤ exp (-t * ε) * mgf X μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsFiniteMeasure μ] (ε : ℝ) (ht : 0 ≤ t) (h_int : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ)
Goal: (μ {ω | ε ≤ X ω}).toReal ≤ rexp (-t * ε) * ProbabilityTheory.mgf X μ t
-- ProbabilityTheory.measure_le_le_exp_mul_mgf
theorem measure_le_le_exp_mul_mgf [IsFiniteMeasure μ] (ε : ℝ) (ht : t ≤ 0)
    (h_int : Integrable (fun ω => exp (t * X ω)) μ) :
    (μ {ω | X ω ≤ ε}).toReal ≤ exp (-t * ε) * mgf X μ t 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} [MeasureTheory.IsFiniteMeasure μ] (ε : ℝ) (ht : t ≤ 0) (h_int : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ)
Goal: (μ {ω | X ω ≤ ε}).toReal ≤ rexp (-t * ε) * ProbabilityTheory.mgf X μ t