ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.IsRatCondKernelCDFAux.nonneg'
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) (q : ℚ)
Goal: ∀ᵐ (c : β) ∂ν a, 0 ≤ f (a, c) q
-- ProbabilityTheory.IsRatCondKernelCDFAux.le_one'
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) (q : ℚ)
Goal: ∀ᵐ (c : β) ∂ν a, f (a, c) q ≤ 1
-- ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_integral_of_monotone
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq) (_hs_tendsto : Filter.Tendsto seq Filter.atTop Filter.atTop)
Goal: Filter.Tendsto (fun m => ∫ (c : β), f (a, c) (seq m) ∂ν a) Filter.atTop (𝓝 ((ν a) Set.univ).toReal)
-- ProbabilityTheory.IsRatCondKernelCDFAux.mono'
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) {q r : ℚ} (_hqr : q ≤ r)
Goal: ∀ᵐ (c : β) ∂ν a, f (a, c) q ≤ f (a, c) r
-- ProbabilityTheory.IsRatCondKernelCDFAux.setIntegral
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ)
Goal: ∫ (c : β) in A, f (a, c) q ∂ν a = ((κ a) (A ×ˢ Set.Iic ↑q)).toReal
-- ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_integral_of_antitone
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq) (_hs_tendsto : Filter.Tendsto seq Filter.atTop Filter.atBot)
Goal: Filter.Tendsto (fun m => ∫ (c : β), f (a, c) (seq m) ∂ν a) Filter.atTop (𝓝 0)
-- ProbabilityTheory.IsRatCondKernelCDFAux.integrable
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α) (q : ℚ)
Goal: MeasureTheory.Integrable (fun c => f (a, c) q) (ν a)
-- ProbabilityTheory.IsRatCondKernelCDFAux.measurable_right
lemma IsRatCondKernelCDFAux.measurable_right (hf : IsRatCondKernelCDFAux f κ ν) (a : α) (q : ℚ) :
    Measurable (fun t ↦ f (a, t) q) 
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.IsRatCondKernelCDFAux f κ ν) (a : α) (q : ℚ)
Goal: Measurable fun t => f (a, t) q
-- ProbabilityTheory.IsRatCondKernelCDFAux.mono
lemma IsRatCondKernelCDFAux.mono (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
    ∀ᵐ c ∂(ν a), Monotone (f (a, c)) 
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.IsRatCondKernelCDFAux f κ ν) (a : α)
Goal: ∀ᵐ (c : β) ∂ν a, Monotone (f (a, c))
-- ProbabilityTheory.IsRatCondKernelCDFAux.nonneg
lemma IsRatCondKernelCDFAux.nonneg (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
    ∀ᵐ c ∂(ν a), ∀ q, 0 ≤ f (a, c) q 
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.IsRatCondKernelCDFAux f κ ν) (a : α)
Goal: ∀ᵐ (c : β) ∂ν a, ∀ (q : ℚ), 0 ≤ f (a, c) q
-- ProbabilityTheory.IsRatCondKernelCDFAux.le_one
lemma IsRatCondKernelCDFAux.le_one (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
    ∀ᵐ c ∂(ν a), ∀ q, f (a, c) q ≤ 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 → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) (a : α)
Goal: ∀ᵐ (c : β) ∂ν a, ∀ (q : ℚ), f (a, c) q ≤ 1
-- ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_zero_of_antitone
lemma IsRatCondKernelCDFAux.tendsto_zero_of_antitone (hf : IsRatCondKernelCDFAux f κ ν)
    [IsFiniteKernel ν] (a : α) (seq : ℕ → ℚ) (hseq : Antitone seq)
    (hseq_tendsto : Tendsto seq atTop atBot) :
    ∀ᵐ c ∂(ν a), Tendsto (fun m ↦ f (a, c) (seq m)) atTop (𝓝 0) 
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.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) (seq : ℕ → ℚ) (hseq : Antitone seq) (hseq_tendsto : Filter.Tendsto seq Filter.atTop Filter.atBot)
Goal: ∀ᵐ (c : β) ∂ν a, Filter.Tendsto (fun m => f (a, c) (seq m)) Filter.atTop (𝓝 0)
-- ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_one_of_monotone
lemma IsRatCondKernelCDFAux.tendsto_one_of_monotone (hf : IsRatCondKernelCDFAux f κ ν)
    [IsFiniteKernel ν] (a : α) (seq : ℕ → ℚ) (hseq : Monotone seq)
    (hseq_tendsto : Tendsto seq atTop atTop) :
    ∀ᵐ c ∂(ν a), Tendsto (fun m ↦ f (a, c) (seq m)) atTop (𝓝 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 → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) (seq : ℕ → ℚ) (hseq : Monotone seq) (hseq_tendsto : Filter.Tendsto seq Filter.atTop Filter.atTop)
Goal: ∀ᵐ (c : β) ∂ν a, Filter.Tendsto (fun m => f (a, c) (seq m)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_atTop_one
lemma IsRatCondKernelCDFAux.tendsto_atTop_one (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel ν]
    (a : α) :
