mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
g : ↥(Lp E 1 ((η ∘ₖ κ) a))
⊢ (fun i ↦ ∫⁻ (x : β), ‖∫ (y : γ), ↑↑i y - ↑↑g y ∂η x‖ₑ ∂κ a) ≤ fun i ↦
    ∫⁻ (x : β), ∫⁻ (y : γ), ‖↑↑i y - ↑↑g y‖ₑ ∂η x ∂κ a

pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
g : ↥(Lp E 1 ((η ∘ₖ κ) a))
⊢ Tendsto (fun i ↦ ∫⁻ (x : β), ∫⁻ (y : γ), ‖↑↑i y - ↑↑g y‖ₑ ∂η x ∂κ a) (𝓝 g) (𝓝 0)

Tactic:
;

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
g : ↥(Lp E 1 ((η ∘ₖ κ) a))
⊢ (fun i ↦ ∫⁻ (x : β), ‖∫ (y : γ), ↑↑i y - ↑↑g y ∂η x‖ₑ ∂κ a) ≤ fun i ↦
    ∫⁻ (x : β), ∫⁻ (y : γ), ‖↑↑i y - ↑↑g y‖ₑ ∂η x ∂κ a

case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
g : ↥(Lp E 1 ((η ∘ₖ κ) a))
⊢ Tendsto (fun i ↦ ∫⁻ (x : β), ∫⁻ (y : γ), ‖↑↑i y - ↑↑g y‖ₑ ∂η x ∂κ a) (𝓝 g) (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h : μ = 0
⊢ ∫ (x : α), rexp (g x) ∂Measure.tilted 0 f = (∫ (x : α), rexp ((f + g) x) ∂0) / ∫ (x : α), rexp (f x) ∂0

Tactic:
;

Goal state after:
case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h : μ = 0
⊢ ∫ (x : α), rexp (g x) ∂Measure.tilted 0 f = (∫ (x : α), rexp ((f + g) x) ∂0) / ∫ (x : α), rexp (f x) ∂0
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
f✝ : γ → E
hE : CompleteSpace E
⊢ Integrable f✝ ((η ∘ₖ κ) a) → ∫ (z : γ), f✝ z ∂(η ∘ₖ κ) a = ∫ (x : β), ∫ (y : γ), f✝ y ∂η x ∂κ a

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
f✝ : γ → E
hE : ¬CompleteSpace E
⊢ Integrable f✝ ((η ∘ₖ κ) a) → ∫ (z : γ), f✝ z ∂(η ∘ₖ κ) a = ∫ (x : β), ∫ (y : γ), f✝ y ∂η x ∂κ a

Tactic:
;

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
f✝ : γ → E
hE : CompleteSpace E
⊢ Integrable f✝ ((η ∘ₖ κ) a) → ∫ (z : γ), f✝ z ∂(η ∘ₖ κ) a = ∫ (x : β), ∫ (y : γ), f✝ y ∂η x ∂κ a

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
f✝ : γ → E
hE : ¬CompleteSpace E
⊢ Integrable f✝ ((η ∘ₖ κ) a) → ∫ (z : γ), f✝ z ∂(η ∘ₖ κ) a = ∫ (x : β), ∫ (y : γ), f✝ y ∂η x ∂κ a
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ μ.tilted (f + -f) = (μ Set.univ)⁻¹ • μ

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ μ.tilted (f + -f) = (μ Set.univ)⁻¹ • μ
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x ↦ rexp (f x)) μ
h : μ = 0
⊢ 0 ≪ 0

Tactic:
;

Goal state after:
case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x ↦ rexp (f x)) μ
h : μ = 0
⊢ 0 ≪ 0
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case inl
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x ↦ rexp (f x)) ν
h : ν = 0
⊢ ∀ᶠ (x : α) in ⊥,
    μ.rnDeriv (Measure.tilted 0 f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂0) * μ.rnDeriv 0 x

Tactic:
;

Goal state after:
case inl
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x ↦ rexp (f x)) ν
h : ν = 0
⊢ ∀ᶠ (x : α) in ⊥,
    μ.rnDeriv (Measure.tilted 0 f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂0) * μ.rnDeriv 0 x
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : SigmaFinite μ
hf : Integrable (fun x ↦ rexp (f x)) μ
h : μ = 0
⊢ ∀ᶠ (x : α) in ⊥, log ((Measure.tilted 0 f).rnDeriv 0 x).toReal = f x - log (∫ (x : α), rexp (f x) ∂0)

Tactic:
;

Goal state after:
case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : SigmaFinite μ
hf : Integrable (fun x ↦ rexp (f x)) μ
h : μ = 0
⊢ ∀ᶠ (x : α) in ⊥, log ((Measure.tilted 0 f).rnDeriv 0 x).toReal = f x - log (∫ (x : α), rexp (f x) ∂0)
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable f μ
h_int : Integrable (llr μ ν) μ
hfμ : Integrable (fun x ↦ rexp (f x)) μ
hfν : AEMeasurable f ν
⊢ Integrable (fun x ↦ f x - log (∫ (x : α), rexp (f x) ∂μ)) μ

pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable f μ
h_int : Integrable (llr μ ν) μ
hfμ : Integrable (fun x ↦ rexp (f x)) μ
hfν : AEMeasurable f ν
⊢ ∫ (a : α), f a - log (∫ (x : α), rexp (f x) ∂μ) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (x : α), f x ∂μ - log (∫ (x : α), rexp (f x) ∂μ) + ∫ (x : α), llr μ ν x ∂μ

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable f μ
h_int : Integrable (llr μ ν) μ
hfμ : Integrable (fun x ↦ rexp (f x)) μ
hfν : AEMeasurable f ν
⊢ Integrable (fun x ↦ f x - log (∫ (x : α), rexp (f x) ∂μ)) μ

α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable f μ
h_int : Integrable (llr μ ν) μ
hfμ : Integrable (fun x ↦ rexp (f x)) μ
hfν : AEMeasurable f ν
⊢ ∫ (a : α), f a - log (∫ (x : α), rexp (f x) ∂μ) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (x : α), f x ∂μ - log (∫ (x : α), rexp (f x) ∂μ) + ∫ (x : α), llr μ ν x ∂μ
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: ⊢ Continuous fun x ↦ x * log x + 1 - x

Tactic:
;

Goal state after:
⊢ Continuous fun x ↦ x * log x + 1 - x
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: ⊢ ¬DifferentiableAt ℝ (fun x ↦ x * log x + 1 - x) 0

Tactic:
;

Goal state after:
⊢ ¬DifferentiableAt ℝ (fun x ↦ x * log x + 1 - x) 0
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: case pos
x : ℝ
h0 : x = 0
⊢ derivWithin klFun (Iio 0) 0 = 0

Tactic:
exact derivWithin_zero_of_not_differentiableWithinAt not_differentiableWithinAt_klFun_Iio_zero

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: ⊢ (fun x ↦ x * log x + 1 - x) = fun x ↦ x * (log x - 1) + 1

Tactic:
;

Goal state after:
⊢ (fun x ↦ x * log x + 1 - x) = fun x ↦ x * (log x - 1) + 1
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: x✝ : ℝ
⊢ x✝ * log x✝ + 1 - x✝ = x✝ * (log x✝ - 1) + 1

Tactic:
;

Goal state after:
x✝ : ℝ
⊢ x✝ * log x✝ + 1 - x✝ = x✝ * (log x✝ - 1) + 1
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ν : Measure α
inst✝ : IsFiniteMeasure ν
⊢ klDiv 0 ν = ν univ

Tactic:
by
  convert! klDiv_of_ac_of_integrable (Measure.AbsolutelyContinuous.zero _) integrable_zero_measure
  simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable (fun x ↦ rexp (f x)) ν
h : ν = 0
s : Set α
a✝ : MeasurableSet s
⊢ μ s = 0 s

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable (fun x ↦ rexp (f x)) ν
h : ν = 0
s : Set α
a✝ : MeasurableSet s
⊢ μ s = 0 s
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : ¬μ ≪ ν
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞

pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞

Tactic:
;

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : ¬μ ≪ ν
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞

case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ -f x + log (∫ (x : α), rexp (f x) ∂ν)) μ

pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a + log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ -f x + log (∫ (x : α), rexp (f x) ∂ν)) μ

α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a + log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun a ↦ -f a) μ

pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a ∂μ + ∫ (a : α), log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun a ↦ -f a) μ

α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a ∂μ + ∫ (a : α), log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f g : ↥(Lp E' p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑f) s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑g) s μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ
hf_meas : AEStronglyMeasurable (↑↑f) μ
hg_meas : AEStronglyMeasurable (↑↑g) μ
h_sub : ↑↑(f - g) =ᵐ[μ] 0
⊢ ↑↑f - ↑↑g =ᵐ[μ] 0

Tactic:
;

Goal state after:
α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f g : ↥(Lp E' p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑f) s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑g) s μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ
hf_meas : AEStronglyMeasurable (↑↑f) μ
hg_meas : AEStronglyMeasurable (↑↑g) μ
h_sub : ↑↑(f - g) =ᵐ[μ] 0
⊢ ↑↑f - ↑↑g =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : Fact (m ≤ m0)
inst✝ : CompleteSpace F
hp : Fact (1 ≤ p)
⊢ CompleteSpace ↥(Lp F p (μ.trim ⋯))

Tactic:
;

Goal state after:
α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : Fact (m ≤ m0)
inst✝ : CompleteSpace F
hp : Fact (1 ≤ p)
⊢ CompleteSpace ↥(Lp F p (μ.trim ⋯))
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : Fact (m ≤ m0)
inst✝ : CompleteSpace F
hp : Fact (1 ≤ p)
⊢ CompleteSpace ↥(lpMeasSubgroup F m p μ)

Tactic:
;

Goal state after:
α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : Fact (m ≤ m0)
inst✝ : CompleteSpace F
hp : Fact (1 ≤ p)
⊢ CompleteSpace ↥(lpMeasSubgroup F m p μ)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ Integrable (↑↑(condExpIndSMul hm hs hμs x)) μ

Tactic:
;

Goal state after:
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ Integrable (↑↑(condExpIndSMul hm hs hμs x)) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ Integrable (↑↑(condExpIndL1Fin hm hs hμs x)) μ

pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ ∫⁻ (x_1 : α), ‖↑↑(condExpIndL1Fin hm hs hμs x) x_1‖ₑ ∂μ ≤ μ s * ENNReal.ofReal ‖x‖

Tactic:
;

Goal state after:
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ Integrable (↑↑(condExpIndL1Fin hm hs hμs x)) μ

α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ ∫⁻ (x_1 : α), ‖↑↑(condExpIndL1Fin hm hs hμs x) x_1‖ₑ ∂μ ≤ μ s * ENNReal.ofReal ‖x‖
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ MemLp (↑↑(condExpIndL1Fin hm hs hμs x)) 1 μ

Tactic:
;

Goal state after:
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ MemLp (↑↑(condExpIndL1Fin hm hs hμs x)) 1 μ
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case hb
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hμ : ¬μ = 0
hν : ¬ν = 0
⊢ ν.real univ ≠ 0

Tactic:
;

Goal state after:
case hb
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hμ : ¬μ = 0
hν : ¬ν = 0
⊢ ν.real univ ≠ 0
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : ¬μ ≪ ν
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

Tactic:
;

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : ¬μ ≪ ν
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : ¬MeasurableSet s
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

pp: case pos
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : MeasurableSet s
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

Tactic:
;

Goal state after:
case neg
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : ¬MeasurableSet s
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

case pos
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : MeasurableSet s
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : ¬MeasurableSet s
⊢ 0 = 0 + 0

Tactic:
;

Goal state after:
case neg
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : ¬MeasurableSet s
⊢ 0 = 0 + 0
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

Tactic:
;

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case pos
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : MeasurableSet s
hμs : μ s = ∞
⊢ 0 = 0 + 0

Tactic:
;

Goal state after:
case pos
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x y : G
hs : MeasurableSet s
hμs : μ s = ∞
⊢ 0 = 0 + 0