ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.Kernel.indep_limsup_atBot_self
theorem Kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
    Indep (limsup s atBot) (limsup s atBot) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.Kernel.iIndep s κ μα)
Goal: ProbabilityTheory.Kernel.Indep (Filter.limsup s Filter.atBot) (Filter.limsup s Filter.atBot) κ μα
-- ProbabilityTheory.indep_limsup_atBot_self
theorem indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
    Indep (limsup s atBot) (limsup s atBot) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ)
Goal: ProbabilityTheory.Indep (Filter.limsup s Filter.atBot) (Filter.limsup s Filter.atBot) μ
-- ProbabilityTheory.condIndep_limsup_atBot_self
theorem condIndep_limsup_atBot_self [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
    CondIndep m (limsup s atBot) (limsup s atBot) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ)
Goal: ProbabilityTheory.CondIndep m (Filter.limsup s Filter.atBot) (Filter.limsup s Filter.atBot) hm μ
-- ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot
theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.Kernel.iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1
-- ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot
theorem measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
    μ t = 0 ∨ μ t = 1 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: μ t = 0 ∨ μ t = 1
-- ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup_atBot
theorem condexp_zero_or_one_of_measurableSet_limsup_atBot [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ] (h_le : ∀ n, s n ≤ m0)
    (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι] [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1
-- MeasureTheory.pdf.IsUniform.aemeasurable
theorem aemeasurable {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
    (hu : IsUniform X s ℙ μ) : AEMeasurable X ℙ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  μ s ≠ 0 → μ s ≠ ⊤ → MeasureTheory.pdf.IsUniform X s ℙ μ → AEMeasurable X ℙ
-- MeasureTheory.pdf.IsUniform.absolutelyContinuous
theorem absolutelyContinuous {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) : map X ℙ ≪ μ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasureTheory.pdf.IsUniform X s ℙ μ → MeasureTheory.Measure.map X ℙ ≪ μ
-- MeasureTheory.pdf.IsUniform.measure_preimage
theorem measure_preimage {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
    (hu : IsUniform X s ℙ μ) {A : Set E} (hA : MeasurableSet A) :
    ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  μ s ≠ 0 →
    μ s ≠ ⊤ → MeasureTheory.pdf.IsUniform X s ℙ μ → ∀ {A : Set E}, MeasurableSet A → ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s
-- MeasureTheory.pdf.IsUniform.isProbabilityMeasure
theorem isProbabilityMeasure {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
    (hu : IsUniform X s ℙ μ) : IsProbabilityMeasure ℙ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  μ s ≠ 0 → μ s ≠ ⊤ → MeasureTheory.pdf.IsUniform X s ℙ μ → MeasureTheory.IsProbabilityMeasure ℙ
-- MeasureTheory.pdf.IsUniform.toMeasurable_iff
theorem toMeasurable_iff {X : Ω → E} {s : Set E} :
    IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasureTheory.pdf.IsUniform X (MeasureTheory.toMeasurable μ s) ℙ μ ↔ MeasureTheory.pdf.IsUniform X s ℙ μ
-- MeasureTheory.pdf.IsUniform.toMeasurable
protected theorem toMeasurable {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) :
    IsUniform X (toMeasurable μ s) ℙ μ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasureTheory.pdf.IsUniform X s ℙ μ → MeasureTheory.pdf.IsUniform X (MeasureTheory.toMeasurable μ s) ℙ μ
-- MeasureTheory.pdf.IsUniform.hasPDF
theorem hasPDF {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
    (hu : IsUniform X s ℙ μ) : HasPDF X ℙ μ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  μ s ≠ 0 → μ s ≠ ⊤ → MeasureTheory.pdf.IsUniform X s ℙ μ → MeasureTheory.HasPDF X ℙ μ
-- MeasureTheory.pdf.IsUniform.pdf_eq_zero_of_measure_eq_zero_or_top
theorem pdf_eq_zero_of_measure_eq_zero_or_top {X : Ω → E} {s : Set E}
    (hu : IsUniform X s ℙ μ) (hμs : μ s = 0 ∨ μ s = ∞) : pdf X ℙ μ =ᵐ[μ] 0 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasureTheory.pdf.IsUniform X s ℙ μ → μ s = 0 ∨ μ s = ⊤ → MeasureTheory.pdf X ℙ μ =ᵐ[μ] 0
-- MeasureTheory.pdf.IsUniform.pdf_eq
theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
    (hu : IsUniform X s ℙ μ) : pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasurableSet s → MeasureTheory.pdf.IsUniform X s ℙ μ → MeasureTheory.pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • 1)
-- MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq
theorem pdf_toReal_ae_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
    (hX : IsUniform X s ℙ μ) :
    (fun x => (pdf X ℙ μ x).toReal) =ᵐ[μ] fun x =>
      (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).toReal 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasurableSet s →
    MeasureTheory.pdf.IsUniform X s ℙ μ →
      (fun x_1 => (MeasureTheory.pdf X ℙ μ x_1).toReal) =ᵐ[μ] fun x => (s.indicator ((μ s)⁻¹ • 1) x).toReal
-- MeasureTheory.pdf.IsUniform.mul_pdf_integrable
theorem mul_pdf_integrable (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
    Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal 
Arguments: {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {s : Set ℝ},
  IsCompact s →
    MeasureTheory.pdf.IsUniform X s ℙ MeasureTheory.volume →
      MeasureTheory.Integrable (fun x_1 => x_1 * (MeasureTheory.pdf X ℙ MeasureTheory.volume x_1).toReal)
        MeasureTheory.volume
-- MeasureTheory.pdf.IsUniform.integral_eq
theorem integral_eq (huX : IsUniform X s ℙ) :
    ∫ x, X x ∂ℙ = (volume s)⁻¹.toReal * ∫ x in s, x 
Arguments: {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {s : Set ℝ},
  MeasureTheory.pdf.IsUniform X s ℙ MeasureTheory.volume →
    ∫ (x : Ω), X x ∂ℙ = (MeasureTheory.volume s)⁻¹.toReal * ∫ (x : ℝ) in s, x
-- MeasureTheory.pdf.IsUniform.cond
lemma IsUniform.cond {s : Set E} :
    IsUniform (id : E → E) s (ProbabilityTheory.cond μ s) μ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {s : Set E}
Goal: MeasureTheory.pdf.IsUniform id s (ProbabilityTheory.cond μ s) μ
-- MeasureTheory.pdf.uniformPDF_eq_pdf
lemma uniformPDF_eq_pdf {s : Set E} (hs : MeasurableSet s) (hu : pdf.IsUniform X s ℙ μ) :
    (fun x ↦ uniformPDF s x μ) =ᵐ[μ] pdf X ℙ μ 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {Ω : Type u_2}
Goal: ∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},
  MeasurableSet s →
    MeasureTheory.pdf.IsUniform X s ℙ μ → (fun x => MeasureTheory.pdf.uniformPDF s x μ) =ᵐ[μ] MeasureTheory.pdf X ℙ μ
-- MeasureTheory.pdf.uniformPDF_ite
lemma uniformPDF_ite {s : Set E} {x : E} :
    uniformPDF s x μ = if x ∈ s then (μ s)⁻¹ else 0 
Arguments: {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {s : Set E} {x : E}
Goal: MeasureTheory.pdf.uniformPDF s x μ = if x ∈ s then (μ s)⁻¹ else 0
-- PMF.uniformOfFinset_apply
@[simp]
theorem uniformOfFinset_apply (a : α) :
    uniformOfFinset s hs a = if a ∈ s then (s.card : ℝ≥0∞)⁻¹ else 0 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty) (a : α)
Goal: (PMF.uniformOfFinset s hs) a = if a ∈ s then (↑s.card)⁻¹ else 0
-- PMF.uniformOfFinset_apply_of_mem
theorem uniformOfFinset_apply_of_mem (ha : a ∈ s) : uniformOfFinset s hs a = (s.card : ℝ≥0∞)⁻¹ 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty) {a : α} (ha : a ∈ s)
Goal: (PMF.uniformOfFinset s hs) a = (↑s.card)⁻¹
-- PMF.uniformOfFinset_apply_of_not_mem
theorem uniformOfFinset_apply_of_not_mem (ha : a ∉ s) : uniformOfFinset s hs a = 0 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty) {a : α} (ha : a ∉ s)
Goal: (PMF.uniformOfFinset s hs) a = 0
-- PMF.support_uniformOfFinset
@[simp]
theorem support_uniformOfFinset : (uniformOfFinset s hs).support = s 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty)
Goal: (PMF.uniformOfFinset s hs).support = ↑s
-- PMF.mem_support_uniformOfFinset_iff
theorem mem_support_uniformOfFinset_iff (a : α) : a ∈ (uniformOfFinset s hs).support ↔ a ∈ s 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty) (a : α)
Goal: a ∈ (PMF.uniformOfFinset s hs).support ↔ a ∈ s
-- PMF.toOuterMeasure_uniformOfFinset_apply
@[simp]
theorem toOuterMeasure_uniformOfFinset_apply :
    (uniformOfFinset s hs).toOuterMeasure t = (s.filter (· ∈ t)).card / s.card 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty) (t : Set α)
Goal: (PMF.uniformOfFinset s hs).toOuterMeasure t = ↑(Finset.filter (fun x => x ∈ t) s).card / ↑s.card
-- PMF.toMeasure_uniformOfFinset_apply
@[simp]
theorem toMeasure_uniformOfFinset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
    (uniformOfFinset s hs).toMeasure t = (s.filter (· ∈ t)).card / s.card 
Arguments: {α : Type u_1} {s : Finset α} (hs : s.Nonempty) (t : Set α) [MeasurableSpace α] (ht : MeasurableSet t)
Goal: (PMF.uniformOfFinset s hs).toMeasure t = ↑(Finset.filter (fun x => x ∈ t) s).card / ↑s.card
-- PMF.uniformOfFintype_apply
@[simp]
theorem uniformOfFintype_apply (a : α) : uniformOfFintype α a = (Fintype.card α : ℝ≥0∞)⁻¹ 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] (a : α)
Goal: (PMF.uniformOfFintype α) a = (↑(Fintype.card α))⁻¹
-- PMF.support_uniformOfFintype
@[simp]
theorem support_uniformOfFintype (α : Type*) [Fintype α] [Nonempty α] :
    (uniformOfFintype α).support = ⊤ 
Arguments: (α : Type u_4) [Fintype α] [Nonempty α]
Goal: (PMF.uniformOfFintype α).support = ⊤
-- PMF.mem_support_uniformOfFintype
theorem mem_support_uniformOfFintype (a : α) : a ∈ (uniformOfFintype α).support 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] (a : α)
Goal: a ∈ (PMF.uniformOfFintype α).support
-- PMF.toOuterMeasure_uniformOfFintype_apply
theorem toOuterMeasure_uniformOfFintype_apply :
    (uniformOfFintype α).toOuterMeasure s = Fintype.card s / Fintype.card α 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] (s : Set α)
Goal: (PMF.uniformOfFintype α).toOuterMeasure s = ↑(Fintype.card ↑s) / ↑(Fintype.card α)
-- PMF.toMeasure_uniformOfFintype_apply
theorem toMeasure_uniformOfFintype_apply [MeasurableSpace α] (hs : MeasurableSet s) :
    (uniformOfFintype α).toMeasure s = Fintype.card s / Fintype.card α 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] (s : Set α) [MeasurableSpace α] (hs : MeasurableSet s)
Goal: (PMF.uniformOfFintype α).toMeasure s = ↑(Fintype.card ↑s) / ↑(Fintype.card α)
-- PMF.ofMultiset_apply
@[simp]
theorem ofMultiset_apply (a : α) : ofMultiset s hs a = s.count a / (Multiset.card s) 
