ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.Kernel.lintegral_compProd₀
theorem lintegral_compProd₀ (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : AEMeasurable f ((κ ⊗ₖ η) a)) :
    ∫⁻ z, f z ∂(κ ⊗ₖ η) a = ∫⁻ x, ∫⁻ y, f (x, y) ∂η (a, x) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : AEMeasurable f ((κ.compProd η) a))
Goal: ∫⁻ (z : β × γ), f z ∂(κ.compProd η) a = ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂η (a, x) ∂κ a
-- ProbabilityTheory.Kernel.setLIntegral_compProd
theorem setLIntegral_compProd (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f) {s : Set β} {t : Set γ}
    (hs : MeasurableSet s) (ht : MeasurableSet t) :
    ∫⁻ z in s ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫⁻ x in s, ∫⁻ y in t, f (x, y) ∂η (a, x) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f) {s : Set β} {t : Set γ} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ∫⁻ (z : β × γ) in s ×ˢ t, f z ∂(κ.compProd η) a = ∫⁻ (x : β) in s, ∫⁻ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a
-- ProbabilityTheory.Kernel.setLIntegral_compProd_univ_right
theorem setLIntegral_compProd_univ_right (κ : Kernel α β) [IsSFiniteKernel κ]
    (η : Kernel (α × β) γ) [IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f)
    {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ z in s ×ˢ Set.univ, f z ∂(κ ⊗ₖ η) a = ∫⁻ x in s, ∫⁻ y, f (x, y) ∂η (a, x) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (z : β × γ) in s ×ˢ Set.univ, f z ∂(κ.compProd η) a = ∫⁻ (x : β) in s, ∫⁻ (y : γ), f (x, y) ∂η (a, x) ∂κ a
-- ProbabilityTheory.Kernel.setLIntegral_compProd_univ_left
theorem setLIntegral_compProd_univ_left (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f) {t : Set γ}
    (ht : MeasurableSet t) :
    ∫⁻ z in Set.univ ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫⁻ x, ∫⁻ y in t, f (x, y) ∂η (a, x) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f) {t : Set γ} (ht : MeasurableSet t)
Goal: ∫⁻ (z : β × γ) in Set.univ ×ˢ t, f z ∂(κ.compProd η) a = ∫⁻ (x : β), ∫⁻ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a
-- ProbabilityTheory.Kernel.compProd_eq_tsum_compProd
theorem compProd_eq_tsum_compProd (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
    (κ ⊗ₖ η) a s = ∑' (n : ℕ) (m : ℕ), (seq κ n ⊗ₖ seq η m) a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (hs : MeasurableSet s)
Goal: ((κ.compProd η) a) s = ∑' (n : ℕ) (m : ℕ), (((κ.seq n).compProd (η.seq m)) a) s
-- ProbabilityTheory.Kernel.compProd_eq_sum_compProd
theorem compProd_eq_sum_compProd (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] : κ ⊗ₖ η = Kernel.sum fun n => Kernel.sum fun m => seq κ n ⊗ₖ seq η m 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η]
Goal: κ.compProd η = ProbabilityTheory.Kernel.sum fun n => ProbabilityTheory.Kernel.sum fun m => (κ.seq n).compProd (η.seq m)
-- ProbabilityTheory.Kernel.compProd_eq_sum_compProd_left
theorem compProd_eq_sum_compProd_left (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ) :
    κ ⊗ₖ η = Kernel.sum fun n => seq κ n ⊗ₖ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ)
Goal: κ.compProd η = ProbabilityTheory.Kernel.sum fun n => (κ.seq n).compProd η
-- ProbabilityTheory.Kernel.compProd_eq_sum_compProd_right
theorem compProd_eq_sum_compProd_right (κ : Kernel α β) (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] : κ ⊗ₖ η = Kernel.sum fun n => κ ⊗ₖ seq η n 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η]
Goal: κ.compProd η = ProbabilityTheory.Kernel.sum fun n => κ.compProd (η.seq n)
-- ProbabilityTheory.Kernel.compProd_apply_univ_le
theorem compProd_apply_univ_le (κ : Kernel α β) (η : Kernel (α × β) γ) [IsFiniteKernel η] (a : α) :
