ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContMDiff.contMDiff_tangentMap
theorem ContMDiff.contMDiff_tangentMap (hf : ContMDiff I I' n f) (hmn : m + 1 ≤ n) :
    ContMDiff I.tangent I'.tangent m (tangentMap I I' f) 
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'} {m n : ℕ∞} (hf : ContMDiff I I' n f) (hmn : m + 1 ≤ n)
Goal: ContMDiff I.tangent I'.tangent m (tangentMap I I' f)
-- ContMDiff.continuous_tangentMap
theorem ContMDiff.continuous_tangentMap (hf : ContMDiff I I' n f) (hmn : 1 ≤ n) :
    Continuous (tangentMap I I' f) 
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'} {n : ℕ∞} (hf : ContMDiff I I' n f) (hmn : 1 ≤ n)
Goal: Continuous (tangentMap I I' f)
-- TangentBundle.tangentMap_tangentBundle_pure
theorem tangentMap_tangentBundle_pure (p : TangentBundle I M) :
    tangentMap I I.tangent (zeroSection E (TangentSpace I)) p = ⟨⟨p.proj, 0⟩, ⟨p.2, 0⟩⟩ 
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] (p : TangentBundle I M)
Goal: tangentMap I I.tangent (Bundle.zeroSection E (TangentSpace I)) p =
  { proj := { proj := p.proj, snd := 0 }, snd := (p.snd, 0) }
-- ContMDiffMap.mdifferentiable'
protected theorem mdifferentiable' (f : C^n⟮I, M; I', M'⟯) (hn : 1 ≤ n) : MDifferentiable I I' f 
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] {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'] {n : ℕ∞} (f : C^n⟮I, M; I', M'⟯) (hn : 1 ≤ n)
Goal: MDifferentiable I I' ⇑f
-- ContMDiffMap.mdifferentiable
protected theorem mdifferentiable (f : C^∞⟮I, M; I', M'⟯) : MDifferentiable I I' f 
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] {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'] (f : C^⊤⟮I, M; I', M'⟯)
Goal: MDifferentiable I I' ⇑f
-- ContMDiffMap.mdifferentiableAt
protected theorem mdifferentiableAt (f : C^∞⟮I, M; I', M'⟯) {x} : MDifferentiableAt I I' f x 
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] {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'] (f : C^⊤⟮I, M; I', M'⟯) {x : M}
Goal: MDifferentiableAt I I' (⇑f) x
-- ProbabilityTheory.Kernel.bind_add
@[simp]
theorem bind_add (μ ν : Measure α) (κ : Kernel α β) : (μ + ν).bind κ = μ.bind κ + ν.bind κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ ν : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β)
Goal: (μ + ν).bind ⇑κ = μ.bind ⇑κ + ν.bind ⇑κ
-- ProbabilityTheory.Kernel.bind_smul
@[simp]
theorem bind_smul (κ : Kernel α β) (μ : Measure α) (r : ℝ≥0∞) : (r • μ).bind κ = r • μ.bind κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (μ : MeasureTheory.Measure α) (r : ℝ≥0∞)
Goal: (r • μ).bind ⇑κ = r • μ.bind ⇑κ
-- ProbabilityTheory.Kernel.const_bind_eq_comp_const
theorem const_bind_eq_comp_const (κ : Kernel α β) (μ : Measure α) :
    const α (μ.bind κ) = κ ∘ₖ const α μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (μ : MeasureTheory.Measure α)
Goal: ProbabilityTheory.Kernel.const α (μ.bind ⇑κ) = κ ∘ₖ ProbabilityTheory.Kernel.const α μ
-- ProbabilityTheory.Kernel.comp_const_apply_eq_bind
theorem comp_const_apply_eq_bind (κ : Kernel α β) (μ : Measure α) (a : α) :
    (κ ∘ₖ const α μ) a = μ.bind κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (μ : MeasureTheory.Measure α) (a : α)
Goal: (κ ∘ₖ ProbabilityTheory.Kernel.const α μ) a = μ.bind ⇑κ
-- ProbabilityTheory.Kernel.Invariant.def
theorem Invariant.def (hκ : Invariant κ μ) : μ.bind κ = μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {κ : ProbabilityTheory.Kernel α α} {μ : MeasureTheory.Measure α} (hκ : κ.Invariant μ)
Goal: μ.bind ⇑κ = μ
-- ProbabilityTheory.Kernel.Invariant.comp_const
theorem Invariant.comp_const (hκ : Invariant κ μ) : κ ∘ₖ const α μ = const α μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {κ : ProbabilityTheory.Kernel α α} {μ : MeasureTheory.Measure α} (hκ : κ.Invariant μ)
Goal: κ ∘ₖ ProbabilityTheory.Kernel.const α μ = ProbabilityTheory.Kernel.const α μ
-- ProbabilityTheory.Kernel.Invariant.comp
theorem Invariant.comp [IsSFiniteKernel κ] (hκ : Invariant κ μ) (hη : Invariant η μ) :
    Invariant (κ ∘ₖ η) μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {κ η : ProbabilityTheory.Kernel α α} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsSFiniteKernel κ] (hκ : κ.Invariant μ) (hη : η.Invariant μ)
Goal: (κ ∘ₖ η).Invariant μ
-- ProbabilityTheory.Kernel.measurable'
structure Kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] where
    toFun : α → Measure β
    measurable' : Measurable toFun
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (self : ProbabilityTheory.Kernel α β)
Goal: Measurable self.toFun
-- ProbabilityTheory.Kernel.measurable
lemma measurable (κ : Kernel α β) : Measurable κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β)
Goal: Measurable ⇑κ
-- ProbabilityTheory.Kernel.coe_zero
@[simp, norm_cast] lemma coe_zero : ⇑(0 : Kernel α β) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
Goal: ⇑0 = 0
-- ProbabilityTheory.Kernel.coe_add
@[simp, norm_cast] lemma coe_add (κ η : Kernel α β) : ⇑(κ + η) = κ + η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ η : ProbabilityTheory.Kernel α β)
Goal: ⇑(κ + η) = ⇑κ + ⇑η
-- ProbabilityTheory.Kernel.coe_nsmul
@[simp, norm_cast] lemma coe_nsmul (n : ℕ) (κ : Kernel α β) : ⇑(n • κ) = n • κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (n : ℕ) (κ : ProbabilityTheory.Kernel α β)
Goal: ⇑(n • κ) = n • ⇑κ
-- ProbabilityTheory.Kernel.zero_apply
@[simp] lemma zero_apply (a : α) : (0 : Kernel α β) a = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (a : α)
Goal: 0 a = 0
-- ProbabilityTheory.Kernel.add_apply
@[simp] lemma add_apply (κ η : Kernel α β) (a : α) : (κ + η) a = κ a + η a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ η : ProbabilityTheory.Kernel α β) (a : α)
Goal: (κ + η) a = κ a + η a
-- ProbabilityTheory.Kernel.nsmul_apply
@[simp] lemma nsmul_apply (n : ℕ) (κ : Kernel α β) (a : α) : (n • κ) a = n • κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (n : ℕ) (κ : ProbabilityTheory.Kernel α β) (a : α)
Goal: (n • κ) a = n • κ a
-- ProbabilityTheory.Kernel.coe_finset_sum
@[simp]
theorem coe_finset_sum (I : Finset ι) (κ : ι → Kernel α β) : ⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (I : Finset ι) (κ : ι → ProbabilityTheory.Kernel α β)
Goal: ⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i)
-- ProbabilityTheory.Kernel.finset_sum_apply
theorem finset_sum_apply (I : Finset ι) (κ : ι → Kernel α β) (a : α) :
    (∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (I : Finset ι) (κ : ι → ProbabilityTheory.Kernel α β) (a : α)
Goal: (∑ i ∈ I, κ i) a = ∑ i ∈ I, (κ i) a
-- ProbabilityTheory.Kernel.finset_sum_apply'
theorem finset_sum_apply' (I : Finset ι) (κ : ι → Kernel α β) (a : α) (s : Set β) :
    (∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (I : Finset ι) (κ : ι → ProbabilityTheory.Kernel α β) (a : α) (s : Set β)
Goal: ((∑ i ∈ I, κ i) a) s = ∑ i ∈ I, ((κ i) a) s
-- ProbabilityTheory.IsMarkovKernel.isProbabilityMeasure
class IsMarkovKernel (κ : Kernel α β) : Prop where
  isProbabilityMeasure : ∀ a, IsProbabilityMeasure (κ a)
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [self : ProbabilityTheory.IsMarkovKernel κ] (a : α)
Goal: MeasureTheory.IsProbabilityMeasure (κ a)
-- ProbabilityTheory.IsFiniteKernel.exists_univ_le
class IsFiniteKernel (κ : Kernel α β) : Prop where
  exists_univ_le : ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ a, κ a Set.univ ≤ C
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [self : ProbabilityTheory.IsFiniteKernel κ]
Goal: ∃ C < ⊤, ∀ (a : α), (κ a) Set.univ ≤ C
-- ProbabilityTheory.IsFiniteKernel.bound_lt_top
theorem IsFiniteKernel.bound_lt_top (κ : Kernel α β) [h : IsFiniteKernel κ] :
    IsFiniteKernel.bound κ < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [h : ProbabilityTheory.IsFiniteKernel κ]
Goal: ProbabilityTheory.IsFiniteKernel.bound κ < ⊤
-- ProbabilityTheory.IsFiniteKernel.bound_ne_top
theorem IsFiniteKernel.bound_ne_top (κ : Kernel α β) [IsFiniteKernel κ] :
    IsFiniteKernel.bound κ ≠ ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel κ]
Goal: ProbabilityTheory.IsFiniteKernel.bound κ ≠ ⊤
-- ProbabilityTheory.Kernel.measure_le_bound
theorem Kernel.measure_le_bound (κ : Kernel α β) [h : IsFiniteKernel κ] (a : α) (s : Set β) :
    κ a s ≤ IsFiniteKernel.bound κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [h : ProbabilityTheory.IsFiniteKernel κ] (a : α) (s : Set β)
Goal: (κ a) s ≤ ProbabilityTheory.IsFiniteKernel.bound κ
-- ProbabilityTheory.isFiniteKernel_of_le
lemma isFiniteKernel_of_le {κ ν : Kernel α β} [hν : IsFiniteKernel ν] (hκν : κ ≤ ν) :
    IsFiniteKernel κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ ν : ProbabilityTheory.Kernel α β} [hν : ProbabilityTheory.IsFiniteKernel ν] (hκν : κ ≤ ν)
Goal: ProbabilityTheory.IsFiniteKernel κ
-- ProbabilityTheory.Kernel.ext
@[ext]
theorem ext {η : Kernel α β} (h : ∀ a, κ a = η a) : κ = η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : ProbabilityTheory.Kernel α β} (h : ∀ (a : α), κ a = η a)
Goal: κ = η
-- ProbabilityTheory.Kernel.ext_iff
theorem ext_iff {η : Kernel α β} : κ = η ↔ ∀ a, κ a = η a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : ProbabilityTheory.Kernel α β}
Goal: κ = η ↔ ∀ (a : α), κ a = η a
-- ProbabilityTheory.Kernel.ext_iff'
theorem ext_iff' {η : Kernel α β} :
    κ = η ↔ ∀ a s, MeasurableSet s → κ a s = η a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : ProbabilityTheory.Kernel α β}
Goal: κ = η ↔ ∀ (a : α) (s : Set β), MeasurableSet s → (κ a) s = (η a) s
-- ProbabilityTheory.Kernel.ext_fun
theorem ext_fun {η : Kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
    κ = η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : ProbabilityTheory.Kernel α β} (h : ∀ (a : α) (f : β → ℝ≥0∞), Measurable f → ∫⁻ (b : β), f b ∂κ a = ∫⁻ (b : β), f b ∂η a)
Goal: κ = η
-- ProbabilityTheory.Kernel.ext_fun_iff
theorem ext_fun_iff {η : Kernel α β} :
    κ = η ↔ ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : ProbabilityTheory.Kernel α β}
Goal: κ = η ↔ ∀ (a : α) (f : β → ℝ≥0∞), Measurable f → ∫⁻ (b : β), f b ∂κ a = ∫⁻ (b : β), f b ∂η a
-- ProbabilityTheory.Kernel.measurable_coe
protected theorem measurable_coe (κ : Kernel α β) {s : Set β} (hs : MeasurableSet s) :
    Measurable fun a => κ a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun a => (κ a) s
-- ProbabilityTheory.Kernel.apply_congr_of_mem_measurableAtom
lemma apply_congr_of_mem_measurableAtom (κ : Kernel α β) {y' y : α} (hy' : y' ∈ measurableAtom y) :
  κ y' = κ y 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) {y' y : α} (hy' : y' ∈ measurableAtom y)
Goal: κ y' = κ y
-- ProbabilityTheory.Kernel.IsFiniteKernel.integrable
lemma IsFiniteKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
    (κ : Kernel α β) [IsFiniteKernel κ] {s : Set β} (hs : MeasurableSet s) :
    Integrable (fun x => (κ x s).toReal) μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel κ] {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (fun x => ((κ x) s).toReal) μ
-- ProbabilityTheory.Kernel.IsMarkovKernel.integrable
lemma IsMarkovKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
    (κ : Kernel α β) [IsMarkovKernel κ] {s : Set β} (hs : MeasurableSet s) :
    Integrable (fun x => (κ x s).toReal) μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ] {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (fun x => ((κ x) s).toReal) μ
-- ProbabilityTheory.Kernel.sum_apply
theorem sum_apply [Countable ι] (κ : ι → Kernel α β) (a : α) :
    Kernel.sum κ a = Measure.sum fun n => κ n a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → ProbabilityTheory.Kernel α β) (a : α)
Goal: (ProbabilityTheory.Kernel.sum κ) a = MeasureTheory.Measure.sum fun n => (κ n) a
-- ProbabilityTheory.Kernel.sum_apply'
theorem sum_apply' [Countable ι] (κ : ι → Kernel α β) (a : α) {s : Set β} (hs : MeasurableSet s) :
    Kernel.sum κ a s = ∑' n, κ n a s 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → ProbabilityTheory.Kernel α β) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ((ProbabilityTheory.Kernel.sum κ) a) s = ∑' (n : ι), ((κ n) a) s
-- ProbabilityTheory.Kernel.sum_zero
@[simp]
theorem sum_zero [Countable ι] : (Kernel.sum fun _ : ι => (0 : Kernel α β)) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι]
Goal: (ProbabilityTheory.Kernel.sum fun x => 0) = 0
-- ProbabilityTheory.Kernel.sum_comm
theorem sum_comm [Countable ι] (κ : ι → ι → Kernel α β) :
    (Kernel.sum fun n => Kernel.sum (κ n)) = Kernel.sum fun m => Kernel.sum fun n => κ n m 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → ι → ProbabilityTheory.Kernel α β)
Goal: (ProbabilityTheory.Kernel.sum fun n => ProbabilityTheory.Kernel.sum (κ n)) =
  ProbabilityTheory.Kernel.sum fun m => ProbabilityTheory.Kernel.sum fun n => κ n m
-- ProbabilityTheory.Kernel.sum_fintype
@[simp]
theorem sum_fintype [Fintype ι] (κ : ι → Kernel α β) : Kernel.sum κ = ∑ i, κ i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Fintype ι] (κ : ι → ProbabilityTheory.Kernel α β)
Goal: ProbabilityTheory.Kernel.sum κ = ∑ i : ι, κ i
-- ProbabilityTheory.Kernel.sum_add
theorem sum_add [Countable ι] (κ η : ι → Kernel α β) :
    (Kernel.sum fun n => κ n + η n) = Kernel.sum κ + Kernel.sum η 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ η : ι → ProbabilityTheory.Kernel α β)
Goal: (ProbabilityTheory.Kernel.sum fun n => κ n + η n) = ProbabilityTheory.Kernel.sum κ + ProbabilityTheory.Kernel.sum η
-- ProbabilityTheory.IsSFiniteKernel.tsum_finite
class _root_.ProbabilityTheory.IsSFiniteKernel (κ : Kernel α β) : Prop where
  tsum_finite : ∃ κs : ℕ → Kernel α β, (∀ n, IsFiniteKernel (κs n)) ∧ κ = Kernel.sum κs
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [self : ProbabilityTheory.IsSFiniteKernel κ]
Goal: ∃ κs, (∀ (n : ℕ), ProbabilityTheory.IsFiniteKernel (κs n)) ∧ κ = ProbabilityTheory.Kernel.sum κs
-- ProbabilityTheory.Kernel.kernel_sum_seq
theorem kernel_sum_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] : Kernel.sum (seq κ) = κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [h : ProbabilityTheory.IsSFiniteKernel κ]
Goal: ProbabilityTheory.Kernel.sum κ.seq = κ
-- ProbabilityTheory.Kernel.measure_sum_seq
theorem measure_sum_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] (a : α) :
    (Measure.sum fun n => seq κ n a) = κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [h : ProbabilityTheory.IsSFiniteKernel κ] (a : α)
Goal: (MeasureTheory.Measure.sum fun n => (κ.seq n) a) = κ a
-- ProbabilityTheory.Kernel.IsSFiniteKernel.finset_sum
theorem IsSFiniteKernel.finset_sum {κs : ι → Kernel α β} (I : Finset ι)
    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i ∈ I, κs i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κs : ι → ProbabilityTheory.Kernel α β} (I : Finset ι) (h : ∀ i ∈ I, ProbabilityTheory.IsSFiniteKernel (κs i))
Goal: ProbabilityTheory.IsSFiniteKernel (∑ i ∈ I, κs i)
-- ProbabilityTheory.Kernel.isSFiniteKernel_sum_of_denumerable
theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → Kernel α β}
    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (Kernel.sum κs) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Denumerable ι] {κs : ι → ProbabilityTheory.Kernel α β} (hκs : ∀ (n : ι), ProbabilityTheory.IsSFiniteKernel (κs n))
Goal: ProbabilityTheory.IsSFiniteKernel (ProbabilityTheory.Kernel.sum κs)
-- ProbabilityTheory.Kernel.isSFiniteKernel_sum
theorem isSFiniteKernel_sum [Countable ι] {κs : ι → Kernel α β}
    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (Kernel.sum κs) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] {κs : ι → ProbabilityTheory.Kernel α β} (hκs : ∀ (n : ι), ProbabilityTheory.IsSFiniteKernel (κs n))
Goal: ProbabilityTheory.IsSFiniteKernel (ProbabilityTheory.Kernel.sum κs)
-- ProbabilityTheory.Kernel.deterministic_apply
theorem deterministic_apply {f : α → β} (hf : Measurable f) (a : α) :
    deterministic f hf a = Measure.dirac (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (hf : Measurable f) (a : α)
Goal: (ProbabilityTheory.Kernel.deterministic f hf) a = MeasureTheory.Measure.dirac (f a)
-- ProbabilityTheory.Kernel.deterministic_apply'
theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : Set β}
    (hs : MeasurableSet s) : deterministic f hf a s = s.indicator (fun _ => 1) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (hf : Measurable f) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ((ProbabilityTheory.Kernel.deterministic f hf) a) s = s.indicator (fun x => 1) (f a)
-- ProbabilityTheory.Kernel.lintegral_deterministic'
theorem lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
    (hf : Measurable f) : ∫⁻ x, f x ∂deterministic g hg a = f (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g) (hf : Measurable f)
Goal: ∫⁻ (x : β), f x ∂(ProbabilityTheory.Kernel.deterministic g hg) a = f (g a)
-- ProbabilityTheory.Kernel.lintegral_deterministic
@[simp]
theorem lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
    [MeasurableSingletonClass β] : ∫⁻ x, f x ∂deterministic g hg a = f (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g) [MeasurableSingletonClass β]
Goal: ∫⁻ (x : β), f x ∂(ProbabilityTheory.Kernel.deterministic g hg) a = f (g a)
-- ProbabilityTheory.Kernel.setLIntegral_deterministic'
theorem setLIntegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
    (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
    ∫⁻ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g) (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)]
Goal: ∫⁻ (x : β) in s, f x ∂(ProbabilityTheory.Kernel.deterministic g hg) a = if g a ∈ s then f (g a) else 0
-- ProbabilityTheory.Kernel.setLIntegral_deterministic
@[simp]
theorem setLIntegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
    [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
    ∫⁻ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g) [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)]
Goal: ∫⁻ (x : β) in s, f x ∂(ProbabilityTheory.Kernel.deterministic g hg) a = if g a ∈ s then f (g a) else 0
-- ProbabilityTheory.Kernel.integral_deterministic'
theorem integral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
    (hf : StronglyMeasurable f) : ∫ x, f x ∂deterministic g hg a = f (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g) (hf : MeasureTheory.StronglyMeasurable f)
Goal: ∫ (x : β), f x ∂(ProbabilityTheory.Kernel.deterministic g hg) a = f (g a)