Arguments: {α : Type u_1} {s : Multiset α} (hs : s ≠ 0) (a : α)
Goal: (PMF.ofMultiset s hs) a = ↑(Multiset.count a s) / ↑(Multiset.card s)
-- PMF.support_ofMultiset
@[simp]
theorem support_ofMultiset : (ofMultiset s hs).support = s.toFinset 
Arguments: {α : Type u_1} {s : Multiset α} (hs : s ≠ 0)
Goal: (PMF.ofMultiset s hs).support = ↑s.toFinset
-- PMF.mem_support_ofMultiset_iff
theorem mem_support_ofMultiset_iff (a : α) : a ∈ (ofMultiset s hs).support ↔ a ∈ s.toFinset 
Arguments: {α : Type u_1} {s : Multiset α} (hs : s ≠ 0) (a : α)
Goal: a ∈ (PMF.ofMultiset s hs).support ↔ a ∈ s.toFinset
-- PMF.ofMultiset_apply_of_not_mem
theorem ofMultiset_apply_of_not_mem {a : α} (ha : a ∉ s) : ofMultiset s hs a = 0 
Arguments: {α : Type u_1} {s : Multiset α} (hs : s ≠ 0) {a : α} (ha : a ∉ s)
Goal: (PMF.ofMultiset s hs) a = 0
-- PMF.toOuterMeasure_ofMultiset_apply
@[simp]
theorem toOuterMeasure_ofMultiset_apply :
    (ofMultiset s hs).toOuterMeasure t =
      (∑' x, (s.filter (· ∈ t)).count x : ℝ≥0∞) / (Multiset.card s) 
Arguments: {α : Type u_1} {s : Multiset α} (hs : s ≠ 0) (t : Set α)
Goal: (PMF.ofMultiset s hs).toOuterMeasure t =
  (∑' (x : α), ↑(Multiset.count x (Multiset.filter (fun x => x ∈ t) s))) / ↑(Multiset.card s)
-- PMF.toMeasure_ofMultiset_apply
@[simp]
theorem toMeasure_ofMultiset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
    (ofMultiset s hs).toMeasure t = (∑' x, (s.filter (· ∈ t)).count x : ℝ≥0∞) / (Multiset.card s) 
Arguments: {α : Type u_1} {s : Multiset α} (hs : s ≠ 0) (t : Set α) [MeasurableSpace α] (ht : MeasurableSet t)
Goal: (PMF.ofMultiset s hs).toMeasure t =
  (∑' (x : α), ↑(Multiset.count x (Multiset.filter (fun x => x ∈ t) s))) / ↑(Multiset.card s)
-- ContMDiffAt.mfderiv
protected theorem ContMDiffAt.mfderiv {x₀ : N} (f : N → M → M') (g : N → M)
    (hf : ContMDiffAt (J.prod I) I' n (Function.uncurry f) (x₀, g x₀)) (hg : ContMDiffAt J I m g x₀)
    (hmn : m + 1 ≤ n) :
    ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m
      (inTangentCoordinates I I' g (fun x => f x (g x)) (fun x => mfderiv I I' (f x) (g x)) x₀)
      x₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [Is : SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [I's : SmoothManifoldWithCorners I' M'] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_9} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_10} [TopologicalSpace N] [ChartedSpace G N] [Js : SmoothManifoldWithCorners J N] {m n : ℕ∞} {x₀ : N} (f : N → M → M') (g : N → M) (hf : ContMDiffAt (J.prod I) I' n (Function.uncurry f) (x₀, g x₀)) (hg : ContMDiffAt J I m g x₀) (hmn : m + 1 ≤ n)
Goal: ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m
  (inTangentCoordinates I I' g (fun x => f x (g x)) (fun x => mfderiv I I' (f x) (g x)) x₀) x₀
-- ContMDiffAt.mfderiv_const
theorem ContMDiffAt.mfderiv_const {x₀ : M} {f : M → M'} (hf : ContMDiffAt I I' n f x₀)
    (hmn : m + 1 ≤ n) :
    ContMDiffAt I 𝓘(𝕜, E →L[𝕜] E') m (inTangentCoordinates I I' id f (mfderiv I I' f) x₀) x₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [Is : SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [I's : SmoothManifoldWithCorners I' M'] {m n : ℕ∞} {x₀ : M} {f : M → M'} (hf : ContMDiffAt I I' n f x₀) (hmn : m + 1 ≤ n)
Goal: ContMDiffAt I 𝓘(𝕜, E →L[𝕜] E') m (inTangentCoordinates I I' id f (mfderiv I I' f) x₀) x₀
-- ContMDiffAt.mfderiv_apply
theorem ContMDiffAt.mfderiv_apply {x₀ : N'} (f : N → M → M') (g : N → M) (g₁ : N' → N) (g₂ : N' → E)
    (hf : ContMDiffAt (J.prod I) I' n (Function.uncurry f) (g₁ x₀, g (g₁ x₀)))
    (hg : ContMDiffAt J I m g (g₁ x₀)) (hg₁ : ContMDiffAt J' J m g₁ x₀)
    (hg₂ : ContMDiffAt J' 𝓘(𝕜, E) m g₂ x₀) (hmn : m + 1 ≤ n) :
    ContMDiffAt J' 𝓘(𝕜, E') m
      (fun x => inTangentCoordinates I I' g (fun x => f x (g x))
        (fun x => mfderiv I I' (f x) (g x)) (g₁ x₀) (g₁ x) (g₂ x)) x₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [Is : SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [I's : SmoothManifoldWithCorners I' M'] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_9} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_10} [TopologicalSpace N] [ChartedSpace G N] [Js : SmoothManifoldWithCorners J N] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_12} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_13} [TopologicalSpace N'] [ChartedSpace G' N'] {m n : ℕ∞} {x₀ : N'} (f : N → M → M') (g : N → M) (g₁ : N' → N) (g₂ : N' → E) (hf : ContMDiffAt (J.prod I) I' n (Function.uncurry f) (g₁ x₀, g (g₁ x₀))) (hg : ContMDiffAt J I m g (g₁ x₀)) (hg₁ : ContMDiffAt J' J m g₁ x₀) (hg₂ : ContMDiffAt J' 𝓘(𝕜, E) m g₂ x₀) (hmn : m + 1 ≤ n)
Goal: ContMDiffAt J' 𝓘(𝕜, E') m
  (fun x =>
    (inTangentCoordinates I I' g (fun x => f x (g x)) (fun x => mfderiv I I' (f x) (g x)) (g₁ x₀) (g₁ x)) (g₂ x))
  x₀
-- ContMDiffOn.continuousOn_tangentMapWithin_aux
theorem ContMDiffOn.continuousOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
    (hf : ContMDiffOn I I' n f s) (hn : 1 ≤ n) (hs : UniqueMDiffOn I s) :
    ContinuousOn (tangentMapWithin I I' f s) (π E (TangentSpace I) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {n : ℕ∞} {f : H → H'} {s : Set H} (hf : ContMDiffOn I I' n f s) (hn : 1 ≤ n) (hs : UniqueMDiffOn I s)
Goal: ContinuousOn (tangentMapWithin I I' f s) (Bundle.TotalSpace.proj ⁻¹' s)
-- ContMDiffOn.contMDiffOn_tangentMapWithin_aux
theorem ContMDiffOn.contMDiffOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
    (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n) (hs : UniqueMDiffOn I s) :
    ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
      (π E (TangentSpace I) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {m n : ℕ∞} {f : H → H'} {s : Set H} (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n) (hs : UniqueMDiffOn I s)
Goal: ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s) (Bundle.TotalSpace.proj ⁻¹' s)
-- ContMDiffOn.contMDiffOn_tangentMapWithin
theorem ContMDiffOn.contMDiffOn_tangentMapWithin (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n)
    (hs : UniqueMDiffOn I s) :
    ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
      (π E (TangentSpace I) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [Is : SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [I's : SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {m n : ℕ∞} (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n) (hs : UniqueMDiffOn I s)
Goal: ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s) (Bundle.TotalSpace.proj ⁻¹' s)
-- ContMDiffOn.continuousOn_tangentMapWithin
theorem ContMDiffOn.continuousOn_tangentMapWithin (hf : ContMDiffOn I I' n f s) (hmn : 1 ≤ n)
    (hs : UniqueMDiffOn I s) :
    ContinuousOn (tangentMapWithin I I' f s) (π E (TangentSpace I) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [Is : SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [I's : SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {n : ℕ∞} (hf : ContMDiffOn I I' n f s) (hmn : 1 ≤ n) (hs : UniqueMDiffOn I s)
Goal: ContinuousOn (tangentMapWithin I I' f s) (Bundle.TotalSpace.proj ⁻¹' s)