    ∀ᵐ t ∂(ν a), Tendsto (f (a, t)) atTop (𝓝 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 → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α)
Goal: ∀ᵐ (t : β) ∂ν a, Filter.Tendsto (f (a, t)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.IsRatCondKernelCDFAux.tendsto_atBot_zero
lemma IsRatCondKernelCDFAux.tendsto_atBot_zero (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel ν]
    (a : α) :
    ∀ᵐ t ∂(ν a), Tendsto (f (a, t)) atBot (𝓝 0) 
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.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α)
Goal: ∀ᵐ (t : β) ∂ν a, Filter.Tendsto (f (a, t)) Filter.atBot (𝓝 0)
-- ProbabilityTheory.IsRatCondKernelCDFAux.bddBelow_range
lemma IsRatCondKernelCDFAux.bddBelow_range (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
    ∀ᵐ t ∂(ν a), ∀ q : ℚ, BddBelow (range fun (r : Ioi q) ↦ f (a, t) r) 
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.IsRatCondKernelCDFAux f κ ν) (a : α)
Goal: ∀ᵐ (t : β) ∂ν a, ∀ (q : ℚ), BddBelow (Set.range fun r => f (a, t) ↑r)
-- ProbabilityTheory.IsRatCondKernelCDFAux.integrable_iInf_rat_gt
lemma IsRatCondKernelCDFAux.integrable_iInf_rat_gt (hf : IsRatCondKernelCDFAux f κ ν)
    [IsFiniteKernel ν] (a : α) (q : ℚ) :
    Integrable (fun t ↦ ⨅ r : Ioi q, f (a, t) r) (ν a) 
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.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) (q : ℚ)
Goal: MeasureTheory.Integrable (fun t => ⨅ r, f (a, t) ↑r) (ν a)
-- MeasureTheory.Measure.iInf_rat_gt_prod_Iic
lemma _root_.MeasureTheory.Measure.iInf_rat_gt_prod_Iic {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ]
    {s : Set α} (hs : MeasurableSet s) (t : ℚ) :
    ⨅ r : { r' : ℚ // t < r' }, ρ (s ×ˢ Iic (r : ℝ)) = ρ (s ×ˢ Iic (t : ℝ)) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {ρ : MeasureTheory.Measure (α × ℝ)} [MeasureTheory.IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) (t : ℚ)
Goal: ⨅ r, ρ (s ×ˢ Set.Iic ↑↑r) = ρ (s ×ˢ Set.Iic ↑t)
-- ProbabilityTheory.IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt
lemma IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt (hf : IsRatCondKernelCDFAux f κ ν)
    [IsFiniteKernel κ] [IsFiniteKernel ν] (a : α) (q : ℚ) {A : Set β} (hA : MeasurableSet A) :
    ∫ t in A, ⨅ r : Ioi q, f (a, t) r ∂(ν a) = (κ a (A ×ˢ Iic (q : ℝ))).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 → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel ν] (a : α) (q : ℚ) {A : Set β} (hA : MeasurableSet A)
Goal: ∫ (t : β) in A, ⨅ r, f (a, t) ↑r ∂ν a = ((κ a) (A ×ˢ Set.Iic ↑q)).toReal
-- ProbabilityTheory.IsRatCondKernelCDFAux.iInf_rat_gt_eq
lemma IsRatCondKernelCDFAux.iInf_rat_gt_eq (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel κ]
    [IsFiniteKernel ν] (a : α) :
    ∀ᵐ t ∂(ν a), ∀ q : ℚ, ⨅ r : Ioi q, f (a, t) r = f (a, t) q 
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.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel ν] (a : α)
Goal: ∀ᵐ (t : β) ∂ν a, ∀ (q : ℚ), ⨅ r, f (a, t) ↑r = f (a, t) q
-- ProbabilityTheory.IsRatCondKernelCDFAux.isRatStieltjesPoint_ae
lemma IsRatCondKernelCDFAux.isRatStieltjesPoint_ae (hf : IsRatCondKernelCDFAux f κ ν)
    [IsFiniteKernel κ] [IsFiniteKernel ν] (a : α) :
    ∀ᵐ t ∂(ν a), IsRatStieltjesPoint f (a, 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 → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel ν] (a : α)
Goal: ∀ᵐ (t : β) ∂ν a, ProbabilityTheory.IsRatStieltjesPoint f (a, t)
-- ProbabilityTheory.IsRatCondKernelCDFAux.isRatCondKernelCDF
lemma IsRatCondKernelCDFAux.isRatCondKernelCDF (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel κ]
    [IsFiniteKernel ν] :
    IsRatCondKernelCDF f 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.IsRatCondKernelCDFAux f κ ν) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel ν]
Goal: ProbabilityTheory.IsRatCondKernelCDF f κ ν
-- ProbabilityTheory.IsCondKernelCDF.integrable
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) (x : ℝ)
Goal: MeasureTheory.Integrable (fun b => ↑(f (a, b)) x) (ν a)
-- ProbabilityTheory.IsCondKernelCDF.tendsto_atBot_zero
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.atBot (𝓝 0)
-- ProbabilityTheory.IsCondKernelCDF.measurable
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) (x : ℝ)
Goal: Measurable fun p => ↑(f p) x