mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
⊢ ∀ (i : ι), AEMeasurable (X i) (P.restrict s)

Tactic:
by

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

Goal state before:
pp: 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
⊢ Integrable (fun x ↦ f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] x)) ν

Tactic:
by

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
h_int : Integrable (llr μ ν) μ
⊢ StronglyMeasurable klFun

Tactic:
by

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
h_int : Integrable (llr μ ν) μ
⊢ ContinuousWithinAt klFun (Ici 0) 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ P⁻[∑' (i : ι), X i | mΩ] =ᵐ[P] ∑' (i : ι), P⁻[X i | mΩ]

Tactic:
by

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
h_int : Integrable (llr μ ν) μ
⊢ Integrable (llr (Measure.map g μ) (Measure.map g ν)) (Measure.map g μ)

Tactic:
by

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
h_int : Integrable (llr μ ν) μ
⊢ Integrable (llr (Measure.map g μ) (Measure.map g ν)) (Measure.map g μ)

Tactic:
by
  rw [← integrable_klFun_rnDeriv_iff (hμν.map hg)]
  refine convexOn_klFun.integrable_comp_rnDeriv_map hμν hg (by fun_prop) (by fun_prop) ?_
  rwa [integrable_klFun_rnDeriv_iff hμν]

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
⊢ StronglyMeasurable fun x ↦ klFun ((∂Measure.map g μ/∂Measure.map g ν) x).toReal

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
s : Finset ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ P⁻[∑ i ∈ s, X i | mΩ] =ᵐ[P] ∑ i ∈ s, P⁻[X i | mΩ]

Tactic:
by

Goal state after:
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
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
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hφ_meas : StronglyMeasurable φ
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable (φ ∘ μ[f | m])

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hφ_meas : StronglyMeasurable φ
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable μ[φ ∘ f | m]

Tactic:
by

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

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hφ_meas : StronglyMeasurable φ
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ φ ∘ μ[f | m] ≤ᵐ[μ.trim hm] μ[φ ∘ f | m]

Tactic:
by

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
⊢ (klDiv (Measure.map g μ) (Measure.map g ν)).toReal =
    ∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x) ∂ν

Tactic:
by

Goal state after:
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ s φ
hφ_cont : UpperSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ μ[φ ∘ f | m] ≤ᵐ[μ] φ ∘ μ[f | m]

Tactic:
by

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
h_int : Integrable (llr μ ν) μ
⊢ klDiv (Measure.map g μ) (Measure.map g ν) =
    ENNReal.ofReal (∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x) ∂ν)

Tactic:
by

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

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hm : m ≤ m𝓧
hμν : μ ≪ ν
⊢ (klDiv (μ.trim hm) (ν.trim hm)).toReal = ∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] x) ∂ν

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ s φ
hφ_cont : UpperSemicontinuousOn φ s
hφ_meas : StronglyMeasurable φ
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable μ[φ ∘ f | m]

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ s φ
hφ_cont : UpperSemicontinuousOn φ s
hφ_meas : StronglyMeasurable φ
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable (φ ∘ μ[f | m])

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ s φ
hφ_cont : UpperSemicontinuousOn φ s
hφ_meas : StronglyMeasurable φ
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ μ[φ ∘ f | m] ≤ᵐ[μ.trim hm] φ ∘ μ[f | m]

Tactic:
by

Goal state after:
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ univ φ
hφ_cont : LowerSemicontinuous φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ ∀ᵐ (a : α) ∂μ, f a ∈ univ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ univ φ
hφ_cont : LowerSemicontinuous φ
hφ_meas : StronglyMeasurable φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable (φ ∘ μ[f | m])

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ univ φ
hφ_cont : LowerSemicontinuous φ
hφ_meas : StronglyMeasurable φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable μ[φ ∘ f | m]

Tactic:
by

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
hg : Measurable g
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ StronglyMeasurable klFun

Tactic:
by

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
hg : Measurable g
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hf : StronglyMeasurable klFun
⊢ ContinuousWithinAt klFun (Ici 0) 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConvexOn ℝ univ φ
hφ_cont : LowerSemicontinuous φ
hφ_meas : StronglyMeasurable φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ φ ∘ μ[f | m] ≤ᵐ[μ.trim hm] μ[φ ∘ f | m]

Tactic:
by

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
hg : Measurable g
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hf : StronglyMeasurable klFun
hf_cont : ContinuousWithinAt klFun (Ici 0) 0
⊢ Integrable (fun x ↦ klFun ((∂μ/∂ν) x).toReal) ν

Tactic:
by

Goal state after:
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hφ_cont : UpperSemicontinuous φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ μ[φ ∘ f | m] ≤ᵐ[μ] φ ∘ μ[f | m]

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hφ_cont : UpperSemicontinuous φ
hφ_meas : StronglyMeasurable φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable μ[φ ∘ f | m]

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hφ_cont : UpperSemicontinuous φ
hφ_meas : StronglyMeasurable φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ StronglyMeasurable (φ ∘ μ[f | m])

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
mE : MeasurableSpace E
inst✝¹ : BorelSpace E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hφ_cont : UpperSemicontinuous φ
hφ_meas : StronglyMeasurable φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ μ[φ ∘ f | m] ≤ᵐ[μ.trim hm] φ ∘ μ[f | m]

