Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: case refine_2
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ c * ((∂μ.trim hm/∂ν.trim hm) x).toReal + c') ≤ᵐ[ν.trim hm] fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal

pp: case refine_3
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal) ≤ᵐ[ν.trim hm] fun x ↦ ν[fun x ↦ f ((∂μ/∂ν) x).toReal | m] x

pp: case refine_4
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ c * ((∂μ.trim hm/∂ν.trim hm) x).toReal + c') (ν.trim hm)

pp: case refine_5
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ ν[fun x ↦ f ((∂μ/∂ν) x).toReal | m] x) (ν.trim hm)

Tactic:
· exact ae_of_all _ (fun x ↦ h _ ENNReal.toReal_nonneg)

Goal state after:
case refine_3
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal) ≤ᵐ[ν.trim hm] fun x ↦ ν[fun x ↦ f ((∂μ/∂ν) x).toReal | m] x

case refine_4
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ c * ((∂μ.trim hm/∂ν.trim hm) x).toReal + c') (ν.trim hm)

case refine_5
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ ν[fun x ↦ f ((∂μ/∂ν) x).toReal | m] x) (ν.trim hm)