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

Goal state before:
pp: case pos
X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
η : Kernel Z T
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
⊢ Kernel.id ∥ₖ κ ∘ₖ (η ∥ₖ Kernel.id) = η ∥ₖ Kernel.id ∘ₖ (Kernel.id ∥ₖ κ)

Tactic:
parallelComp_id_left_comp_parallelComp,

Goal state after:
case pos
X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
η : Kernel Z T
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
⊢ η ∥ₖ (κ ∘ₖ Kernel.id) = η ∥ₖ Kernel.id ∘ₖ (Kernel.id ∥ₖ κ)
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ Kernel.id = deterministic Prod.fst ⋯ ×ₖ deterministic Prod.snd ⋯

Tactic:
rw [deterministic_prod_deterministic]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ Kernel.id = deterministic (fun a ↦ (a.1, a.2)) ⋯
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: 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 : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
a : α
⊢ ((κ ×ₖ ξ ×ₖ η).map ⇑MeasurableEquiv.prodAssoc) a = (κ ×ₖ (ξ ×ₖ η)) a

Tactic:
map_apply _ (by fun_prop),

Goal state after:
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 : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
a : α
⊢ Measure.map (⇑MeasurableEquiv.prodAssoc) ((κ ×ₖ ξ ×ₖ η) a) = (κ ×ₖ (ξ ×ₖ η)) a
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : f (μ'.real univ) ≤ ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν'
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν

Tactic:
rw [h_eq, mul_comm, ← div_le_iff₀, div_eq_inv_mul, inv_inv] at h

Goal state after:
case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : ν.real univ * f (μ'.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν

case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : f (μ'.real univ) ≤ (∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν) * (ν.real univ)⁻¹
⊢ 0 < (ν.real univ)⁻¹
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case neg
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
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

Tactic:
integral_undef hf'

Goal state after:
case neg
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
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / 0) * g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case neg
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
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

Tactic:
integral_undef hf'

Goal state after:
case neg
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
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / 0) * g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
⊢ ∫⁻ (x : α), g x ∂μ.tilted f = ∫⁻ (x : α), ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

Tactic:
← setLIntegral_univ,

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
⊢ ∫⁻ (x : α) in Set.univ, g x ∂μ.tilted f = ∫⁻ (x : α), ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g 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
ι : Type u_4
inst✝¹ : Countable ι
κ : Kernel α β
η : ι → Kernel (α × β) γ
inst✝ : ∀ (i : ι), IsSFiniteKernel (η i)
hκ : IsSFiniteKernel κ
a : α
s : Set (β × γ)
hs : MeasurableSet s
⊢ ∫⁻ (b : β), (Measure.sum fun n ↦ (η n) (a, b)) (Prod.mk b ⁻¹' s) ∂κ a =
    ∑' (i : ι), ∫⁻ (b : β), ((η i) (a, b)) (Prod.mk b ⁻¹' s) ∂κ a

Tactic:
← lintegral_tsum

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
ι : Type u_4
inst✝¹ : Countable ι
κ : Kernel α β
η : ι → Kernel (α × β) γ
inst✝ : ∀ (i : ι), IsSFiniteKernel (η i)
hκ : IsSFiniteKernel κ
a : α
s : Set (β × γ)
hs : MeasurableSet s
⊢ ∫⁻ (b : β), (Measure.sum fun n ↦ (η n) (a, b)) (Prod.mk b ⁻¹' s) ∂κ a =
    ∫⁻ (a_1 : β), ∑' (i : ι), ((η i) (a, a_1)) (Prod.mk a_1 ⁻¹' s) ∂κ a

case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
ι : Type u_4
inst✝¹ : Countable ι
κ : Kernel α β
η : ι → Kernel (α × β) γ
inst✝ : ∀ (i : ι), IsSFiniteKernel (η i)
hκ : IsSFiniteKernel κ
a : α
s : Set (β × γ)
hs : MeasurableSet s
⊢ ∀ (i : ι), AEMeasurable (fun b ↦ ((η i) (a, b)) (Prod.mk b ⁻¹' s)) (κ a)
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: 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✝ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
f : γ → E
hf : StronglyMeasurable f
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ (∀ᵐ (x : β) ∂κ a, ∫⁻ (a : γ), ‖f a‖ₑ ∂η x < ∞) →
    (∫⁻ (b : β), ∫⁻ (c : γ), ‖f c‖ₑ ∂η b ∂κ a < ∞ ↔
      ∫⁻ (a : β), ENNReal.ofReal (∫⁻ (a : γ), ‖f a‖ₑ ∂η a).toReal ∂κ a < ∞)

pp: 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✝ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
f : γ → E
hf : StronglyMeasurable f
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ‖f c‖ₑ ∂η b ∂κ a < ∞ → ∀ᵐ (x : β) ∂κ a, ∫⁻ (a : γ), ‖f a‖ₑ ∂η x < ∞

Tactic:
· intro h
    rw [lintegral_congr_ae]
    filter_upwards [h] with x hx
    rw [ofReal_toReal]
    finiteness

Goal state after:
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✝ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
f : γ → E
hf : StronglyMeasurable f
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ‖f c‖ₑ ∂η b ∂κ a < ∞ → ∀ᵐ (x : β) ∂κ a, ∫⁻ (a : γ), ‖f a‖ₑ ∂η x < ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: case neg
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
a : 𝓧
ha : μ.rnDeriv ν a ≠ 0 → ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = (κ a) univ
h0 : ¬μ.rnDeriv ν a = 0
⊢ μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = μ.rnDeriv ν a

Tactic:
· simp [ha h0]

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

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ
g : α → E
s : Set α
hs : MeasurableSet s
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ

Tactic:
integral_undef hf'

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ
g : α → E
s : Set α
hs : MeasurableSet s
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / 0) • g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ

Tactic:
integral_undef hf'

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / 0) • g x ∂μ
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
⊢ -llr μ ν =ᵐ[μ] llr ν μ

Tactic:
filter_upwards [Measure.inv_rnDeriv hμν] with x hx

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
x : α
hx : (μ.rnDeriv ν)⁻¹ x = ν.rnDeriv μ x
⊢ (-llr μ ν) x = llr ν μ x
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
⊢ ∫ (x : α), f x ∂μ - log (∫ (x : α), rexp (f x) ∂μ) + ∫ (x : α), llr μ ν x ∂μ =
    ∫ (x : α), llr μ ν x ∂μ + ∫ (x : α), f x ∂μ - log (∫ (x : α), rexp (f x) ∂μ)

Tactic:
abel

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: case refine_3
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
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun a ↦ f (μ.rnDeriv ν a).toReal) ≤ᵐ[ν] fun x ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, b)).toReal ∂η x

pp: case refine_4
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
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, b)).toReal ∂η x) ν

Tactic:
· exact hf_cvx.apply_rnDeriv_ae_le_integral hf hf_cont_at hf_int hκη

Goal state after:
case refine_4
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
hf_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, b)).toReal ∂η x) ν
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
⊢ -∫ (x : α), f x ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ =
    ∫ (x : α), llr μ ν x ∂μ - ∫ (x : α), f x ∂μ + log (∫ (x : α), rexp (f x) ∂ν)

Tactic:
abel

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

Goal state before:
pp: case refine_2
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 : ↥(lpMeas E' 𝕜 m 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
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (μ.trim hm)
hfg : ↑↑↑f =ᵐ[μ] g
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
hfg_restrict : ↑↑↑f =ᵐ[μ.restrict s] g
⊢ ∫ (a : α) in s, ↑↑↑f a ∂μ = 0

Tactic:
exact hf_zero s hs hμs

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

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_7
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp E 2 μ)
⊢ ‖↑((condExpL2 E 𝕜 hm) f)‖ ≤ ‖f‖

Tactic:
by
  rw [Lp.norm_def, Lp.norm_def]
  exact ENNReal.toReal_mono (Lp.eLpNorm_ne_top _) (eLpNorm_condExpL2_le hm f)

Goal state after:
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)
x y : G
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

Tactic:
by_cases hs : MeasurableSet s
  swap; · simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [zero_add]
  by_cases hμs : μ s = ∞
  · simp_rw [condExpIndL1_of_measure_eq_top hμs]; rw [zero_add]
  · simp_rw [condExpIndL1_of_measurableSet_of_measure_ne_top hs hμs]
    exact condExpIndL1Fin_add hs hμs x y

Goal state after:
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 ≠ ∞
c : G
x : α
hx : ↑↑(indicatorConstLp 2 ⋯ hμs 1) x = s.indicator (fun x ↦ 1) x
⊢ s.indicator (fun x ↦ 1) x • c = s.indicator (fun x ↦ c) x

Tactic:
by_cases hx_mem : x ∈ s

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)
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : G
x : α
hx : ↑↑(indicatorConstLp 2 ⋯ hμs 1) x = s.indicator (fun x ↦ 1) x
hx_mem : x ∈ s
⊢ s.indicator (fun x ↦ 1) x • c = s.indicator (fun x ↦ c) x

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)
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : G
x : α
hx : ↑↑(indicatorConstLp 2 ⋯ hμs 1) x = s.indicator (fun x ↦ 1) x
hx_mem : x ∉ s
⊢ s.indicator (fun x ↦ 1) x • c = s.indicator (fun x ↦ c) x
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_3
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : InnerProductSpace 𝕜 E'
inst✝⁶ : CompleteSpace E'
inst✝⁵ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : RCLike 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2 μ)
h_coe : ∀ᵐ (a : α) ∂μ, ↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f)) a = T (↑↑↑((condExpL2 E' 𝕜 hm) f) a)
⊢ AEStronglyMeasurable (↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f))) μ

Tactic:
← EventuallyEq

Goal state after:
case refine_3
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : InnerProductSpace 𝕜 E'
inst✝⁶ : CompleteSpace E'
inst✝⁵ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : RCLike 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2 μ)
h_coe : ↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f)) =ᵐ[μ] fun a ↦ T (↑↑↑((condExpL2 E' 𝕜 hm) f) a)
⊢ AEStronglyMeasurable (↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f))) μ
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'
h_comp :
  ∀ᵐ (a : α) ∂μ,
    ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) a =
      (toSpanSingleton ℝ x) (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a)
⊢ (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) =ᵐ[μ]
    ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)))

Tactic:
← EventuallyEq

Goal state after:
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'
h_comp :
  ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) =ᵐ[μ] fun a ↦
    (toSpanSingleton ℝ x) (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a)
⊢ (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) =ᵐ[μ]
    ↑↑((toSpanSingleton ℝ x).compLp ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)))
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✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
hf : Integrable f μ
⊢ ∫ (x : α), μ[f | m] x ∂μ = ∫ (x : α), f x ∂μ

Tactic:
suffices ∫ x in Set.univ, (μ[f | m]) x ∂μ = ∫ x in Set.univ, f x ∂μ by
      simp_rw [setIntegral_univ] at this; exact this

Goal state after:
case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
hf : Integrable f μ
⊢ ∫ (x : α) in Set.univ, μ[f | m] x ∂μ = ∫ (x : α) in Set.univ, f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_2
α : Type u_1
G : Type u_5
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
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 → μ s_1 ≠ ∞ → ∫⁻ (x_1 : α) in s_1, ‖↑↑(condExpIndSMul hm hs hμs x) x_1‖ₑ ∂μ ≤ μ s * ↑‖x‖₊

Tactic:
refine fun t ht hμt => (setLIntegral_nnnorm_condExpIndSMul_le hm hs hμs x ht hμt).trans ?_

Goal state after:
case refine_2
α : Type u_1
G : Type u_5
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
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ μ (s ∩ t) * ↑‖x‖₊ ≤ μ s * ↑‖x‖₊
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsProbabilityMeasure μ
f : α → E
⊢ μ[f | ⊥] = fun x ↦ ∫ (x : α), f x ∂μ

Tactic:
refine (condExp_bot' f).trans ?_

Goal state after:
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsProbabilityMeasure μ
f : α → E
⊢ (fun x ↦ (μ.real Set.univ)⁻¹ • ∫ (x : α), f x ∂μ) = fun x ↦ ∫ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict sᶜ] 0
⊢ μ[s.indicator f | m] =ᵐ[μ] s.indicator μ[f | m]

Tactic:
by_cases hm : m ≤ m0
  swap; · simp_rw [condExp_of_not_le hm, Set.indicator_zero']; rfl
  have hsf_zero : ∀ g : α → E, g =ᵐ[μ.restrict sᶜ] 0 → s.indicator g =ᵐ[μ] g := fun g =>
    indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs)
  refine ((hsf_zero (μ[f | m]) (condExp_ae_eq_restrict_zero hs.compl hf)).trans ?_).symm
  exact condExp_congr_ae (hsf_zero f hf).symm

Goal state after: