mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ Integrable (fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) (μ.restrict s)

pp: case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 ≤ᵐ[μ.restrict s] fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ

Tactic:
· exact hf.integrableOn.div_const _

Goal state after:
case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 ≤ᵐ[μ.restrict s] fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
⊢ IsZeroOrProbabilityMeasure (μ.tilted f)

Tactic:
by
  rcases eq_zero_or_neZero μ with hμ | hμ
  · simp only [hμ, tilted_zero_measure]
    infer_instance
  by_cases hf : Integrable (fun x ↦ exp (f x)) μ
  · have := isProbabilityMeasure_tilted hf
    infer_instance
  · simp only [hf, not_false_eq_true, tilted_of_not_integrable]
    infer_instance

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

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α) in s, ((fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) * g) a ∂μ =
    ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

pp: case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

pp: case pos.hs
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ MeasurableSet s

pp: case pos.h'f
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) < ∞

Tactic:
· simp only [Pi.mul_apply]

Goal state after:
case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

case pos.hs
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ MeasurableSet s

case pos.h'f
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) < ∞
Module: Mathlib.Probability.Kernel.Composition.CompProd

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel (α × β) γ
ξ : Kernel (α × β × γ) δ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
hξ : IsSFiniteKernel ξ
a : α
s : Set ((β × γ) × δ)
hs : MeasurableSet s
⊢ ∫⁻ (b : β),
      ((η ⊗ₖ ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯) (a, b)) (Prod.mk b ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s) ∂κ a =
    ∫⁻ (b : β × γ), (ξ (a, b)) (Prod.mk b ⁻¹' s) ∂(κ ⊗ₖ η) a

Tactic:
lintegral_compProd

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel (α × β) γ
ξ : Kernel (α × β × γ) δ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
hξ : IsSFiniteKernel ξ
a : α
s : Set ((β × γ) × δ)
hs : MeasurableSet s
⊢ ∫⁻ (b : β),
      ((η ⊗ₖ ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯) (a, b)) (Prod.mk b ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s) ∂κ a =
    ∫⁻ (b : β), ∫⁻ (c : γ), (ξ (a, b, c)) (Prod.mk (b, c) ⁻¹' s) ∂η (a, b) ∂κ a

case pos.hf
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel (α × β) γ
ξ : Kernel (α × β × γ) δ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
hξ : IsSFiniteKernel ξ
a : α
s : Set ((β × γ) × δ)
hs : MeasurableSet s
⊢ Measurable fun b ↦ (ξ (a, b)) (Prod.mk b ⁻¹' s)
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α) in s, ((fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) * g) a ∂μ =
    ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

pp: case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

pp: case pos.h'f
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) < ∞

Tactic:
· simp only [Pi.mul_apply]

Goal state after:
case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

case pos.h'f
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) < ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int :
  (∀ᵐ (x : 𝓧) ∂ν, Integrable (fun y ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) (η x)) ∧
    Integrable (fun x ↦ ∫ (y : 𝓨), ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖ ∂η x) ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν, Integrable (fun y ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) (η x)) ∧
    Integrable (fun x ↦ ∫ (y : 𝓨), ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖ ∂η x) ν
h_ae1 : ∀ᵐ (a : 𝓧) ∂ν, μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = μ.rnDeriv ν a
⊢ ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)

Tactic:
have h_compProd : (fun p ↦ μ.rnDeriv ν p.1 * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) p) =ᵐ[ν ⊗ₘ η]
        (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) := (rnDeriv_compProd hκη ν).symm

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int :
  (∀ᵐ (x : 𝓧) ∂ν, Integrable (fun y ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) (η x)) ∧
    Integrable (fun x ↦ ∫ (y : 𝓨), ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖ ∂η x) ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν, Integrable (fun y ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) (η x)) ∧
    Integrable (fun x ↦ ∫ (y : 𝓨), ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖ ∂η x) ν
h_ae1 : ∀ᵐ (a : 𝓧) ∂ν, μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = μ.rnDeriv ν a
h_compProd : (fun p ↦ μ.rnDeriv ν p.1 * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) p) =ᵐ[ν ⊗ₘ η] (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η)
⊢ ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ∞
⊢ llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log c.toReal

Tactic:
by
  simp only [llr, llr_def]
  have h := Measure.rnDeriv_smul_left_of_ne_top μ ν hc_ne_top
  filter_upwards [hμν.ae_le h, Measure.rnDeriv_pos hμν, hμν.ae_le (Measure.rnDeriv_lt_top μ ν)]
    with x hx_eq hx_pos hx_ne_top
  rw [hx_eq]
  simp only [Pi.smul_apply, smul_eq_mul, ENNReal.toReal_mul]
  rw [log_mul]
  rotate_left
  · rw [ENNReal.toReal_ne_zero]
    simp [hc, hc_ne_top]
  · rw [ENNReal.toReal_ne_zero]
    simp [hx_pos.ne', hx_ne_top.ne]
  ring

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0
hc : c ≠ 0
⊢ llr (↑c • μ) (↑c • ν) =ᵐ[μ] llr μ ν

Tactic:
filter_upwards [llr_smul_same hμν (c : ℝ≥0∞) (by simpa) (by simp)] with x hx

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0
hc : c ≠ 0
x : α
hx : llr (↑c • μ) (↑c • ν) x = llr μ ν x
⊢ llr (↑c • μ) (↑c • ν) x = llr μ ν x
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 ∞

Tactic:
simp [hμν]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedSpace ℝ E
hp : Fact (1 ≤ p)
inst✝¹ : CompleteSpace ↥(Lp E p μ)
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp ℝ p μ)
hf : f ≠ 0
m : E →L[ℝ] ↥(Lp E p μ) := (compLpL₂ p μ (lsmul ℝ ℝ).flip).flip f
u : ℕ → E
hu : CauchySeq u
g : ↥(Lp E p μ)
hg : Tendsto (⇑m ∘ u) atTop (𝓝 g)
f' : ℕ → α → E := fun n ↦ ↑↑((⇑m ∘ u) n)
ns : ℕ → ℕ
hns : StrictMono ns
nslim : ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f' (ns i) x) atTop (𝓝 (↑↑g x))
this : (ae (μ.restrict (Function.support ↑↑f))).NeBot
A : ∀ᵐ (x : α) ∂μ.restrict (Function.support ↑↑f), Tendsto (fun i ↦ f' (ns i) x) atTop (𝓝 (↑↑g x))
B : ∀ᵐ (x : α) ∂μ.restrict (Function.support ↑↑f), x ∈ Function.support ↑↑f
n : ℕ
x : α
hx : ↑↑((toSpanSingleton ℝ (u n)).compLp f) x = (toSpanSingleton ℝ (u n)) (↑↑f x)
⊢ ↑↑(m (u n)) x = ↑↑f x • u n

Tactic:
by simp [m, hx]

Goal state after:
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
hμ : ¬μ = 0
hν : ¬ν = 0
this : NeZero ν
ν' : Measure α := (ν univ)⁻¹ • ν
⊢ 0 ≤ ∫ (x : α), llr μ ν x ∂μ + μ.real univ * log (ν.real univ) + 1 - μ.real univ

Tactic:
have hμν' : μ ≪ ν' := hμν.trans (Measure.absolutelyContinuous_smul (by simp))

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
hμ : ¬μ = 0
hν : ¬ν = 0
this : NeZero ν
ν' : Measure α := (ν univ)⁻¹ • ν
hμν' : μ ≪ ν'
⊢ 0 ≤ ∫ (x : α), llr μ ν x ∂μ + μ.real univ * log (ν.real univ) + 1 - μ.real univ
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

Tactic:
simp [hμν]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : ¬Integrable f μ
⊢ 0 =ᵐ[μ] ↑↑0

Tactic:
exact (coeFn_zero _ _ _).symm

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
a : α
ha :
  ↑↑↑((condExpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) a =
    (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a
x✝ : a ∈ t
⊢ ↑‖↑↑↑((condExpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) a‖₊ =
    ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊

Tactic:
by rw [ha]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
a : α
ha : ↑↑(condExpIndSMul hm hs hμs x) a = (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a
x✝ : a ∈ t
⊢ ↑‖↑↑(condExpIndSMul hm hs hμs x) a‖₊ = ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊

Tactic:
by rw [ha]

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

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)
h_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_int_μν : Integrable (llr μ ν) μ
h_int_κη : Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)
h_int1 : Integrable (fun p ↦ log ((∂μ/∂ν) p.1).toReal) (μ ⊗ₘ κ)
⊢ ∫ (p : 𝓧 × 𝓨), log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal ∂μ ⊗ₘ κ =
    ∫ (p : 𝓧 × 𝓨), log ((∂μ/∂ν) p.1).toReal ∂μ ⊗ₘ κ + ∫ (p : 𝓧 × 𝓨), log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal ∂μ ⊗ₘ κ

Tactic:
by
    rw [integral_add h_int1]
    exact h_int_κη.ofReal

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ ∫ (x : α) in t, s.indicator (μ.restrict s)[f | m] x ∂μ = ∫ (x : α) in t, (μ.restrict s)[f | m] x ∂μ.restrict s

Tactic:
integral_indicator (hm _ hs_m),

Goal state after:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ ∫ (x : α) in s, (μ.restrict s)[f | m] x ∂μ.restrict t = ∫ (x : α) in t, (μ.restrict s)[f | m] x ∂μ.restrict s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
⊢ ∀ᵐ (a : α) ∂μ, f a ∈ Y

Tactic:
filter_upwards [htt] with a ha using
      (Submodule.closure_subset_topologicalClosure_span t) (subset_closure ha)

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ ∫ (x : α) in t, f x ∂μ.restrict s = ∫ (x : α) in t, s.indicator f x ∂μ

Tactic:
integral_indicator (hm _ hs_m),

Goal state after:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
⊢ ∫ (x : α) in t, f x ∂μ.restrict s = ∫ (x : α) in s, f x ∂μ.restrict t
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
inst✝ : NeZero P
X : Ω → ℝ≥0∞
⊢ P⁻[X | ⊥] = fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P

Tactic:
by_cases hP : IsFiniteMeasure P

Goal state after:
case pos
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
inst✝ : NeZero P
X : Ω → ℝ≥0∞
hP : IsFiniteMeasure P
⊢ P⁻[X | ⊥] = fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P

case neg
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
inst✝ : NeZero P
X : Ω → ℝ≥0∞
hP : ¬IsFiniteMeasure P
⊢ P⁻[X | ⊥] = fun x ↦ (P Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂P
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos.refine_1
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : Integrable f μ
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
this : ∫ (x : α) in t, μ[s.indicator f | m] x ∂μ = ∫ (x : α) in s ∩ t, μ[s.indicator f | m] x ∂μ
hst_m : MeasurableSet (s ∩ t)
⊢ ∫ (x : α) in s ∩ t, s.indicator f x ∂μ = ∫ (x : α) in t, s.indicator f x ∂μ

Tactic:
integral_indicator (hm _ hs_m),

Goal state after:
case pos.refine_1
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : α → E
s : Set α
m m₂ m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hm₂ : m₂ ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : SigmaFinite (μ.trim hm₂)
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)
hs_m₂ : MeasurableSet s
hf_int : Integrable f μ
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
this : ∫ (x : α) in t, μ[s.indicator f | m] x ∂μ = ∫ (x : α) in s ∩ t, μ[s.indicator f | m] x ∂μ
hst_m : MeasurableSet (s ∩ t)
⊢ ∫ (x : α) in s, f x ∂μ.restrict (s ∩ t) = ∫ (x : α) in s, f x ∂μ.restrict t
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X ≤ᵐ[P] Y
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] ≤ᵐ[P] P⁻[Y | mΩ]

pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X ≤ᵐ[P] Y
hm : mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] ≤ᵐ[P] P⁻[Y | mΩ]

Tactic:
· simp_rw [condLExp_of_not_le hm, Filter.EventuallyLE.rfl]

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X ≤ᵐ[P] Y
hm : mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] ≤ᵐ[P] P⁻[Y | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PartialOrder E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
hf : 0 ≤ᵐ[μ] f
hfint : Integrable f μ
⊢ 0 ≤ᵐ[μ] μ[f | m]

pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PartialOrder E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
hf : 0 ≤ᵐ[μ] f
hfint : ¬Integrable f μ
⊢ 0 ≤ᵐ[μ] μ[f | m]

Tactic:
· rw [(condExp_zero.symm : (0 : α → E) = μ[0 | m])]
    exact condExp_mono (integrable_zero _ _ _) hfint hf

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PartialOrder E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsOrderedModule ℝ E
hf : 0 ≤ᵐ[μ] f
hfint : ¬Integrable f μ
⊢ 0 ≤ᵐ[μ] μ[f | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
⊢ ∀ᵐ (a : α) ∂μ, f a ∈ Y

Tactic:
filter_upwards [htt] with a ha using
      (Submodule.closure_subset_topologicalClosure_span t) (subset_closure ha)

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
⊢ ∫⁻ (ω : Ω) in s, X ω ∂P + ∫⁻ (ω : Ω) in s, Y ω ∂P = ∫⁻ (ω : Ω) in s, X ω + Y ω ∂P

Tactic:
rw [lintegral_add_left' (by fun_prop)]

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

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
a : 𝓧
ha : ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | MeasurableSpace.comap g m𝓨] a
⊢ klFun ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal =
    klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] a)

Tactic:
by rw [ha]

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

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : 𝓧 → 𝓨
hg : Measurable g
hμν : ¬μ ≪ ν
⊢ klDiv (Measure.map g μ) (Measure.map g ν) ≤ klDiv μ ν

Tactic:
simp [hμν]

Goal state after: