ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.IsCondKernelCDF.setIntegral
structure IsCondKernelCDF (f : α × β → StieltjesFunction) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable (x : ℝ) : Measurable fun p ↦ f p x
  integrable (a : α) (x : ℝ) : Integrable (fun b ↦ f (a, b) x) (ν a)
  tendsto_atTop_one (p : α × β) : Tendsto (f p) atTop (𝓝 1)
  tendsto_atBot_zero (p : α × β) : Tendsto (f p) atBot (𝓝 0)
  setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (x : ℝ) :
    ∫ b in s, f (a, b) x ∂(ν a) = (κ a (s ×ˢ Iic x)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → StieltjesFunction} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) {s : Set β} (_hs : MeasurableSet s) (x : ℝ)
Goal: ∫ (b : β) in s, ↑(f (a, b)) x ∂ν a = ((κ a) (s ×ˢ Set.Iic x)).toReal
-- ProbabilityTheory.IsCondKernelCDF.tendsto_atTop_one
structure IsCondKernelCDF (f : α × β → StieltjesFunction) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable (x : ℝ) : Measurable fun p ↦ f p x
  integrable (a : α) (x : ℝ) : Integrable (fun b ↦ f (a, b) x) (ν a)
  tendsto_atTop_one (p : α × β) : Tendsto (f p) atTop (𝓝 1)
  tendsto_atBot_zero (p : α × β) : Tendsto (f p) atBot (𝓝 0)
  setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (x : ℝ) :
    ∫ b in s, f (a, b) x ∂(ν a) = (κ a (s ×ˢ Iic x)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → StieltjesFunction} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsCondKernelCDF f κ ν) (p : α × β)
Goal: Filter.Tendsto (↑(f p)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.IsCondKernelCDF.nonneg
lemma IsCondKernelCDF.nonneg (hf : IsCondKernelCDF f κ ν) (p : α × β) (x : ℝ) : 0 ≤ f p x 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (p : α × β) (x : ℝ)
Goal: 0 ≤ ↑(f p) x
-- ProbabilityTheory.IsCondKernelCDF.le_one
lemma IsCondKernelCDF.le_one (hf : IsCondKernelCDF f κ ν) (p : α × β) (x : ℝ) : f p x ≤ 1 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (p : α × β) (x : ℝ)
Goal: ↑(f p) x ≤ 1
-- ProbabilityTheory.IsCondKernelCDF.integral
lemma IsCondKernelCDF.integral
    {f : α × β → StieltjesFunction} (hf : IsCondKernelCDF f κ ν) (a : α) (x : ℝ) :
    ∫ b, f (a, b) x ∂(ν a) = (κ a (univ ×ˢ Iic x)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) (x : ℝ)
Goal: ∫ (b : β), ↑(f (a, b)) x ∂ν a = ((κ a) (Set.univ ×ˢ Set.Iic x)).toReal
-- ProbabilityTheory.IsCondKernelCDF.setLIntegral
lemma IsCondKernelCDF.setLIntegral [IsFiniteKernel κ]
    {f : α × β → StieltjesFunction} (hf : IsCondKernelCDF f κ ν)
    (a : α) {s : Set β} (hs : MeasurableSet s) (x : ℝ) :
    ∫⁻ b in s, ENNReal.ofReal (f (a, b) x) ∂(ν a) = κ a (s ×ˢ Iic x) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsFiniteKernel κ] {f : α × β → StieltjesFunction} (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) {s : Set β} (hs : MeasurableSet s) (x : ℝ)
Goal: ∫⁻ (b : β) in s, ENNReal.ofReal (↑(f (a, b)) x) ∂ν a = (κ a) (s ×ˢ Set.Iic x)
-- ProbabilityTheory.IsCondKernelCDF.lintegral
lemma IsCondKernelCDF.lintegral [IsFiniteKernel κ]
    {f : α × β → StieltjesFunction} (hf : IsCondKernelCDF f κ ν) (a : α) (x : ℝ) :
    ∫⁻ b, ENNReal.ofReal (f (a, b) x) ∂(ν a) = κ a (univ ×ˢ Iic x) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsFiniteKernel κ] {f : α × β → StieltjesFunction} (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) (x : ℝ)
Goal: ∫⁻ (b : β), ENNReal.ofReal (↑(f (a, b)) x) ∂ν a = (κ a) (Set.univ ×ˢ Set.Iic x)
-- ProbabilityTheory.isCondKernelCDF_stieltjesOfMeasurableRat
lemma isCondKernelCDF_stieltjesOfMeasurableRat {f : α × β → ℚ → ℝ} (hf : IsRatCondKernelCDF f κ ν)
    [IsFiniteKernel κ] :
    IsCondKernelCDF (stieltjesOfMeasurableRat f hf.measurable) κ ν 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) [ProbabilityTheory.IsFiniteKernel κ]
Goal: ProbabilityTheory.IsCondKernelCDF (ProbabilityTheory.stieltjesOfMeasurableRat f ⋯) κ ν
-- ProbabilityTheory.StieltjesFunction.measurable_measure
lemma StieltjesFunction.measurable_measure {f : α → StieltjesFunction}
    (hf : ∀ q, Measurable fun a ↦ f a q)
    (hf_bot : ∀ a, Tendsto (f a) atBot (𝓝 0))
    (hf_top : ∀ a, Tendsto (f a) atTop (𝓝 1)) :
    Measurable fun a ↦ (f a).measure 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {f : α → StieltjesFunction} (hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q) (hf_bot : ∀ (a : α), Filter.Tendsto (↑(f a)) Filter.atBot (𝓝 0)) (hf_top : ∀ (a : α), Filter.Tendsto (↑(f a)) Filter.atTop (𝓝 1))
Goal: Measurable fun a => (f a).measure
-- ProbabilityTheory.IsCondKernelCDF.toKernel_apply
lemma IsCondKernelCDF.toKernel_apply {hf : IsCondKernelCDF f κ ν} (p : α × β) :
    hf.toKernel f p = (f p).measure 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace β} {f : α × β → StieltjesFunction} {κ : ProbabilityTheory.Kernel α (β × ℝ)}
  {ν : ProbabilityTheory.Kernel α β} {hf : ProbabilityTheory.IsCondKernelCDF f κ ν} (p : α × β),
  (ProbabilityTheory.IsCondKernelCDF.toKernel f hf) p = (f p).measure
-- ProbabilityTheory.IsCondKernelCDF.toKernel_Iic
lemma IsCondKernelCDF.toKernel_Iic {hf : IsCondKernelCDF f κ ν} (p : α × β) (x : ℝ) :
    hf.toKernel f p (Iic x) = ENNReal.ofReal (f p x) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace β} {f : α × β → StieltjesFunction} {κ : ProbabilityTheory.Kernel α (β × ℝ)}
  {ν : ProbabilityTheory.Kernel α β} {hf : ProbabilityTheory.IsCondKernelCDF f κ ν} (p : α × β) (x_1 : ℝ),
  ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) p) (Set.Iic x_1) = ENNReal.ofReal (↑(f p) x_1)
-- ProbabilityTheory.setLIntegral_toKernel_Iic
lemma setLIntegral_toKernel_Iic [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
    (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ b in s, hf.toKernel f (a, b) (Iic x) ∂(ν a) = κ a (s ×ˢ Iic x) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (b : β) in s, ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) (Set.Iic x) ∂ν a = (κ a) (s ×ˢ Set.Iic x)
-- ProbabilityTheory.setLIntegral_toKernel_univ
lemma setLIntegral_toKernel_univ [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ b in s, hf.toKernel f (a, b) univ ∂(ν a) = κ a (s ×ˢ univ) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (b : β) in s, ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) Set.univ ∂ν a = (κ a) (s ×ˢ Set.univ)
-- ProbabilityTheory.lintegral_toKernel_univ
lemma lintegral_toKernel_univ [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν) (a : α) :
    ∫⁻ b, hf.toKernel f (a, b) univ ∂(ν a) = κ a univ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α)
Goal: ∫⁻ (b : β), ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) Set.univ ∂ν a = (κ a) Set.univ
-- ProbabilityTheory.setLIntegral_toKernel_prod
lemma setLIntegral_toKernel_prod [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
    (a : α) {s : Set β} (hs : MeasurableSet s) {t : Set ℝ} (ht : MeasurableSet t) :
    ∫⁻ b in s, hf.toKernel f (a, b) t ∂(ν a) = κ a (s ×ˢ t) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) {s : Set β} (hs : MeasurableSet s) {t : Set ℝ} (ht : MeasurableSet t)
Goal: ∫⁻ (b : β) in s, ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) t ∂ν a = (κ a) (s ×ˢ t)
-- ProbabilityTheory.lintegral_toKernel_mem
lemma lintegral_toKernel_mem [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
    (a : α) {s : Set (β × ℝ)} (hs : MeasurableSet s) :
    ∫⁻ b, hf.toKernel f (a, b) {y | (b, y) ∈ s} ∂(ν a) = κ a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) {s : Set (β × ℝ)} (hs : MeasurableSet s)
Goal: ∫⁻ (b : β), ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) {y | (b, y) ∈ s} ∂ν a = (κ a) s
-- ProbabilityTheory.compProd_toKernel
lemma compProd_toKernel [IsFiniteKernel κ] [IsSFiniteKernel ν] (hf : IsCondKernelCDF f κ ν) :
    ν ⊗ₖ hf.toKernel f = κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel ν] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν)
Goal: ν ⊗ₖ ProbabilityTheory.IsCondKernelCDF.toKernel f hf = κ
-- PMF.monad_map_eq_map
theorem monad_map_eq_map {α β : Type u} (f : α → β) (p : PMF α) : f <$> p = p.map f 
Arguments: {α β : Type u} (f : α → β) (p : PMF α)
Goal: f <$> p = PMF.map f p
-- PMF.map_apply
@[simp]
theorem map_apply : (map f p) b = ∑' a, if b = f a then p a else 0 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (p : PMF α) (b : β)
Goal: (PMF.map f p) b = ∑' (a : α), if b = f a then p a else 0
-- PMF.support_map
@[simp]
theorem support_map : (map f p).support = f '' p.support 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (p : PMF α)
Goal: (PMF.map f p).support = f '' p.support
-- PMF.mem_support_map_iff
theorem mem_support_map_iff : b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (p : PMF α) (b : β)
Goal: b ∈ (PMF.map f p).support ↔ ∃ a ∈ p.support, f a = b
-- PMF.bind_pure_comp
theorem bind_pure_comp : bind p (pure ∘ f) = map f p 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (p : PMF α)
Goal: p.bind (PMF.pure ∘ f) = PMF.map f p
-- PMF.map_id
theorem map_id : map id p = p 
Arguments: {α : Type u_1} (p : PMF α)
Goal: PMF.map id p = p
-- PMF.map_comp
theorem map_comp (g : β → γ) : (p.map f).map g = p.map (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β) (p : PMF α) (g : β → γ)
Goal: PMF.map g (PMF.map f p) = PMF.map (g ∘ f) p
-- PMF.pure_map
theorem pure_map (a : α) : (pure a).map f = pure (f a) 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (a : α)
Goal: PMF.map f (PMF.pure a) = PMF.pure (f a)
-- PMF.map_bind
theorem map_bind (q : α → PMF β) (f : β → γ) : (p.bind q).map f = p.bind fun a => (q a).map f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : PMF α) (q : α → PMF β) (f : β → γ)
Goal: PMF.map f (p.bind q) = p.bind fun a => PMF.map f (q a)
-- PMF.bind_map
@[simp]
theorem bind_map (p : PMF α) (f : α → β) (q : β → PMF γ) : (p.map f).bind q = p.bind (q ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : PMF α) (f : α → β) (q : β → PMF γ)
Goal: (PMF.map f p).bind q = p.bind (q ∘ f)
-- PMF.map_const
@[simp]
theorem map_const : p.map (Function.const α b) = pure b 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (b : β)
Goal: PMF.map (Function.const α b) p = PMF.pure b
-- PMF.toOuterMeasure_map_apply
@[simp]
theorem toOuterMeasure_map_apply : (p.map f).toOuterMeasure s = p.toOuterMeasure (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (p : PMF α) (s : Set β)
Goal: (PMF.map f p).toOuterMeasure s = p.toOuterMeasure (f ⁻¹' s)
-- PMF.toMeasure_map_apply
@[simp]
theorem toMeasure_map_apply [MeasurableSpace α] [MeasurableSpace β] (hf : Measurable f)
    (hs : MeasurableSet s) : (p.map f).toMeasure s = p.toMeasure (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (p : PMF α) (s : Set β) [MeasurableSpace α] [MeasurableSpace β] (hf : Measurable f) (hs : MeasurableSet s)
Goal: (PMF.map f p).toMeasure s = p.toMeasure (f ⁻¹' s)
-- PMF.monad_seq_eq_seq
theorem monad_seq_eq_seq {α β : Type u} (q : PMF (α → β)) (p : PMF α) : q <*> p = q.seq p 
Arguments: {α β : Type u} (q : PMF (α → β)) (p : PMF α)
Goal: (Seq.seq q fun x => p) = q.seq p
-- PMF.seq_apply
@[simp]
theorem seq_apply : (seq q p) b = ∑' (f : α → β) (a : α), if b = f a then q f * p a else 0 
Arguments: {α : Type u_1} {β : Type u_2} (q : PMF (α → β)) (p : PMF α) (b : β)
Goal: (q.seq p) b = ∑' (f : α → β) (a : α), if b = f a then q f * p a else 0
-- PMF.support_seq
@[simp]
theorem support_seq : (seq q p).support = ⋃ f ∈ q.support, f '' p.support 
Arguments: {α : Type u_1} {β : Type u_2} (q : PMF (α → β)) (p : PMF α)
Goal: (q.seq p).support = ⋃ f ∈ q.support, f '' p.support
-- PMF.mem_support_seq_iff
theorem mem_support_seq_iff : b ∈ (seq q p).support ↔ ∃ f ∈ q.support, b ∈ f '' p.support 
Arguments: {α : Type u_1} {β : Type u_2} (q : PMF (α → β)) (p : PMF α) (b : β)
Goal: b ∈ (q.seq p).support ↔ ∃ f ∈ q.support, b ∈ f '' p.support
-- PMF.ofFinset_apply
@[simp]
theorem ofFinset_apply (a : α) : ofFinset f s h h' a = f a 
Arguments: {α : Type u_1} {f : α → ENNReal} {s : Finset α} (h : ∑ a ∈ s, f a = 1) (h' : ∀ a ∉ s, f a = 0) (a : α)
Goal: (PMF.ofFinset f s h h') a = f a
-- PMF.support_ofFinset
@[simp]
theorem support_ofFinset : (ofFinset f s h h').support = ↑s ∩ Function.support f 
Arguments: {α : Type u_1} {f : α → ENNReal} {s : Finset α} (h : ∑ a ∈ s, f a = 1) (h' : ∀ a ∉ s, f a = 0)
Goal: (PMF.ofFinset f s h h').support = ↑s ∩ Function.support f
-- PMF.mem_support_ofFinset_iff
theorem mem_support_ofFinset_iff (a : α) : a ∈ (ofFinset f s h h').support ↔ a ∈ s ∧ f a ≠ 0 
Arguments: {α : Type u_1} {f : α → ENNReal} {s : Finset α} (h : ∑ a ∈ s, f a = 1) (h' : ∀ a ∉ s, f a = 0) (a : α)
Goal: a ∈ (PMF.ofFinset f s h h').support ↔ a ∈ s ∧ f a ≠ 0
-- PMF.ofFinset_apply_of_not_mem
theorem ofFinset_apply_of_not_mem {a : α} (ha : a ∉ s) : ofFinset f s h h' a = 0 
Arguments: {α : Type u_1} {f : α → ENNReal} {s : Finset α} (h : ∑ a ∈ s, f a = 1) (h' : ∀ a ∉ s, f a = 0) {a : α} (ha : a ∉ s)
Goal: (PMF.ofFinset f s h h') a = 0
-- PMF.toOuterMeasure_ofFinset_apply
@[simp]
theorem toOuterMeasure_ofFinset_apply :
    (ofFinset f s h h').toOuterMeasure t = ∑' x, t.indicator f x 
Arguments: {α : Type u_1} {f : α → ENNReal} {s : Finset α} (h : ∑ a ∈ s, f a = 1) (h' : ∀ a ∉ s, f a = 0) (t : Set α)
Goal: (PMF.ofFinset f s h h').toOuterMeasure t = ∑' (x : α), t.indicator f x
-- PMF.toMeasure_ofFinset_apply
@[simp]
theorem toMeasure_ofFinset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
    (ofFinset f s h h').toMeasure t = ∑' x, t.indicator f x 
Arguments: {α : Type u_1} {f : α → ENNReal} {s : Finset α} (h : ∑ a ∈ s, f a = 1) (h' : ∀ a ∉ s, f a = 0) (t : Set α) [MeasurableSpace α] (ht : MeasurableSet t)
Goal: (PMF.ofFinset f s h h').toMeasure t = ∑' (x : α), t.indicator f x
-- PMF.ofFintype_apply
@[simp]
theorem ofFintype_apply (a : α) : ofFintype f h a = f a 
Arguments: {α : Type u_1} [Fintype α] {f : α → ENNReal} (h : ∑ a : α, f a = 1) (a : α)
Goal: (PMF.ofFintype f h) a = f a
-- PMF.support_ofFintype
@[simp]
theorem support_ofFintype : (ofFintype f h).support = Function.support f 
Arguments: {α : Type u_1} [Fintype α] {f : α → ENNReal} (h : ∑ a : α, f a = 1)
Goal: (PMF.ofFintype f h).support = Function.support f
-- PMF.mem_support_ofFintype_iff
theorem mem_support_ofFintype_iff (a : α) : a ∈ (ofFintype f h).support ↔ f a ≠ 0 
Arguments: {α : Type u_1} [Fintype α] {f : α → ENNReal} (h : ∑ a : α, f a = 1) (a : α)
Goal: a ∈ (PMF.ofFintype f h).support ↔ f a ≠ 0
-- PMF.toOuterMeasure_ofFintype_apply
@[simp high]
theorem toOuterMeasure_ofFintype_apply : (ofFintype f h).toOuterMeasure s = ∑' x, s.indicator f x 
Arguments: {α : Type u_1} [Fintype α] {f : α → ENNReal} (h : ∑ a : α, f a = 1) (s : Set α)
Goal: (PMF.ofFintype f h).toOuterMeasure s = ∑' (x : α), s.indicator f x
-- PMF.toMeasure_ofFintype_apply
@[simp]
theorem toMeasure_ofFintype_apply [MeasurableSpace α] (hs : MeasurableSet s) :
    (ofFintype f h).toMeasure s = ∑' x, s.indicator f x 
Arguments: {α : Type u_1} [Fintype α] {f : α → ENNReal} (h : ∑ a : α, f a = 1) (s : Set α) [MeasurableSpace α] (hs : MeasurableSet s)
Goal: (PMF.ofFintype f h).toMeasure s = ∑' (x : α), s.indicator f x
-- PMF.normalize_apply
@[simp]
theorem normalize_apply (a : α) : (normalize f hf0 hf) a = f a * (∑' x, f x)⁻¹ 
Arguments: {α : Type u_1} {f : α → ENNReal} (hf0 : tsum f ≠ 0) (hf : tsum f ≠ ⊤) (a : α)
Goal: (PMF.normalize f hf0 hf) a = f a * (∑' (x : α), f x)⁻¹
-- PMF.support_normalize
@[simp]
theorem support_normalize : (normalize f hf0 hf).support = Function.support f 
Arguments: {α : Type u_1} {f : α → ENNReal} (hf0 : tsum f ≠ 0) (hf : tsum f ≠ ⊤)
Goal: (PMF.normalize f hf0 hf).support = Function.support f
-- PMF.mem_support_normalize_iff
theorem mem_support_normalize_iff (a : α) : a ∈ (normalize f hf0 hf).support ↔ f a ≠ 0 
Arguments: {α : Type u_1} {f : α → ENNReal} (hf0 : tsum f ≠ 0) (hf : tsum f ≠ ⊤) (a : α)
Goal: a ∈ (PMF.normalize f hf0 hf).support ↔ f a ≠ 0
-- PMF.filter_apply
@[simp]
theorem filter_apply (a : α) :
    (p.filter s h) a = s.indicator p a * (∑' a', (s.indicator p) a')⁻¹ 
Arguments: {α : Type u_1} {p : PMF α} {s : Set α} (h : ∃ a ∈ s, a ∈ p.support) (a : α)
Goal: (p.filter s h) a = s.indicator (⇑p) a * (∑' (a' : α), s.indicator (⇑p) a')⁻¹
-- PMF.filter_apply_eq_zero_of_not_mem
theorem filter_apply_eq_zero_of_not_mem {a : α} (ha : a ∉ s) : (p.filter s h) a = 0 
Arguments: {α : Type u_1} {p : PMF α} {s : Set α} (h : ∃ a ∈ s, a ∈ p.support) {a : α} (ha : a ∉ s)
Goal: (p.filter s h) a = 0
-- PMF.mem_support_filter_iff
theorem mem_support_filter_iff {a : α} : a ∈ (p.filter s h).support ↔ a ∈ s ∧ a ∈ p.support 
Arguments: {α : Type u_1} {p : PMF α} {s : Set α} (h : ∃ a ∈ s, a ∈ p.support) {a : α}
Goal: a ∈ (p.filter s h).support ↔ a ∈ s ∧ a ∈ p.support
-- PMF.support_filter
@[simp]
theorem support_filter : (p.filter s h).support = s ∩ p.support 
Arguments: {α : Type u_1} {p : PMF α} {s : Set α} (h : ∃ a ∈ s, a ∈ p.support)
Goal: (p.filter s h).support = s ∩ p.support
-- PMF.filter_apply_eq_zero_iff
theorem filter_apply_eq_zero_iff (a : α) : (p.filter s h) a = 0 ↔ a ∉ s ∨ a ∉ p.support 
Arguments: {α : Type u_1} {p : PMF α} {s : Set α} (h : ∃ a ∈ s, a ∈ p.support) (a : α)
Goal: (p.filter s h) a = 0 ↔ a ∉ s ∨ a ∉ p.support
-- PMF.filter_apply_ne_zero_iff
theorem filter_apply_ne_zero_iff (a : α) : (p.filter s h) a ≠ 0 ↔ a ∈ s ∧ a ∈ p.support 
Arguments: {α : Type u_1} {p : PMF α} {s : Set α} (h : ∃ a ∈ s, a ∈ p.support) (a : α)
Goal: (p.filter s h) a ≠ 0 ↔ a ∈ s ∧ a ∈ p.support
-- PMF.bernoulli_apply
@[simp]
theorem bernoulli_apply : bernoulli p h b = cond b p (1 - p) 
Arguments: {p : ENNReal} (h : p ≤ 1) (b : Bool)
Goal: (PMF.bernoulli p h) b = bif b then p else 1 - p
-- PMF.support_bernoulli
@[simp]
theorem support_bernoulli : (bernoulli p h).support = { b | cond b (p ≠ 0) (p ≠ 1) } 
Arguments: {p : ENNReal} (h : p ≤ 1)
Goal: (PMF.bernoulli p h).support = {b | bif b then p ≠ 0 else p ≠ 1}
-- PMF.mem_support_bernoulli_iff
theorem mem_support_bernoulli_iff : b ∈ (bernoulli p h).support ↔ cond b (p ≠ 0) (p ≠ 1) 
Arguments: {p : ENNReal} (h : p ≤ 1) (b : Bool)
Goal: b ∈ (PMF.bernoulli p h).support ↔ bif b then p ≠ 0 else p ≠ 1
-- MeasureTheory.martingale_const
theorem martingale_const (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ] (x : E) :
    Martingale (fun _ _ => x) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (ℱ : MeasureTheory.Filtration ι m0) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (x : E)
Goal: MeasureTheory.Martingale (fun x_1 x_2 => x) ℱ μ
-- MeasureTheory.martingale_const_fun
theorem martingale_const_fun [OrderBot ι] (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ]
    {f : Ω → E} (hf : StronglyMeasurable[ℱ ⊥] f) (hfint : Integrable f μ) :
    Martingale (fun _ => f) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [OrderBot ι] (ℱ : MeasureTheory.Filtration ι m0) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {f : Ω → E} (hf : MeasureTheory.StronglyMeasurable f) (hfint : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Martingale (fun x => f) ℱ μ
-- MeasureTheory.martingale_zero
theorem martingale_zero (ℱ : Filtration ι m0) (μ : Measure Ω) : Martingale (0 : ι → Ω → E) ℱ μ 
Arguments: {Ω : Type u_1} (E : Type u_2) {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (ℱ : MeasureTheory.Filtration ι m0) (μ : MeasureTheory.Measure Ω)
Goal: MeasureTheory.Martingale 0 ℱ μ
-- MeasureTheory.Martingale.adapted
protected theorem adapted (hf : Martingale f ℱ μ) : Adapted ℱ f 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ)
Goal: MeasureTheory.Adapted ℱ f
-- MeasureTheory.Martingale.stronglyMeasurable
protected theorem stronglyMeasurable (hf : Martingale f ℱ μ) (i : ι) :
    StronglyMeasurable[ℱ i] (f i) 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ) (i : ι)
Goal: MeasureTheory.StronglyMeasurable (f i)
-- MeasureTheory.Martingale.condexp_ae_eq
theorem condexp_ae_eq (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j) : μ[f j|ℱ i] =ᵐ[μ] f i 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
Goal: μ[f j|↑ℱ i] =ᵐ[μ] f i
-- MeasureTheory.Martingale.integrable
protected theorem integrable (hf : Martingale f ℱ μ) (i : ι) : Integrable (f i) μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ) (i : ι)
Goal: MeasureTheory.Integrable (f i) μ