ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.intervalIntegral_pos_of_pos_on
theorem intervalIntegral_pos_of_pos_on {f : ℝ → ℝ} {a b : ℝ} (hfi : IntervalIntegrable f volume a b)
    (hpos : ∀ x : ℝ, x ∈ Ioo a b → 0 < f x) (hab : a < b) : 0 < ∫ x : ℝ in a..b, f x 
Arguments: {f : ℝ → ℝ} {a b : ℝ} (hfi : IntervalIntegrable f MeasureTheory.volume a b) (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) (hab : a < b)
Goal: 0 < ∫ (x : ℝ) in a..b, f x
-- intervalIntegral.intervalIntegral_pos_of_pos
theorem intervalIntegral_pos_of_pos {f : ℝ → ℝ} {a b : ℝ}
    (hfi : IntervalIntegrable f MeasureSpace.volume a b) (hpos : ∀ x, 0 < f x) (hab : a < b) :
    0 < ∫ x in a..b, f x 
Arguments: {f : ℝ → ℝ} {a b : ℝ} (hfi : IntervalIntegrable f MeasureTheory.volume a b) (hpos : ∀ (x : ℝ), 0 < f x) (hab : a < b)
Goal: 0 < ∫ (x : ℝ) in a..b, f x
-- intervalIntegral.integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero
theorem integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero (hab : a ≤ b)
    (hfi : IntervalIntegrable f μ a b) (hgi : IntervalIntegrable g μ a b)
    (hle : f ≤ᵐ[μ.restrict (Ioc a b)] g) (hlt : μ.restrict (Ioc a b) {x | f x < g x} ≠ 0) :
    (∫ x in a..b, f x ∂μ) < ∫ x in a..b, g x ∂μ 
Arguments: {f g : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hfi : IntervalIntegrable f μ a b) (hgi : IntervalIntegrable g μ a b) (hle : f ≤ᵐ[μ.restrict (Set.Ioc a b)] g) (hlt : (μ.restrict (Set.Ioc a b)) {x | f x < g x} ≠ 0)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ < ∫ (x : ℝ) in a..b, g x ∂μ
-- intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt
theorem integral_lt_integral_of_continuousOn_of_le_of_exists_lt {f g : ℝ → ℝ} {a b : ℝ}
    (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hgc : ContinuousOn g (Icc a b))
    (hle : ∀ x ∈ Ioc a b, f x ≤ g x) (hlt : ∃ c ∈ Icc a b, f c < g c) :
    (∫ x in a..b, f x) < ∫ x in a..b, g x 
Arguments: {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hgc : ContinuousOn g (Set.Icc a b)) (hle : ∀ x ∈ Set.Ioc a b, f x ≤ g x) (hlt : ∃ c ∈ Set.Icc a b, f c < g c)
Goal: ∫ (x : ℝ) in a..b, f x < ∫ (x : ℝ) in a..b, g x
-- intervalIntegral.integral_nonneg_of_ae_restrict
theorem integral_nonneg_of_ae_restrict (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Icc a b)] f) :
    0 ≤ ∫ u in a..b, f u ∂μ 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Set.Icc a b)] f)
Goal: 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
-- intervalIntegral.integral_nonneg_of_ae
theorem integral_nonneg_of_ae (hab : a ≤ b) (hf : 0 ≤ᵐ[μ] f) : 0 ≤ ∫ u in a..b, f u ∂μ 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : 0 ≤ᵐ[μ] f)
Goal: 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
-- intervalIntegral.integral_nonneg_of_forall
theorem integral_nonneg_of_forall (hab : a ≤ b) (hf : ∀ u, 0 ≤ f u) : 0 ≤ ∫ u in a..b, f u ∂μ 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : ∀ (u : ℝ), 0 ≤ f u)
Goal: 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
-- intervalIntegral.integral_nonneg
theorem integral_nonneg (hab : a ≤ b) (hf : ∀ u, u ∈ Icc a b → 0 ≤ f u) : 0 ≤ ∫ u in a..b, f u ∂μ 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : ∀ u ∈ Set.Icc a b, 0 ≤ f u)
Goal: 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ
-- intervalIntegral.abs_integral_le_integral_abs
theorem abs_integral_le_integral_abs (hab : a ≤ b) :
    |∫ x in a..b, f x ∂μ| ≤ ∫ x in a..b, |f x| ∂μ 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b)
Goal: |∫ (x : ℝ) in a..b, f x ∂μ| ≤ ∫ (x : ℝ) in a..b, |f x| ∂μ
-- intervalIntegral.integral_mono_ae_restrict
theorem integral_mono_ae_restrict (h : f ≤ᵐ[μ.restrict (Icc a b)] g) :
    (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ 
Arguments: {f g : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : f ≤ᵐ[μ.restrict (Set.Icc a b)] g)
Goal: ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ
-- intervalIntegral.integral_mono_ae
theorem integral_mono_ae (h : f ≤ᵐ[μ] g) : (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ 
Arguments: {f g : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : f ≤ᵐ[μ] g)
Goal: ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ
-- intervalIntegral.integral_mono_on
theorem integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) :
    (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ 
Arguments: {f g : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : ∀ x ∈ Set.Icc a b, f x ≤ g x)
Goal: ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ
-- intervalIntegral.integral_mono
theorem integral_mono (h : f ≤ g) : (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ 
Arguments: {f g : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : f ≤ g)
Goal: ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ
-- intervalIntegral.integral_mono_interval
theorem integral_mono_interval {c d} (hca : c ≤ a) (hab : a ≤ b) (hbd : b ≤ d)
    (hf : 0 ≤ᵐ[μ.restrict (Ioc c d)] f) (hfi : IntervalIntegrable f μ c d) :
    (∫ x in a..b, f x ∂μ) ≤ ∫ x in c..d, f x ∂μ 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {c d : ℝ} (hca : c ≤ a) (hab : a ≤ b) (hbd : b ≤ d) (hf : 0 ≤ᵐ[μ.restrict (Set.Ioc c d)] f) (hfi : IntervalIntegrable f μ c d)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ ≤ ∫ (x : ℝ) in c..d, f x ∂μ
-- intervalIntegral.abs_integral_mono_interval
theorem abs_integral_mono_interval {c d} (h : Ι a b ⊆ Ι c d) (hf : 0 ≤ᵐ[μ.restrict (Ι c d)] f)
    (hfi : IntervalIntegrable f μ c d) : |∫ x in a..b, f x ∂μ| ≤ |∫ x in c..d, f x ∂μ| 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {c d : ℝ} (h : Ι a b ⊆ Ι c d) (hf : 0 ≤ᵐ[μ.restrict (Ι c d)] f) (hfi : IntervalIntegrable f μ c d)
Goal: |∫ (x : ℝ) in a..b, f x ∂μ| ≤ |∫ (x : ℝ) in c..d, f x ∂μ|
-- MeasureTheory.Integrable.hasSum_intervalIntegral
theorem _root_.MeasureTheory.Integrable.hasSum_intervalIntegral (hfi : Integrable f μ) (y : ℝ) :
    HasSum (fun n : ℤ => ∫ x in y + n..y + n + 1, f x ∂μ) (∫ x, f x ∂μ) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} (hfi : MeasureTheory.Integrable f μ) (y : ℝ)
Goal: HasSum (fun n => ∫ (x : ℝ) in y + ↑n..y + ↑n + 1, f x ∂μ) (∫ (x : ℝ), f x ∂μ)
-- MeasureTheory.Integrable.hasSum_intervalIntegral_comp_add_int
theorem _root_.MeasureTheory.Integrable.hasSum_intervalIntegral_comp_add_int (hfi : Integrable f) :
    HasSum (fun n : ℤ => ∫ x in (0 : ℝ)..(1 : ℝ), f (x + n)) (∫ x, f x) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} (hfi : MeasureTheory.Integrable f MeasureTheory.volume)
Goal: HasSum (fun n => ∫ (x : ℝ) in 0 ..1, f (x + ↑n)) (∫ (x : ℝ), f x)
-- borel_eq_generateFrom_Iio
theorem borel_eq_generateFrom_Iio : borel α = .generateFrom (range Iio) 
Arguments: (α : Type u_1) [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [OrderTopology α]
Goal: borel α = MeasurableSpace.generateFrom (Set.range Set.Iio)
-- borel_eq_generateFrom_Ioi
theorem borel_eq_generateFrom_Ioi : borel α = .generateFrom (range Ioi) 
Arguments: (α : Type u_1) [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [OrderTopology α]
Goal: borel α = MeasurableSpace.generateFrom (Set.range Set.Ioi)
-- borel_eq_generateFrom_Iic
theorem borel_eq_generateFrom_Iic :
    borel α = MeasurableSpace.generateFrom (range Iic) 
Arguments: (α : Type u_1) [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [OrderTopology α]
Goal: borel α = MeasurableSpace.generateFrom (Set.range Set.Iic)
-- borel_eq_generateFrom_Ici
theorem borel_eq_generateFrom_Ici : borel α = MeasurableSpace.generateFrom (range Ici) 
Arguments: (α : Type u_1) [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [OrderTopology α]
Goal: borel α = MeasurableSpace.generateFrom (Set.range Set.Ici)
-- measurableSet_Ici
@[simp, measurability]
theorem measurableSet_Ici : MeasurableSet (Ici a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α}
Goal: MeasurableSet (Set.Ici a)
-- measurableSet_Iic
@[simp, measurability]
theorem measurableSet_Iic : MeasurableSet (Iic a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α}
Goal: MeasurableSet (Set.Iic a)
-- measurableSet_Icc
@[simp, measurability]
theorem measurableSet_Icc : MeasurableSet (Icc a b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a b : α}
Goal: MeasurableSet (Set.Icc a b)
-- measurableSet_le'
@[measurability]
theorem measurableSet_le' : MeasurableSet { p : α × α | p.1 ≤ p.2 } 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [PartialOrder α] [OrderClosedTopology α] [SecondCountableTopology α]
Goal: MeasurableSet {p | p.1 ≤ p.2}
-- measurableSet_le
@[measurability]
theorem measurableSet_le {f g : δ → α} (hf : Measurable f) (hg : Measurable g) :
    MeasurableSet { a | f a ≤ g a } 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [PartialOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {f g : δ → α} (hf : Measurable f) (hg : Measurable g)
Goal: MeasurableSet {a | f a ≤ g a}
-- measurableSet_Iio
@[simp, measurability]
theorem measurableSet_Iio : MeasurableSet (Iio a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α}
Goal: MeasurableSet (Set.Iio a)
-- measurableSet_Ioi
@[simp, measurability]
theorem measurableSet_Ioi : MeasurableSet (Ioi a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α}
Goal: MeasurableSet (Set.Ioi a)
-- measurableSet_Ioo
@[simp, measurability]
theorem measurableSet_Ioo : MeasurableSet (Ioo a b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: MeasurableSet (Set.Ioo a b)
-- measurableSet_Ioc
@[simp, measurability]
theorem measurableSet_Ioc : MeasurableSet (Ioc a b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: MeasurableSet (Set.Ioc a b)
-- measurableSet_Ico
@[simp, measurability]
theorem measurableSet_Ico : MeasurableSet (Ico a b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: MeasurableSet (Set.Ico a b)
-- measurableSet_lt'
@[measurability]
theorem measurableSet_lt' [SecondCountableTopology α] : MeasurableSet { p : α × α | p.1 < p.2 } 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α]
Goal: MeasurableSet {p | p.1 < p.2}
-- measurableSet_lt
@[measurability]
theorem measurableSet_lt [SecondCountableTopology α] {f g : δ → α} (hf : Measurable f)
    (hg : Measurable g) : MeasurableSet { a | f a < g a } 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {f g : δ → α} (hf : Measurable f) (hg : Measurable g)
Goal: MeasurableSet {a | f a < g a}
-- nullMeasurableSet_lt
theorem nullMeasurableSet_lt [SecondCountableTopology α] {μ : Measure δ} {f g : δ → α}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : NullMeasurableSet { a | f a < g a } μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {μ : MeasureTheory.Measure δ} {f g : δ → α} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: MeasureTheory.NullMeasurableSet {a | f a < g a} μ
-- nullMeasurableSet_lt'
theorem nullMeasurableSet_lt' [SecondCountableTopology α] {μ : Measure (α × α)} :
    NullMeasurableSet { p : α × α | p.1 < p.2 } μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {μ : MeasureTheory.Measure (α × α)}
Goal: MeasureTheory.NullMeasurableSet {p | p.1 < p.2} μ
-- nullMeasurableSet_le
theorem nullMeasurableSet_le [SecondCountableTopology α] {μ : Measure δ}
    {f g : δ → α} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    NullMeasurableSet { a | f a ≤ g a } μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {μ : MeasureTheory.Measure δ} {f g : δ → α} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: MeasureTheory.NullMeasurableSet {a | f a ≤ g a} μ
-- Set.OrdConnected.measurableSet
theorem Set.OrdConnected.measurableSet (h : OrdConnected s) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] (h : s.OrdConnected)
Goal: MeasurableSet s
-- IsPreconnected.measurableSet
theorem IsPreconnected.measurableSet (h : IsPreconnected s) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] (h : IsPreconnected s)
Goal: MeasurableSet s
-- generateFrom_Ico_mem_le_borel
theorem generateFrom_Ico_mem_le_borel {α : Type*} [TopologicalSpace α] [LinearOrder α]
    [OrderClosedTopology α] (s t : Set α) :
    MeasurableSpace.generateFrom { S | ∃ l ∈ s, ∃ u ∈ t, l < u ∧ Ico l u = S }
      ≤ borel α 
Arguments: {α : Type u_5} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] (s t : Set α)
Goal: MeasurableSpace.generateFrom {S | ∃ l ∈ s, ∃ u ∈ t, l < u ∧ Set.Ico l u = S} ≤ borel α
-- Dense.borel_eq_generateFrom_Ico_mem_aux
theorem Dense.borel_eq_generateFrom_Ico_mem_aux {α : Type*} [TopologicalSpace α] [LinearOrder α]
    [OrderTopology α] [SecondCountableTopology α] {s : Set α} (hd : Dense s)
    (hbot : ∀ x, IsBot x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ → y ∈ s) :
    borel α = .generateFrom { S : Set α | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S } 
Arguments: {α : Type u_5} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {s : Set α} (hd : Dense s) (hbot : ∀ (x : α), IsBot x → x ∈ s) (hIoo : ∀ (x y : α), x < y → Set.Ioo x y = ∅ → y ∈ s)
Goal: borel α = MeasurableSpace.generateFrom {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Set.Ico l u = S}
-- Dense.borel_eq_generateFrom_Ico_mem
theorem Dense.borel_eq_generateFrom_Ico_mem {α : Type*} [TopologicalSpace α] [LinearOrder α]
    [OrderTopology α] [SecondCountableTopology α] [DenselyOrdered α] [NoMinOrder α] {s : Set α}
    (hd : Dense s) :
    borel α = .generateFrom { S : Set α | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S } 
Arguments: {α : Type u_5} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [DenselyOrdered α] [NoMinOrder α] {s : Set α} (hd : Dense s)
Goal: borel α = MeasurableSpace.generateFrom {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Set.Ico l u = S}
-- borel_eq_generateFrom_Ico
theorem borel_eq_generateFrom_Ico (α : Type*) [TopologicalSpace α] [SecondCountableTopology α]
    [LinearOrder α] [OrderTopology α] :
    borel α = .generateFrom { S : Set α | ∃ (l u : α), l < u ∧ Ico l u = S } 
Arguments: (α : Type u_5) [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [OrderTopology α]
Goal: borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Set.Ico l u = S}
-- Dense.borel_eq_generateFrom_Ioc_mem_aux
theorem Dense.borel_eq_generateFrom_Ioc_mem_aux {α : Type*} [TopologicalSpace α] [LinearOrder α]
    [OrderTopology α] [SecondCountableTopology α] {s : Set α} (hd : Dense s)
    (hbot : ∀ x, IsTop x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ → x ∈ s) :
    borel α = .generateFrom { S : Set α | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ioc l u = S } 
Arguments: {α : Type u_5} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {s : Set α} (hd : Dense s) (hbot : ∀ (x : α), IsTop x → x ∈ s) (hIoo : ∀ (x y : α), x < y → Set.Ioo x y = ∅ → x ∈ s)
Goal: borel α = MeasurableSpace.generateFrom {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Set.Ioc l u = S}
-- Dense.borel_eq_generateFrom_Ioc_mem
theorem Dense.borel_eq_generateFrom_Ioc_mem {α : Type*} [TopologicalSpace α] [LinearOrder α]
    [OrderTopology α] [SecondCountableTopology α] [DenselyOrdered α] [NoMaxOrder α] {s : Set α}
    (hd : Dense s) :
    borel α = .generateFrom { S : Set α | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ioc l u = S } 
Arguments: {α : Type u_5} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [DenselyOrdered α] [NoMaxOrder α] {s : Set α} (hd : Dense s)
Goal: borel α = MeasurableSpace.generateFrom {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Set.Ioc l u = S}
-- borel_eq_generateFrom_Ioc
theorem borel_eq_generateFrom_Ioc (α : Type*) [TopologicalSpace α] [SecondCountableTopology α]
    [LinearOrder α] [OrderTopology α] :
    borel α = .generateFrom { S : Set α | ∃ l u, l < u ∧ Ioc l u = S } 
Arguments: (α : Type u_5) [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [OrderTopology α]
Goal: borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Set.Ioc l u = S}
-- MeasureTheory.Measure.ext_of_Ico_finite
theorem ext_of_Ico_finite {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
    [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : Measure α)
    [IsFiniteMeasure μ] (hμν : μ univ = ν univ) (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) :
    μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {m : MeasurableSpace α} [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (hμν : μ Set.univ = ν Set.univ) (h : ∀ ⦃a b : α⦄, a < b → μ (Set.Ico a b) = ν (Set.Ico a b))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Ioc_finite
theorem ext_of_Ioc_finite {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
    [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : Measure α)
    [IsFiniteMeasure μ] (hμν : μ univ = ν univ) (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) :
    μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {m : MeasurableSpace α} [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (hμν : μ Set.univ = ν Set.univ) (h : ∀ ⦃a b : α⦄, a < b → μ (Set.Ioc a b) = ν (Set.Ioc a b))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Ico'
theorem ext_of_Ico' {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
    [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] [NoMaxOrder α]
    (μ ν : Measure α) (hμ : ∀ ⦃a b⦄, a < b → μ (Ico a b) ≠ ∞)
    (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {m : MeasurableSpace α} [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] [NoMaxOrder α] (μ ν : MeasureTheory.Measure α) (hμ : ∀ ⦃a b : α⦄, a < b → μ (Set.Ico a b) ≠ ⊤) (h : ∀ ⦃a b : α⦄, a < b → μ (Set.Ico a b) = ν (Set.Ico a b))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Ioc'
theorem ext_of_Ioc' {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
    [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] [NoMinOrder α]
    (μ ν : Measure α) (hμ : ∀ ⦃a b⦄, a < b → μ (Ioc a b) ≠ ∞)
    (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {m : MeasurableSpace α} [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] [NoMinOrder α] (μ ν : MeasureTheory.Measure α) (hμ : ∀ ⦃a b : α⦄, a < b → μ (Set.Ioc a b) ≠ ⊤) (h : ∀ ⦃a b : α⦄, a < b → μ (Set.Ioc a b) = ν (Set.Ioc a b))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Ico
theorem ext_of_Ico {α : Type*} [TopologicalSpace α] {_m : MeasurableSpace α}
    [SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
    [BorelSpace α] [NoMaxOrder α] (μ ν : Measure α) [IsLocallyFiniteMeasure μ]
    (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {_m : MeasurableSpace α} [SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [BorelSpace α] [NoMaxOrder α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ ⦃a b : α⦄, a < b → μ (Set.Ico a b) = ν (Set.Ico a b))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Ioc
theorem ext_of_Ioc {α : Type*} [TopologicalSpace α] {_m : MeasurableSpace α}
    [SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
    [BorelSpace α] [NoMinOrder α] (μ ν : Measure α) [IsLocallyFiniteMeasure μ]
    (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {_m : MeasurableSpace α} [SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [BorelSpace α] [NoMinOrder α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ ⦃a b : α⦄, a < b → μ (Set.Ioc a b) = ν (Set.Ioc a b))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Iic
theorem ext_of_Iic {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
    [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : Measure α)
    [IsFiniteMeasure μ] (h : ∀ a, μ (Iic a) = ν (Iic a)) : μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {m : MeasurableSpace α} [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (h : ∀ (a : α), μ (Set.Iic a) = ν (Set.Iic a))
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_Ici
theorem ext_of_Ici {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
    [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : Measure α)
    [IsFiniteMeasure μ] (h : ∀ a, μ (Ici a) = ν (Ici a)) : μ = ν 
Arguments: {α : Type u_5} [TopologicalSpace α] {m : MeasurableSpace α} [SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (h : ∀ (a : α), μ (Set.Ici a) = ν (Set.Ici a))
Goal: μ = ν
-- measurableSet_uIcc
@[measurability]
theorem measurableSet_uIcc : MeasurableSet (uIcc a b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: MeasurableSet (Set.uIcc a b)
-- measurableSet_uIoc
@[measurability]
theorem measurableSet_uIoc : MeasurableSet (uIoc a b) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a b : α}
Goal: MeasurableSet (Ι a b)
-- Measurable.max
@[measurability]
theorem Measurable.max {f g : δ → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun a => max (f a) (g a) 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {f g : δ → α} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => max (f a) (g a)
-- AEMeasurable.max
@[measurability]
nonrec theorem AEMeasurable.max {f g : δ → α} {μ : Measure δ} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : AEMeasurable (fun a => max (f a) (g a)) μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {f g : δ → α} {μ : MeasureTheory.Measure δ} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => max (f a) (g a)) μ