    (κ ⊗ₖ η) a Set.univ ≤ κ a Set.univ * IsFiniteKernel.bound η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: ((κ.compProd η) a) Set.univ ≤ (κ a) Set.univ * ProbabilityTheory.IsFiniteKernel.bound η
-- ProbabilityTheory.Kernel.compProd_add_left
lemma compProd_add_left (μ κ : Kernel α β) (η : Kernel (α × β) γ)
    [IsSFiniteKernel μ] [IsSFiniteKernel κ] [IsSFiniteKernel η] :
    (μ + κ) ⊗ₖ η = μ ⊗ₖ η + κ ⊗ₖ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (μ κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel μ] [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η]
Goal: (μ + κ).compProd η = μ.compProd η + κ.compProd η
-- ProbabilityTheory.Kernel.compProd_add_right
lemma compProd_add_right (μ : Kernel α β) (κ η : Kernel (α × β) γ)
    [IsSFiniteKernel μ] [IsSFiniteKernel κ] [IsSFiniteKernel η] :
    μ ⊗ₖ (κ + η) = μ ⊗ₖ κ + μ ⊗ₖ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (μ : ProbabilityTheory.Kernel α β) (κ η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel μ] [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η]
Goal: μ.compProd (κ + η) = μ.compProd κ + μ.compProd η
-- ProbabilityTheory.Kernel.comapRight_compProd_id_prod
lemma comapRight_compProd_id_prod {δ : Type*} {mδ : MeasurableSpace δ}
    (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ) [IsSFiniteKernel η]
    {f : δ → γ} (hf : MeasurableEmbedding f) :
    comapRight (κ ⊗ₖ η) (MeasurableEmbedding.id.prod_mk hf) = κ ⊗ₖ (comapRight η hf) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {δ : Type u_5} {mδ : MeasurableSpace δ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] {f : δ → γ} (hf : MeasurableEmbedding f)
Goal: (κ.compProd η).comapRight ⋯ = κ.compProd (η.comapRight hf)
-- ProbabilityTheory.Kernel.map_apply
theorem map_apply (κ : Kernel α β) (hf : Measurable f) (a : α) : map κ f hf a = (κ a).map f 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {f : β → γ} (κ : ProbabilityTheory.Kernel α β) (hf : Measurable f) (a : α)
Goal: (κ.map f hf) a = MeasureTheory.Measure.map f (κ a)
-- ProbabilityTheory.Kernel.map_apply'
theorem map_apply' (κ : Kernel α β) (hf : Measurable f) (a : α) {s : Set γ} (hs : MeasurableSet s) :
    map κ f hf a s = κ a (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {f : β → γ} (κ : ProbabilityTheory.Kernel α β) (hf : Measurable f) (a : α) {s : Set γ} (hs : MeasurableSet s)
Goal: ((κ.map f hf) a) s = (κ a) (f ⁻¹' s)
-- ProbabilityTheory.Kernel.map_zero
@[simp]
lemma map_zero (hf : Measurable f) : Kernel.map (0 : Kernel α β) f hf = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {f : β → γ} (hf : Measurable f)
Goal: ProbabilityTheory.Kernel.map 0 f hf = 0
-- ProbabilityTheory.Kernel.map_id
@[simp]
lemma map_id (κ : Kernel α β) : map κ id measurable_id = κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β)
Goal: κ.map id ⋯ = κ
-- ProbabilityTheory.Kernel.map_id'
@[simp]
lemma map_id' (κ : Kernel α β) : map κ (fun a ↦ a) measurable_id = κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β)
Goal: κ.map (fun a => a) ⋯ = κ
-- ProbabilityTheory.Kernel.lintegral_map
nonrec theorem lintegral_map (κ : Kernel α β) (hf : Measurable f) (a : α) {g' : γ → ℝ≥0∞}
    (hg : Measurable g') : ∫⁻ b, g' b ∂map κ f hf a = ∫⁻ a, g' (f a) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {f : β → γ} (κ : ProbabilityTheory.Kernel α β) (hf : Measurable f) (a : α) {g' : γ → ℝ≥0∞} (hg : Measurable g')
Goal: ∫⁻ (b : γ), g' b ∂(κ.map f hf) a = ∫⁻ (a : β), g' (f a) ∂κ a
-- ProbabilityTheory.Kernel.sum_map_seq
theorem sum_map_seq (κ : Kernel α β) [IsSFiniteKernel κ] (hf : Measurable f) :
    (Kernel.sum fun n => map (seq κ n) f hf) = map κ f hf 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {f : β → γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hf : Measurable f)
Goal: (ProbabilityTheory.Kernel.sum fun n => (κ.seq n).map f hf) = κ.map f hf
-- ProbabilityTheory.Kernel.map_const
@[simp]
lemma map_const (μ : Measure α) {f : α → β} (hf : Measurable f) :
    map (const γ μ) f hf = const γ (μ.map f) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (μ : MeasureTheory.Measure α) {f : α → β} (hf : Measurable f)
Goal: (ProbabilityTheory.Kernel.const γ μ).map f hf = ProbabilityTheory.Kernel.const γ (MeasureTheory.Measure.map f μ)
-- ProbabilityTheory.Kernel.comap_apply
theorem comap_apply (κ : Kernel α β) (hg : Measurable g) (c : γ) : comap κ g hg c = κ (g c) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {g : γ → α} (κ : ProbabilityTheory.Kernel α β) (hg : Measurable g) (c : γ)
Goal: (κ.comap g hg) c = κ (g c)
-- ProbabilityTheory.Kernel.comap_apply'
theorem comap_apply' (κ : Kernel α β) (hg : Measurable g) (c : γ) (s : Set β) :
    comap κ g hg c s = κ (g c) s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {g : γ → α} (κ : ProbabilityTheory.Kernel α β) (hg : Measurable g) (c : γ) (s : Set β)
Goal: ((κ.comap g hg) c) s = (κ (g c)) s
-- ProbabilityTheory.Kernel.comap_zero
@[simp]
lemma comap_zero (hg : Measurable g) : Kernel.comap (0 : Kernel α β) g hg = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {g : γ → α} (hg : Measurable g)
Goal: ProbabilityTheory.Kernel.comap 0 g hg = 0
-- ProbabilityTheory.Kernel.comap_id
@[simp]
lemma comap_id (κ : Kernel α β) : comap κ id measurable_id = κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β)
Goal: κ.comap id ⋯ = κ
-- ProbabilityTheory.Kernel.comap_id'
@[simp]
lemma comap_id' (κ : Kernel α β) : comap κ (fun a ↦ a) measurable_id = κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β)
Goal: κ.comap (fun a => a) ⋯ = κ
-- ProbabilityTheory.Kernel.lintegral_comap
theorem lintegral_comap (κ : Kernel α β) (hg : Measurable g) (c : γ) (g' : β → ℝ≥0∞) :
    ∫⁻ b, g' b ∂comap κ g hg c = ∫⁻ b, g' b ∂κ (g c) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {g : γ → α} (κ : ProbabilityTheory.Kernel α β) (hg : Measurable g) (c : γ) (g' : β → ℝ≥0∞)
Goal: ∫⁻ (b : β), g' b ∂(κ.comap g hg) c = ∫⁻ (b : β), g' b ∂κ (g c)
-- ProbabilityTheory.Kernel.sum_comap_seq
theorem sum_comap_seq (κ : Kernel α β) [IsSFiniteKernel κ] (hg : Measurable g) :
    (Kernel.sum fun n => comap (seq κ n) g hg) = comap κ g hg 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {g : γ → α} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hg : Measurable g)
Goal: (ProbabilityTheory.Kernel.sum fun n => (κ.seq n).comap g hg) = κ.comap g hg
-- ProbabilityTheory.Kernel.comap_map_comm
lemma comap_map_comm (κ : Kernel β γ) {f : α → β} {g : γ → δ}
    (hf : Measurable f) (hg : Measurable g) :
    comap (map κ g hg) f hf = map (comap κ f hf) g hg 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {δ : Type u_5} {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ} (κ : ProbabilityTheory.Kernel β γ) {f : α → β} {g : γ → δ} (hf : Measurable f) (hg : Measurable g)
Goal: (κ.map g hg).comap f hf = (κ.comap f hf).map g hg
-- ProbabilityTheory.Kernel.prodMkLeft_apply
@[simp]
theorem prodMkLeft_apply (κ : Kernel α β) (ca : γ × α) : prodMkLeft γ κ ca = κ ca.snd 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (ca : γ × α)
Goal: (ProbabilityTheory.Kernel.prodMkLeft γ κ) ca = κ ca.2
-- ProbabilityTheory.Kernel.prodMkRight_apply
@[simp]
theorem prodMkRight_apply (κ : Kernel α β) (ca : α × γ) : prodMkRight γ κ ca = κ ca.fst 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (ca : α × γ)