Tactic:
by

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
hg : Measurable g
⊢ klDiv (Measure.map g μ) (Measure.map g ν) ≤ klDiv μ ν

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
m mα : MeasurableSpace α
μ : Measure α
f : α → E
⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]

Tactic:
by

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

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hm : m ≤ m𝓧
⊢ klDiv (μ.trim hm) (ν.trim hm) ≤ klDiv μ ν

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
p : ℝ
hp : 1 ≤ p
hfint : Integrable (fun x ↦ ‖f x‖ ^ p) μ
⊢ 0 < p

Tactic:
by

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✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
κ : Kernel 𝓧 𝓨
inst✝ : IsMarkovKernel κ
⊢ klDiv (⇑κ ∘ₘ μ) (⇑κ ∘ₘ ν) ≤ klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
p : ℝ
hp : 1 ≤ p
hfint : Integrable (fun x ↦ ‖f x‖ ^ p) μ
hp' : 0 < p
hm : m ≤ mα
hμm : SigmaFinite (μ.trim hm)
hf_int : Integrable f μ
hl : LowerSemicontinuousOn (fun x ↦ x ^ p) (Ici 0)
⊢ ∀ᵐ (a : α) ∂μ, ‖f a‖ ∈ Ici 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
p : ℝ
hp : 1 ≤ p
hfint : Integrable (fun x ↦ ‖f x‖ ^ p) μ
⊢ (fun x ↦ ‖μ[f | m] x‖ ^ p) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ ^ p | m]

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
inst✝¹ : FiniteDimensional ℝ E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
⊢ μ[φ ∘ f | m] ≤ᵐ[μ] φ ∘ μ[f | m]

Tactic:
by

Goal state after:
Module: Mathlib.Lean.Thunk

Goal state before:
pp: α : Type u
β : Type v
inst✝ : DecidableEq α
a b : Thunk α
⊢ a = b → a.get = b.get

Tactic:
by

Goal state after:
Module: Mathlib.Lean.Thunk

Goal state before:
pp: α : Type u
β : Type v
inst✝ : DecidableEq α
a b : Thunk α
⊢ a.get = b.get → a = b

Tactic:
by

Goal state after:
Module: Mathlib.Lean.Thunk

Goal state before:
pp: α : Type u
β : Type v
inst✝ : DecidableEq α
⊢ DecidableEq (Thunk α)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.MidpointZero

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing R
inst✝³ : CharZero R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
a b : P
⊢ (lineMap a b) (1 / 2) = midpoint R a b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.MidpointZero

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Field k
inst✝³ : CharZero k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AddTorsor V P
a b : P
⊢ (homothety a (1 / 2)) b = midpoint k a b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Ring k
inst✝¹ : Module k V
b : AffineBasis ι k P
inst✝ : DecidableEq ι
⊢ b.toMatrix ⇑b = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Ring k
inst✝¹ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝ : Fintype ι
q : ι' → P
i : ι'
⊢ ∑ j, b.toMatrix q i j = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
⊢ b.flag 0 = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
⊢ b.flag (Fin.last n) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
k : Fin n
⊢ b.flag k.succ = R ∙ b k ⊔ b.flag k.castSucc

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁶ : AddCommGroup V
inst✝⁵ : AffineSpace V P
inst✝⁴ : Ring k
inst✝³ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝² : Fintype ι
inst✝¹ : Finite ι'
inst✝ : DecidableEq ι'
p : ι' → P
A : Matrix ι ι' k
hA : b.toMatrix p * A = 1
val✝ : Fintype ι'
w₁ w₂ : ι' → k
hw₁ : ∑ i, w₁ i = 1
hw₂ : ∑ i, w₂ i = 1
hweq : (Finset.affineCombination k Finset.univ p) w₁ = (Finset.affineCombination k Finset.univ p) w₂
⊢ w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
k : Fin n
⊢ b.flag k.castSucc ≤ b.flag k.succ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁶ : AddCommGroup V
inst✝⁵ : AffineSpace V P
inst✝⁴ : Ring k
inst✝³ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝² : Fintype ι
inst✝¹ : Finite ι'
inst✝ : DecidableEq ι'
p : ι' → P
A : Matrix ι ι' k
hA : b.toMatrix p * A = 1
⊢ AffineIndependent k p

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
n : ℕ
inst✝ : Nontrivial R
b : Basis (Fin n) R M
x✝ : Fin n
⊢ b.flag x✝.castSucc < b.flag x✝.succ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
n : ℕ
inst✝ : Nontrivial R
b : Basis (Fin n) R M
k : Fin (n + 1)
l : Fin n
⊢ b.flag k ≤ (b.coord l).ker ↔ k ≤ l.castSucc

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁷ : AddCommGroup V
inst✝⁶ : AffineSpace V P
inst✝⁵ : Ring k
inst✝⁴ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝³ : Finite ι
inst✝² : Fintype ι'
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial k
p : ι' → P
A : Matrix ι ι' k
hA : A * b.toMatrix p = 1
val✝ : Fintype ι
this : ∀ (i : ι), b i ∈ affineSpan k (range p)
⊢ affineSpan k (range p) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
k : Fin (n + 1)
l : Fin n
h : k ≤ l.castSucc
⊢ b.flag k ≤ (b.coord l).ker

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
k : Fin n
⊢ b.flag k.castSucc ≤ LinearMap.ker (b.dualBasis k)

Tactic:
by

Goal state after: