ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.measure_ge_le_exp_cgf
theorem measure_ge_le_exp_cgf [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 + cgf 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.cgf X μ t)
-- ProbabilityTheory.measure_le_le_exp_cgf
theorem measure_le_le_exp_cgf [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 + cgf 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.cgf X μ t)
-- ProbabilityTheory.condCount_empty_meas
@[simp]
theorem condCount_empty_meas : (condCount ∅ : Measure Ω) = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: ProbabilityTheory.condCount ∅ = 0
-- ProbabilityTheory.condCount_empty
theorem condCount_empty {s : Set Ω} : condCount s ∅ = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω}
Goal: (ProbabilityTheory.condCount s) ∅ = 0
-- ProbabilityTheory.finite_of_condCount_ne_zero
theorem finite_of_condCount_ne_zero {s t : Set Ω} (h : condCount s t ≠ 0) : s.Finite 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {s t : Set Ω} (h : (ProbabilityTheory.condCount s) t ≠ 0)
Goal: s.Finite
-- ProbabilityTheory.condCount_univ
theorem condCount_univ [Fintype Ω] {s : Set Ω} :
    condCount Set.univ s = Measure.count s / Fintype.card Ω 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [Fintype Ω] {s : Set Ω}
Goal: (ProbabilityTheory.condCount Set.univ) s = MeasureTheory.Measure.count s / ↑(Fintype.card Ω)
-- ProbabilityTheory.condCount_isProbabilityMeasure
theorem condCount_isProbabilityMeasure {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) :
    IsProbabilityMeasure (condCount s) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty)
Goal: MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.condCount s)
-- ProbabilityTheory.condCount_singleton
theorem condCount_singleton (ω : Ω) (t : Set Ω) [Decidable (ω ∈ t)] :
    condCount {ω} t = if ω ∈ t then 1 else 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] (ω : Ω) (t : Set Ω) [Decidable (ω ∈ t)]
Goal: (ProbabilityTheory.condCount {ω}) t = if ω ∈ t then 1 else 0
-- ProbabilityTheory.condCount_inter_self
theorem condCount_inter_self (hs : s.Finite) : condCount s (s ∩ t) = condCount s t 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (hs : s.Finite)
Goal: (ProbabilityTheory.condCount s) (s ∩ t) = (ProbabilityTheory.condCount s) t
-- ProbabilityTheory.condCount_self
theorem condCount_self (hs : s.Finite) (hs' : s.Nonempty) : condCount s s = 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty)
Goal: (ProbabilityTheory.condCount s) s = 1
-- ProbabilityTheory.condCount_eq_one_of
theorem condCount_eq_one_of (hs : s.Finite) (hs' : s.Nonempty) (ht : s ⊆ t) :
    condCount s t = 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) (ht : s ⊆ t)
Goal: (ProbabilityTheory.condCount s) t = 1
-- ProbabilityTheory.pred_true_of_condCount_eq_one
theorem pred_true_of_condCount_eq_one (h : condCount s t = 1) : s ⊆ t 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (h : (ProbabilityTheory.condCount s) t = 1)
Goal: s ⊆ t
-- ProbabilityTheory.condCount_eq_zero_iff
theorem condCount_eq_zero_iff (hs : s.Finite) : condCount s t = 0 ↔ s ∩ t = ∅ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (hs : s.Finite)
Goal: (ProbabilityTheory.condCount s) t = 0 ↔ s ∩ t = ∅
-- ProbabilityTheory.condCount_of_univ
theorem condCount_of_univ (hs : s.Finite) (hs' : s.Nonempty) : condCount s Set.univ = 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty)
Goal: (ProbabilityTheory.condCount s) Set.univ = 1
-- ProbabilityTheory.condCount_inter
theorem condCount_inter (hs : s.Finite) :
    condCount s (t ∩ u) = condCount (s ∩ t) u * condCount s t 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite)
Goal: (ProbabilityTheory.condCount s) (t ∩ u) = (ProbabilityTheory.condCount (s ∩ t)) u * (ProbabilityTheory.condCount s) t
-- ProbabilityTheory.condCount_inter'
theorem condCount_inter' (hs : s.Finite) :
    condCount s (t ∩ u) = condCount (s ∩ u) t * condCount s u 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite)
Goal: (ProbabilityTheory.condCount s) (t ∩ u) = (ProbabilityTheory.condCount (s ∩ u)) t * (ProbabilityTheory.condCount s) u
-- ProbabilityTheory.condCount_union
theorem condCount_union (hs : s.Finite) (htu : Disjoint t u) :
    condCount s (t ∪ u) = condCount s t + condCount s u 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite) (htu : Disjoint t u)
Goal: (ProbabilityTheory.condCount s) (t ∪ u) = (ProbabilityTheory.condCount s) t + (ProbabilityTheory.condCount s) u
-- ProbabilityTheory.condCount_compl
theorem condCount_compl (t : Set Ω) (hs : s.Finite) (hs' : s.Nonempty) :
    condCount s t + condCount s tᶜ = 1 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (t : Set Ω) (hs : s.Finite) (hs' : s.Nonempty)
Goal: (ProbabilityTheory.condCount s) t + (ProbabilityTheory.condCount s) tᶜ = 1
-- ProbabilityTheory.condCount_disjoint_union
theorem condCount_disjoint_union (hs : s.Finite) (ht : t.Finite) (hst : Disjoint s t) :
    condCount s u * condCount (s ∪ t) s + condCount t u * condCount (s ∪ t) t =
      condCount (s ∪ t) u 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite) (ht : t.Finite) (hst : Disjoint s t)
Goal: (ProbabilityTheory.condCount s) u * (ProbabilityTheory.condCount (s ∪ t)) s +
    (ProbabilityTheory.condCount t) u * (ProbabilityTheory.condCount (s ∪ t)) t =
  (ProbabilityTheory.condCount (s ∪ t)) u
-- ProbabilityTheory.condCount_add_compl_eq
theorem condCount_add_compl_eq (u t : Set Ω) (hs : s.Finite) :
    condCount (s ∩ u) t * condCount s u + condCount (s ∩ uᶜ) t * condCount s uᶜ =
      condCount s t 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (u t : Set Ω) (hs : s.Finite)
Goal: (ProbabilityTheory.condCount (s ∩ u)) t * (ProbabilityTheory.condCount s) u +
    (ProbabilityTheory.condCount (s ∩ uᶜ)) t * (ProbabilityTheory.condCount s) uᶜ =
  (ProbabilityTheory.condCount s) t
-- MeasureTheory.predictablePart_zero
@[simp]
theorem predictablePart_zero : predictablePart f ℱ μ 0 = 0 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0}
Goal: MeasureTheory.predictablePart f ℱ μ 0 = 0
-- MeasureTheory.adapted_predictablePart
theorem adapted_predictablePart : Adapted ℱ fun n => predictablePart f ℱ μ (n + 1) 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0}
Goal: MeasureTheory.Adapted ℱ fun n => MeasureTheory.predictablePart f ℱ μ (n + 1)
-- MeasureTheory.adapted_predictablePart'
theorem adapted_predictablePart' : Adapted ℱ fun n => predictablePart f ℱ μ n 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0}
Goal: MeasureTheory.Adapted ℱ fun n => MeasureTheory.predictablePart f ℱ μ n
-- MeasureTheory.martingalePart_add_predictablePart
theorem martingalePart_add_predictablePart (ℱ : Filtration ℕ m0) (μ : Measure Ω) (f : ℕ → Ω → E) :
    martingalePart f ℱ μ + predictablePart f ℱ μ = f 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (ℱ : MeasureTheory.Filtration ℕ m0) (μ : MeasureTheory.Measure Ω) (f : ℕ → Ω → E)
Goal: MeasureTheory.martingalePart f ℱ μ + MeasureTheory.predictablePart f ℱ μ = f
-- MeasureTheory.martingalePart_eq_sum
theorem martingalePart_eq_sum : martingalePart f ℱ μ = fun n =>
    f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|ℱ i]) 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0}
Goal: MeasureTheory.martingalePart f ℱ μ = fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])
-- MeasureTheory.adapted_martingalePart
theorem adapted_martingalePart (hf : Adapted ℱ f) : Adapted ℱ (martingalePart f ℱ μ) 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f)
Goal: MeasureTheory.Adapted ℱ (MeasureTheory.martingalePart f ℱ μ)
-- MeasureTheory.integrable_martingalePart
theorem integrable_martingalePart (hf_int : ∀ n, Integrable (f n) μ) (n : ℕ) :
    Integrable (martingalePart f ℱ μ n) μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0} (hf_int : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) (n : ℕ)
Goal: MeasureTheory.Integrable (MeasureTheory.martingalePart f ℱ μ n) μ
-- MeasureTheory.martingale_martingalePart
theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
    [SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f) (hf_int : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) [MeasureTheory.SigmaFiniteFiltration μ ℱ]
Goal: MeasureTheory.Martingale (MeasureTheory.martingalePart f ℱ μ) ℱ μ
-- MeasureTheory.martingalePart_add_ae_eq
theorem martingalePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
    (hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
    (hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : martingalePart (f + g) ℱ μ n =ᵐ[μ] f n 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E} (hf : MeasureTheory.Martingale f ℱ μ) (hg : MeasureTheory.Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0) (hgint : ∀ (n : ℕ), MeasureTheory.Integrable (g n) μ) (n : ℕ)
Goal: MeasureTheory.martingalePart (f + g) ℱ μ n =ᵐ[μ] f n
-- MeasureTheory.predictablePart_add_ae_eq
theorem predictablePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
    (hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
    (hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : predictablePart (f + g) ℱ μ n =ᵐ[μ] g n 
Arguments: {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E} (hf : MeasureTheory.Martingale f ℱ μ) (hg : MeasureTheory.Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0) (hgint : ∀ (n : ℕ), MeasureTheory.Integrable (g n) μ) (n : ℕ)
Goal: MeasureTheory.predictablePart (f + g) ℱ μ n =ᵐ[μ] g n
-- MeasureTheory.predictablePart_bdd_difference
theorem predictablePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, ∀ i, |predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω| ≤ R 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : MeasureTheory.Filtration ℕ m0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |MeasureTheory.predictablePart f ℱ μ (i + 1) ω - MeasureTheory.predictablePart f ℱ μ i ω| ≤ ↑R
-- MeasureTheory.martingalePart_bdd_difference
theorem martingalePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, ∀ i, |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : MeasureTheory.Filtration ℕ m0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ,
  ∀ (i : ℕ), |MeasureTheory.martingalePart f ℱ μ (i + 1) ω - MeasureTheory.martingalePart f ℱ μ i ω| ≤ ↑(2 * R)
-- MeasureTheory.Martingale.condexp_stopping_time_ae_eq_restrict_eq_const
theorem condexp_stopping_time_ae_eq_restrict_eq_const
    [(Filter.atTop : Filter ι).IsCountablyGenerated] (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) [SigmaFinite (μ.trim hτ.measurableSpace_le)] (hin : i ≤ n) :
    μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ : Ω → ι} {f : ι → Ω → E} {i n : ι} [Filter.atTop.IsCountablyGenerated] (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (hin : i ≤ n)
Goal: μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i
-- MeasureTheory.Martingale.condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const
theorem condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hτ_le : ∀ x, τ x ≤ n)
    [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] (i : ι) :
    μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ : Ω → ι} {f : ι → Ω → E} {n : ι} (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hτ_le : ∀ (x : Ω), τ x ≤ n) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (i : ι)
Goal: μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i
-- MeasureTheory.Martingale.stoppedValue_ae_eq_restrict_eq
theorem stoppedValue_ae_eq_restrict_eq (h : Martingale f ℱ μ) (hτ : IsStoppingTime ℱ τ)
    (hτ_le : ∀ x, τ x ≤ n) [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] (i : ι) :
    stoppedValue f τ =ᵐ[μ.restrict {x | τ x = i}] μ[f n|hτ.measurableSpace] 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ : Ω → ι} {f : ι → Ω → E} {n : ι} (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hτ_le : ∀ (x : Ω), τ x ≤ n) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (i : ι)
Goal: MeasureTheory.stoppedValue f τ =ᵐ[μ.restrict {x | τ x = i}] μ[f n|hτ.measurableSpace]
-- MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_const_of_countable_range
theorem stoppedValue_ae_eq_condexp_of_le_const_of_countable_range (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hτ_le : ∀ x, τ x ≤ n) (h_countable_range : (Set.range τ).Countable)
    [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] :
    stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace] 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ : Ω → ι} {f : ι → Ω → E} {n : ι} (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hτ_le : ∀ (x : Ω), τ x ≤ n) (h_countable_range : (Set.range τ).Countable) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: MeasureTheory.stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace]
-- MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_const
theorem stoppedValue_ae_eq_condexp_of_le_const [Countable ι] (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hτ_le : ∀ x, τ x ≤ n)
    [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] :
    stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace] 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ : Ω → ι} {f : ι → Ω → E} {n : ι} [Countable ι] (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hτ_le : ∀ (x : Ω), τ x ≤ n) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: MeasureTheory.stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace]
-- MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_of_countable_range
theorem stoppedValue_ae_eq_condexp_of_le_of_countable_range (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) (hσ_le_τ : σ ≤ τ) (hτ_le : ∀ x, τ x ≤ n)
    (hτ_countable_range : (Set.range τ).Countable) (hσ_countable_range : (Set.range σ).Countable)
    [SigmaFinite (μ.trim (hσ.measurableSpace_le_of_le fun x => (hσ_le_τ x).trans (hτ_le x)))] :
    stoppedValue f σ =ᵐ[μ] μ[stoppedValue f τ|hσ.measurableSpace] 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ σ : Ω → ι} {f : ι → Ω → E} {n : ι} (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hσ : MeasureTheory.IsStoppingTime ℱ σ) (hσ_le_τ : σ ≤ τ) (hτ_le : ∀ (x : Ω), τ x ≤ n) (hτ_countable_range : (Set.range τ).Countable) (hσ_countable_range : (Set.range σ).Countable) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: MeasureTheory.stoppedValue f σ =ᵐ[μ] μ[MeasureTheory.stoppedValue f τ|hσ.measurableSpace]
-- MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le
theorem stoppedValue_ae_eq_condexp_of_le [Countable ι] (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) (hσ_le_τ : σ ≤ τ) (hτ_le : ∀ x, τ x ≤ n)
    [SigmaFinite (μ.trim hσ.measurableSpace_le)] :
    stoppedValue f σ =ᵐ[μ] μ[stoppedValue f τ|hσ.measurableSpace] 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ σ : Ω → ι} {f : ι → Ω → E} {n : ι} [Countable ι] (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hσ : MeasureTheory.IsStoppingTime ℱ σ) (hσ_le_τ : σ ≤ τ) (hτ_le : ∀ (x : Ω), τ x ≤ n) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: MeasureTheory.stoppedValue f σ =ᵐ[μ] μ[MeasureTheory.stoppedValue f τ|hσ.measurableSpace]
-- MeasureTheory.Martingale.condexp_stoppedValue_stopping_time_ae_eq_restrict_le
theorem condexp_stoppedValue_stopping_time_ae_eq_restrict_le (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) [SigmaFinite (μ.trim hσ.measurableSpace_le)]
    (hτ_le : ∀ x, τ x ≤ n) :
    μ[stoppedValue f τ|hσ.measurableSpace] =ᵐ[μ.restrict {x : Ω | τ x ≤ σ x}] stoppedValue f τ 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι] [TopologicalSpace ι] [DiscreteTopology ι] [MeasurableSpace ι] [BorelSpace ι] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] {ℱ : MeasureTheory.Filtration ι m} {τ σ : Ω → ι} {f : ι → Ω → E} {n : ι} (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hσ : MeasureTheory.IsStoppingTime ℱ σ) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (hτ_le : ∀ (x : Ω), τ x ≤ n)
Goal: μ[MeasureTheory.stoppedValue f τ|hσ.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}] MeasureTheory.stoppedValue f τ
-- MeasureTheory.Martingale.stoppedValue_min_ae_eq_condexp
theorem stoppedValue_min_ae_eq_condexp [SigmaFiniteFiltration μ ℱ] (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) {n : ι} (hτ_le : ∀ x, τ x ≤ n)
    [h_sf_min : SigmaFinite (μ.trim (hτ.min hσ).measurableSpace_le)] :
    (stoppedValue f fun x => min (σ x) (τ x)) =ᵐ[μ] μ[stoppedValue f τ|hσ.measurableSpace] 
Arguments: {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι] [TopologicalSpace ι] [DiscreteTopology ι] [MeasurableSpace ι] [BorelSpace ι] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] {ℱ : MeasureTheory.Filtration ι m} {τ σ : Ω → ι} {f : ι → Ω → E} [MeasureTheory.SigmaFiniteFiltration μ ℱ] (h : MeasureTheory.Martingale f ℱ μ) (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hσ : MeasureTheory.IsStoppingTime ℱ σ) {n : ι} (hτ_le : ∀ (x : Ω), τ x ≤ n) [h_sf_min : MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: (MeasureTheory.stoppedValue f fun x => min (σ x) (τ x)) =ᵐ[μ] μ[MeasureTheory.stoppedValue f τ|hσ.measurableSpace]
-- ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt
theorem iIndepFun.indep_comap_natural_of_lt (hf : ∀ i, StronglyMeasurable (f i))
    (hfi : iIndepFun (fun _ => mβ) f μ) (hij : i < j) :
    Indep (MeasurableSpace.comap (f j) mβ) (Filtration.natural f hf i) μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} {β : Type u_3} [LinearOrder ι] [mβ : MeasurableSpace β] [NormedAddCommGroup β] [BorelSpace β] {f : ι → Ω → β} {i j : ι} (hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (hfi : ProbabilityTheory.iIndepFun (fun x => mβ) f μ) (hij : i < j)
Goal: ProbabilityTheory.Indep (MeasurableSpace.comap (f j) mβ) (↑(MeasureTheory.Filtration.natural f hf) i) μ