Goal: (ProbabilityTheory.Kernel.prodMkRight γ κ) ca = κ ca.1
-- ProbabilityTheory.Kernel.prodMkLeft_apply'
theorem prodMkLeft_apply' (κ : Kernel α β) (ca : γ × α) (s : Set β) :
    prodMkLeft γ κ ca s = κ ca.snd s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (ca : γ × α) (s : Set β)
Goal: ((ProbabilityTheory.Kernel.prodMkLeft γ κ) ca) s = (κ ca.2) s
-- ProbabilityTheory.Kernel.prodMkRight_apply'
theorem prodMkRight_apply' (κ : Kernel α β) (ca : α × γ) (s : Set β) :
    prodMkRight γ κ ca s = κ ca.fst s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (ca : α × γ) (s : Set β)
Goal: ((ProbabilityTheory.Kernel.prodMkRight γ κ) ca) s = (κ ca.1) s
-- ProbabilityTheory.Kernel.prodMkLeft_zero
@[simp]
lemma prodMkLeft_zero : Kernel.prodMkLeft α (0 : Kernel β γ) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ}
Goal: ProbabilityTheory.Kernel.prodMkLeft α 0 = 0
-- ProbabilityTheory.Kernel.prodMkRight_zero
@[simp]
lemma prodMkRight_zero : Kernel.prodMkRight α (0 : Kernel β γ) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ}
Goal: ProbabilityTheory.Kernel.prodMkRight α 0 = 0
-- ProbabilityTheory.Kernel.prodMkLeft_add
@[simp]
lemma prodMkLeft_add (κ η : Kernel α β) :
    prodMkLeft γ (κ + η) = prodMkLeft γ κ + prodMkLeft γ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ η : ProbabilityTheory.Kernel α β)
Goal: ProbabilityTheory.Kernel.prodMkLeft γ (κ + η) =
  ProbabilityTheory.Kernel.prodMkLeft γ κ + ProbabilityTheory.Kernel.prodMkLeft γ η
-- ProbabilityTheory.Kernel.prodMkRight_add
@[simp]
lemma prodMkRight_add (κ η : Kernel α β) :
    prodMkRight γ (κ + η) = prodMkRight γ κ + prodMkRight γ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ η : ProbabilityTheory.Kernel α β)
Goal: ProbabilityTheory.Kernel.prodMkRight γ (κ + η) =
  ProbabilityTheory.Kernel.prodMkRight γ κ + ProbabilityTheory.Kernel.prodMkRight γ η
-- ProbabilityTheory.Kernel.lintegral_prodMkLeft
theorem lintegral_prodMkLeft (κ : Kernel α β) (ca : γ × α) (g : β → ℝ≥0∞) :
    ∫⁻ b, g b ∂prodMkLeft γ κ ca = ∫⁻ b, g b ∂κ ca.snd 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (ca : γ × α) (g : β → ℝ≥0∞)
Goal: ∫⁻ (b : β), g b ∂(ProbabilityTheory.Kernel.prodMkLeft γ κ) ca = ∫⁻ (b : β), g b ∂κ ca.2
-- ProbabilityTheory.Kernel.lintegral_prodMkRight
theorem lintegral_prodMkRight (κ : Kernel α β) (ca : α × γ) (g : β → ℝ≥0∞) :
    ∫⁻ b, g b ∂prodMkRight γ κ ca = ∫⁻ b, g b ∂κ ca.fst 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (ca : α × γ) (g : β → ℝ≥0∞)
Goal: ∫⁻ (b : β), g b ∂(ProbabilityTheory.Kernel.prodMkRight γ κ) ca = ∫⁻ (b : β), g b ∂κ ca.1
-- ProbabilityTheory.Kernel.isSFiniteKernel_prodMkLeft_unit
lemma isSFiniteKernel_prodMkLeft_unit {κ : Kernel α β} :
    IsSFiniteKernel (prodMkLeft Unit κ) ↔ IsSFiniteKernel κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β}
Goal: ProbabilityTheory.IsSFiniteKernel (ProbabilityTheory.Kernel.prodMkLeft Unit κ) ↔ ProbabilityTheory.IsSFiniteKernel κ
-- ProbabilityTheory.Kernel.isSFiniteKernel_prodMkRight_unit
lemma isSFiniteKernel_prodMkRight_unit {κ : Kernel α β} :
    IsSFiniteKernel (prodMkRight Unit κ) ↔ IsSFiniteKernel κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β}
Goal: ProbabilityTheory.IsSFiniteKernel (ProbabilityTheory.Kernel.prodMkRight Unit κ) ↔ ProbabilityTheory.IsSFiniteKernel κ
-- ProbabilityTheory.Kernel.map_prodMkLeft
lemma map_prodMkLeft (γ : Type*) [MeasurableSpace γ] (κ : Kernel α β)
    {f : β → δ} (hf : Measurable f) :
    map (prodMkLeft γ κ) f hf = prodMkLeft γ (map κ f hf) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {δ : Type u_4} {mδ : MeasurableSpace δ} (γ : Type u_6) [MeasurableSpace γ] (κ : ProbabilityTheory.Kernel α β) {f : β → δ} (hf : Measurable f)
Goal: (ProbabilityTheory.Kernel.prodMkLeft γ κ).map f hf = ProbabilityTheory.Kernel.prodMkLeft γ (κ.map f hf)
-- ProbabilityTheory.Kernel.map_prodMkRight
lemma map_prodMkRight (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ]
    {f : β → δ} (hf : Measurable f) :
    map (prodMkRight γ κ) f hf = prodMkRight γ (map κ f hf) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {δ : Type u_4} {mδ : MeasurableSpace δ} (κ : ProbabilityTheory.Kernel α β) (γ : Type u_6) [MeasurableSpace γ] {f : β → δ} (hf : Measurable f)
Goal: (ProbabilityTheory.Kernel.prodMkRight γ κ).map f hf = ProbabilityTheory.Kernel.prodMkRight γ (κ.map f hf)
-- ProbabilityTheory.Kernel.swapLeft_apply
@[simp]
theorem swapLeft_apply (κ : Kernel (α × β) γ) (a : β × α) : swapLeft κ a = κ a.swap 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel (α × β) γ) (a : β × α)
Goal: κ.swapLeft a = κ a.swap
-- ProbabilityTheory.Kernel.swapLeft_apply'
theorem swapLeft_apply' (κ : Kernel (α × β) γ) (a : β × α) (s : Set γ) :
    swapLeft κ a s = κ a.swap s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel (α × β) γ) (a : β × α) (s : Set γ)
Goal: (κ.swapLeft a) s = (κ a.swap) s
-- ProbabilityTheory.Kernel.lintegral_swapLeft
theorem lintegral_swapLeft (κ : Kernel (α × β) γ) (a : β × α) (g : γ → ℝ≥0∞) :
    ∫⁻ c, g c ∂swapLeft κ a = ∫⁻ c, g c ∂κ a.swap 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel (α × β) γ) (a : β × α) (g : γ → ℝ≥0∞)
Goal: ∫⁻ (c : γ), g c ∂κ.swapLeft a = ∫⁻ (c : γ), g c ∂κ a.swap
-- ProbabilityTheory.Kernel.swapLeft_prodMkLeft
@[simp] lemma swapLeft_prodMkLeft (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ] :
    swapLeft (prodMkLeft γ κ) = prodMkRight γ κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (γ : Type u_6) [MeasurableSpace γ]
Goal: (ProbabilityTheory.Kernel.prodMkLeft γ κ).swapLeft = ProbabilityTheory.Kernel.prodMkRight γ κ
-- ProbabilityTheory.Kernel.swapLeft_prodMkRight
@[simp] lemma swapLeft_prodMkRight (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ] :
    swapLeft (prodMkRight γ κ) = prodMkLeft γ κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (γ : Type u_6) [MeasurableSpace γ]
Goal: (ProbabilityTheory.Kernel.prodMkRight γ κ).swapLeft = ProbabilityTheory.Kernel.prodMkLeft γ κ
-- ProbabilityTheory.Kernel.swapRight_apply
theorem swapRight_apply (κ : Kernel α (β × γ)) (a : α) : swapRight κ a = (κ a).map Prod.swap 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α)
Goal: κ.swapRight a = MeasureTheory.Measure.map Prod.swap (κ a)
-- ProbabilityTheory.Kernel.swapRight_apply'
theorem swapRight_apply' (κ : Kernel α (β × γ)) (a : α) {s : Set (γ × β)} (hs : MeasurableSet s) :
    swapRight κ a s = κ a {p | p.swap ∈ s} 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α) {s : Set (γ × β)} (hs : MeasurableSet s)
Goal: (κ.swapRight a) s = (κ a) {p | p.swap ∈ s}
-- ProbabilityTheory.Kernel.lintegral_swapRight
theorem lintegral_swapRight (κ : Kernel α (β × γ)) (a : α) {g : γ × β → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ c, g c ∂swapRight κ a = ∫⁻ bc : β × γ, g bc.swap ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α) {g : γ × β → ℝ≥0∞} (hg : Measurable g)
Goal: ∫⁻ (c : γ × β), g c ∂κ.swapRight a = ∫⁻ (bc : β × γ), g bc.swap ∂κ a
-- ProbabilityTheory.Kernel.fst_apply
theorem fst_apply (κ : Kernel α (β × γ)) (a : α) : fst κ a = (κ a).map Prod.fst 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_5} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α)
Goal: κ.fst a = MeasureTheory.Measure.map Prod.fst (κ a)