ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Valuation.supp_quot
theorem supp_quot {J : Ideal R} (hJ : J ≤ supp v) :
    supp (v.onQuot hJ) = (supp v).map (Ideal.Quotient.mk J) 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {J : Ideal R} (hJ : J ≤ v.supp)
Goal: (v.onQuot hJ).supp = Ideal.map (Ideal.Quotient.mk J) v.supp
-- Valuation.supp_quot_supp
theorem supp_quot_supp : supp (v.onQuot le_rfl) = 0 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: (v.onQuot ⋯).supp = 0
-- AddValuation.onQuot_comap_eq
@[simp]
theorem onQuot_comap_eq {J : Ideal R} (hJ : J ≤ supp v) :
    (v.onQuot hJ).comap (Ideal.Quotient.mk J) = v 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {J : Ideal R} (hJ : J ≤ v.supp)
Goal: AddValuation.comap (Ideal.Quotient.mk J) (v.onQuot hJ) = v
-- AddValuation.comap_supp
theorem comap_supp {S : Type*} [CommRing S] (f : S →+* R) :
    supp (v.comap f) = Ideal.comap f v.supp 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {S : Type u_3} [CommRing S] (f : S →+* R)
Goal: (AddValuation.comap f v).supp = Ideal.comap f v.supp
-- AddValuation.self_le_supp_comap
theorem self_le_supp_comap (J : Ideal R) (v : AddValuation (R ⧸ J) Γ₀) :
    J ≤ (v.comap (Ideal.Quotient.mk J)).supp 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedAddCommMonoidWithTop Γ₀] (J : Ideal R) (v : AddValuation (R ⧸ J) Γ₀)
Goal: J ≤ (AddValuation.comap (Ideal.Quotient.mk J) v).supp
-- AddValuation.comap_onQuot_eq
@[simp]
theorem comap_onQuot_eq (J : Ideal R) (v : AddValuation (R ⧸ J) Γ₀) :
    (v.comap (Ideal.Quotient.mk J)).onQuot (v.self_le_supp_comap J) = v 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedAddCommMonoidWithTop Γ₀] (J : Ideal R) (v : AddValuation (R ⧸ J) Γ₀)
Goal: (AddValuation.comap (Ideal.Quotient.mk J) v).onQuot ⋯ = v
-- AddValuation.supp_quot
theorem supp_quot {J : Ideal R} (hJ : J ≤ supp v) :
    supp (v.onQuot hJ) = (supp v).map (Ideal.Quotient.mk J) 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {J : Ideal R} (hJ : J ≤ v.supp)
Goal: (v.onQuot hJ).supp = Ideal.map (Ideal.Quotient.mk J) v.supp
-- AddValuation.supp_quot_supp
theorem supp_quot_supp : supp ((Valuation.onQuot v) le_rfl) = 0 
Arguments: {R : Type u_1} {Γ₀ : Type u_2} [CommRing R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀)
Goal: AddValuation.supp (Valuation.onQuot v ⋯) = 0
-- MeasureTheory.isStoppingTime_const
theorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) :
    IsStoppingTime f fun _ => i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] (f : MeasureTheory.Filtration ι m) (i : ι)
Goal: MeasureTheory.IsStoppingTime f fun x => i
-- MeasureTheory.IsStoppingTime.measurableSet_le
protected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | τ ω ≤ i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω ≤ i}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_of_pred
theorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [PredOrder ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range
protected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ)
    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [PartialOrder ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable
protected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [PartialOrder ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range
protected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ)
    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [PartialOrder ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable
protected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [PartialOrder ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range
protected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → ι}
    {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) :
    MeasurableSet[f i] {ω | i ≤ τ ω} 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ι : Type u_4} [LinearOrder ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι)
Goal: MeasurableSet {ω | i ≤ τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable
protected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → ι} {f : Filtration ι m}
    [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω | i ≤ τ ω} 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ι : Type u_4} [LinearOrder ι] {τ : Ω → ι} {f : MeasureTheory.Filtration ι m} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | i ≤ τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_gt
theorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | i < τ ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | i < τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_of_isLUB
theorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι)
    (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι) (h_lub : IsLUB (Set.Iio i) i)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_lt
theorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_ge
theorem IsStoppingTime.measurableSet_ge (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | i ≤ τ ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | i ≤ τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_eq
theorem IsStoppingTime.measurableSet_eq (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[f i] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_le
theorem IsStoppingTime.measurableSet_eq_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) :
    MeasurableSet[f j] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) {i j : ι} (hle : i ≤ j)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_le
theorem IsStoppingTime.measurableSet_lt_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) :
    MeasurableSet[f j] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) {i j : ι} (hle : i ≤ j)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.isStoppingTime_of_measurableSet_eq
theorem isStoppingTime_of_measurableSet_eq [Preorder ι] [Countable ι] {f : Filtration ι m}
    {τ : Ω → ι} (hτ : ∀ i, MeasurableSet[f i] {ω | τ ω = i}) : IsStoppingTime f τ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] [Countable ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i})
Goal: MeasureTheory.IsStoppingTime f τ
-- MeasureTheory.IsStoppingTime.max
protected theorem max [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : IsStoppingTime f fun ω => max (τ ω) (π ω) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasureTheory.IsStoppingTime f fun ω => max (τ ω) (π ω)
-- MeasureTheory.IsStoppingTime.max_const
protected theorem max_const [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι}
    (hτ : IsStoppingTime f τ) (i : ι) : IsStoppingTime f fun ω => max (τ ω) i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasureTheory.IsStoppingTime f fun ω => max (τ ω) i
-- MeasureTheory.IsStoppingTime.min
protected theorem min [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : IsStoppingTime f fun ω => min (τ ω) (π ω) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasureTheory.IsStoppingTime f fun ω => min (τ ω) (π ω)
-- MeasureTheory.IsStoppingTime.min_const
protected theorem min_const [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι}
    (hτ : IsStoppingTime f τ) (i : ι) : IsStoppingTime f fun ω => min (τ ω) i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasureTheory.IsStoppingTime f fun ω => min (τ ω) i
-- MeasureTheory.IsStoppingTime.add_const
theorem add_const [AddGroup ι] [Preorder ι] [CovariantClass ι ι (Function.swap (· + ·)) (· ≤ ·)]
    [CovariantClass ι ι (· + ·) (· ≤ ·)] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ)
    {i : ι} (hi : 0 ≤ i) : IsStoppingTime f fun ω => τ ω + i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [AddGroup ι] [Preorder ι] [CovariantClass ι ι (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) {i : ι} (hi : 0 ≤ i)
Goal: MeasureTheory.IsStoppingTime f fun ω => τ ω + i
-- MeasureTheory.IsStoppingTime.add_const_nat
theorem add_const_nat {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) {i : ℕ} :
    IsStoppingTime f fun ω => τ ω + i 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {f : MeasureTheory.Filtration ℕ m} {τ : Ω → ℕ} (hτ : MeasureTheory.IsStoppingTime f τ) {i : ℕ}
Goal: MeasureTheory.IsStoppingTime f fun ω => τ ω + i
-- MeasureTheory.IsStoppingTime.add
theorem add {f : Filtration ℕ m} {τ π : Ω → ℕ} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    IsStoppingTime f (τ + π) 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {f : MeasureTheory.Filtration ℕ m} {τ π : Ω → ℕ} (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π)
Goal: MeasureTheory.IsStoppingTime f (τ + π)
-- MeasureTheory.IsStoppingTime.measurableSet
protected theorem measurableSet (hτ : IsStoppingTime f τ) (s : Set Ω) :
    MeasurableSet[hτ.measurableSpace] s ↔ ∀ i : ι, MeasurableSet[f i] (s ∩ {ω | τ ω ≤ i}) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (s : Set Ω)
Goal: MeasurableSet s ↔ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
-- MeasureTheory.IsStoppingTime.measurableSpace_mono
theorem measurableSpace_mono (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) (hle : τ ≤ π) :
    hτ.measurableSpace ≤ hπ.measurableSpace 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ π : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (hπ : MeasureTheory.IsStoppingTime f π) (hle : τ ≤ π)
Goal: hτ.measurableSpace ≤ hπ.measurableSpace
-- MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable
theorem measurableSpace_le_of_countable [Countable ι] (hτ : IsStoppingTime f τ) :
    hτ.measurableSpace ≤ m 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: hτ.measurableSpace ≤ m
-- MeasureTheory.IsStoppingTime.measurableSpace_le'
theorem measurableSpace_le' [IsCountablyGenerated (atTop : Filter ι)] [(atTop : Filter ι).NeBot]
    (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [Filter.atTop.IsCountablyGenerated] [Filter.atTop.NeBot] (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: hτ.measurableSpace ≤ m
-- MeasureTheory.IsStoppingTime.measurableSpace_le
theorem measurableSpace_le {ι} [SemilatticeSup ι] {f : Filtration ι m} {τ : Ω → ι}
    [IsCountablyGenerated (atTop : Filter ι)] (hτ : IsStoppingTime f τ) :
    hτ.measurableSpace ≤ m 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ι : Type u_4} [SemilatticeSup ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [Filter.atTop.IsCountablyGenerated] (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: hτ.measurableSpace ≤ m
-- MeasureTheory.IsStoppingTime.measurableSpace_const
@[simp]
theorem measurableSpace_const (f : Filtration ι m) (i : ι) :
    (isStoppingTime_const f i).measurableSpace = f i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] (f : MeasureTheory.Filtration ι m) (i : ι)
Goal: ⋯.measurableSpace = ↑f i
-- MeasureTheory.IsStoppingTime.measurableSet_inter_eq_iff
theorem measurableSet_inter_eq_iff (hτ : IsStoppingTime f τ) (s : Set Ω) (i : ι) :
    MeasurableSet[hτ.measurableSpace] (s ∩ {ω | τ ω = i}) ↔
      MeasurableSet[f i] (s ∩ {ω | τ ω = i}) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (s : Set Ω) (i : ι)
Goal: MeasurableSet (s ∩ {ω | τ ω = i}) ↔ MeasurableSet (s ∩ {ω | τ ω = i})
-- MeasureTheory.IsStoppingTime.measurableSpace_le_of_le_const
theorem measurableSpace_le_of_le_const (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) :
    hτ.measurableSpace ≤ f i 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) {i : ι} (hτ_le : ∀ (ω : Ω), τ ω ≤ i)
Goal: hτ.measurableSpace ≤ ↑f i
-- MeasureTheory.IsStoppingTime.measurableSpace_le_of_le
theorem measurableSpace_le_of_le (hτ : IsStoppingTime f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) :
    hτ.measurableSpace ≤ m 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) {n : ι} (hτ_le : ∀ (ω : Ω), τ ω ≤ n)
Goal: hτ.measurableSpace ≤ m
-- MeasureTheory.IsStoppingTime.le_measurableSpace_of_const_le
theorem le_measurableSpace_of_const_le (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, i ≤ τ ω) :
    f i ≤ hτ.measurableSpace 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) {i : ι} (hτ_le : ∀ (ω : Ω), i ≤ τ ω)
Goal: ↑f i ≤ hτ.measurableSpace
-- MeasureTheory.IsStoppingTime.measurableSet_le'
protected theorem measurableSet_le' (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω ≤ i}
-- MeasureTheory.IsStoppingTime.measurableSet_gt'
protected theorem measurableSet_gt' (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | i < τ ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | i < τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_eq'
protected theorem measurableSet_eq' [TopologicalSpace ι] [OrderTopology ι]
    [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_ge'
protected theorem measurableSet_ge' [TopologicalSpace ι] [OrderTopology ι]
    [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | i ≤ τ ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | i ≤ τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_lt'
protected theorem measurableSet_lt' [TopologicalSpace ι] [OrderTopology ι]
    [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range'
protected theorem measurableSet_eq_of_countable_range' (hτ : IsStoppingTime f τ)
    (h_countable : (Set.range τ).Countable) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable'
protected theorem measurableSet_eq_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω = i}
-- MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range'
protected theorem measurableSet_ge_of_countable_range' (hτ : IsStoppingTime f τ)
    (h_countable : (Set.range τ).Countable) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | i ≤ τ ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι)
Goal: MeasurableSet {ω | i ≤ τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable'
protected theorem measurableSet_ge_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | i ≤ τ ω} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | i ≤ τ ω}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
protected theorem measurableSet_lt_of_countable_range' (hτ : IsStoppingTime f τ)
    (h_countable : (Set.range τ).Countable) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable'
protected theorem measurableSet_lt_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω < i} 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [Countable ι] (hτ : MeasureTheory.IsStoppingTime f τ) (i : ι)
Goal: MeasurableSet {ω | τ ω < i}
-- MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable_range
protected theorem measurableSpace_le_of_countable_range (hτ : IsStoppingTime f τ)
    (h_countable : (Set.range τ).Countable) : hτ.measurableSpace ≤ m 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} (hτ : MeasureTheory.IsStoppingTime f τ) (h_countable : (Set.range τ).Countable)
Goal: hτ.measurableSpace ≤ m
-- MeasureTheory.IsStoppingTime.measurable
protected theorem measurable [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι]
    [OrderTopology ι] [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) :
    Measurable[hτ.measurableSpace] τ 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {f : MeasureTheory.Filtration ι m} {τ : Ω → ι} [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [SecondCountableTopology ι] (hτ : MeasureTheory.IsStoppingTime f τ)
Goal: Measurable τ