Datasets:

roseking
/

openAI-math-dataset

import OAI.Analysis.HenonEmden.MeasureShuffle

namespace OAI

universe uZ uA uB uC uA2 uB2 uC2 uA3 uB3 uC3

section LayerSplit

open MeasureTheory Set
open scoped ENNReal
namespace HenonLaneEmden.LayerTonelli

variable {Z : Type uZ} [MeasurableSpace Z]

def tripleSwap {A : Type uA} {B : Type uB} {C : Type uC} [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] :
    (A × B) × C ≃ᵐ (A × C) × B :=
  MeasurableEquiv.prodAssoc.trans
    ((MeasurableEquiv.prodCongr (MeasurableEquiv.refl A) MeasurableEquiv.prodComm).trans
      MeasurableEquiv.prodAssoc.symm)

@[simp] lemma tripleSwap_apply {A : Type uA2} {B : Type uB2} {C : Type uC2} [MeasurableSpace A]
    [MeasurableSpace B] [MeasurableSpace C] (x : (A × B) × C) :
    tripleSwap x = ((x.1.1,x.2),x.1.2) := rfl

lemma measurePreserving_tripleSwap {A : Type uA3} {B : Type uB3} {C : Type uC3} [MeasurableSpace A]
    [MeasurableSpace B] [MeasurableSpace C] (μ : Measure A) (ν : Measure B) (ρ : Measure C)
    [SFinite ν] [SFinite ρ] [SFinite μ] :
    MeasurePreserving (tripleSwap : (A × B) × C ≃ᵐ (A × C) × B)
      ((μ.prod ν).prod ρ) ((μ.prod ρ).prod ν) := by
  exact ((MeasurePreserving.symm MeasurableEquiv.prodAssoc
    (measurePreserving_prodAssoc μ ρ ν)).comp
    ((MeasurePreserving.id μ).prod (Measure.measurePreserving_swap (μ := ν) (ν := ρ)))).comp
    (measurePreserving_prodAssoc μ ν ρ)

def parameterLayer (f : Z × ℝ → ℝ) : Set ((Z × ℝ) × ℝ) :=
  {z | 0 < z.1.2 ∧ z.1.2 < f (z.1.1,z.2)}

lemma measurableSet_parameterLayer {f : Z × ℝ → ℝ} (hf : Measurable f) :
    MeasurableSet (parameterLayer f) :=
  (measurableSet_lt measurable_const measurable_fst.snd).inter
    (measurableSet_lt measurable_fst.snd (hf.comp (measurable_fst.fst.prodMk measurable_snd)))

noncomputable def parameterLayerMeasure (μ : Measure Z) (f : Z × ℝ → ℝ) :
    Measure ((Z × ℝ) × ℝ) :=
  ((μ.prod volume).prod volume).restrict (parameterLayer f)

instance parameterLayerMeasure_sfinite (μ : Measure Z) [SFinite μ] (f : Z × ℝ → ℝ) :
    SFinite (parameterLayerMeasure μ f) := inferInstanceAs (SFinite (Measure.restrict _ _))

lemma measurePreserving_layer_tripleSwap (μ : Measure Z) [SFinite μ]
    {f : Z × ℝ → ℝ} (hf : Measurable f) :
    MeasurePreserving (tripleSwap : (Z × ℝ) × ℝ ≃ᵐ (Z × ℝ) × ℝ)
      (layerMeasure (μ.prod volume) f) (parameterLayerMeasure μ f) := by
  exact (measurePreserving_tripleSwap μ volume volume).restrict_preimage
    (measurableSet_parameterLayer hf)

/-- After permuting the real position and level coordinates, every pair layer
integral is the iterated nonnegative integral over two transverse/level parameters
and then the two real line layers. This is valid before proving finiteness. -/
lemma lintegral_pair_layer_split (μ : Measure Z) [SFinite μ]
    {f g : Z × ℝ → ℝ} (hf : Measurable f) (hg : Measurable g)
    {F : ((Z × ℝ) × ℝ) × ((Z × ℝ) × ℝ) → ℝ≥0∞} (hF : Measurable F) :
    (∫⁻ z, F z ∂(layerMeasure (μ.prod volume) f).prod (layerMeasure (μ.prod volume) g)) =
      ∫⁻ p : (Z × ℝ) × (Z × ℝ),
        ∫⁻ s : ℝ × ℝ in {s | (p.1,s) ∈ parameterLayer f} ×ˢ {s | (p.2,s) ∈ parameterLayer g},
          F (((p.1.1,s.1),p.1.2),((p.2.1,s.2),p.2.2))
        ∂volume.prod volume ∂(μ.prod volume).prod (μ.prod volume) := by
  let V := parameterLayer f ×ˢ parameterLayer g
  let G : ((Z × ℝ) × ℝ) × ((Z × ℝ) × ℝ) → ℝ≥0∞ :=
    fun z => F (((z.1.1.1,z.1.2),z.1.1.2),((z.2.1.1,z.2.2),z.2.1.2))
  have hG : Measurable G := by fun_prop
  have hV : MeasurableSet V := (measurableSet_parameterLayer hf).prod (measurableSet_parameterLayer hg)
  calc
    _ = ∫⁻ z, G z ∂(parameterLayerMeasure μ f).prod (parameterLayerMeasure μ g) := by
      exact ((measurePreserving_layer_tripleSwap μ hf).prod
        (measurePreserving_layer_tripleSwap μ hg)).lintegral_comp hG
    _ = ∫⁻ z, V.indicator G z ∂((μ.prod volume).prod volume).prod ((μ.prod volume).prod volume) := by
      rw [parameterLayerMeasure,parameterLayerMeasure,Measure.prod_restrict]
      exact (lintegral_indicator hV G).symm
    _ = ∫⁻ z : ((Z × ℝ) × (Z × ℝ)) × (ℝ × ℝ),
        V.indicator G ((z.1.1,z.2.1),(z.1.2,z.2.2))
        ∂((μ.prod volume).prod (μ.prod volume)).prod (volume.prod volume) :=
      lintegral_prod_prod_shuffle _ _ _ _ (hG.indicator hV)
    _ = _ := by
      rw [lintegral_prod _ (by fun_prop)]
      apply lintegral_congr
      intro p
      have hset : MeasurableSet ({s | (p.1,s) ∈ parameterLayer f} ×ˢ {s | (p.2,s) ∈ parameterLayer g}) :=
        ((measurableSet_parameterLayer hf).preimage measurable_prodMk_left).prod
          ((measurableSet_parameterLayer hg).preimage measurable_prodMk_left)
      rw [← lintegral_indicator hset]
      apply lintegral_congr; intro s
      rfl

end HenonLaneEmden.LayerTonelli
end LayerSplit

end OAI

	import OAI.Analysis.HenonEmden.MeasureShuffle

	namespace OAI

	universe uZ uA uB uC uA2 uB2 uC2 uA3 uB3 uC3

	section LayerSplit

	open MeasureTheory Set
	open scoped ENNReal
	namespace HenonLaneEmden.LayerTonelli

	variable {Z : Type uZ} [MeasurableSpace Z]

	def tripleSwap {A : Type uA} {B : Type uB} {C : Type uC} [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] :
	(A × B) × C ≃ᵐ (A × C) × B :=
	MeasurableEquiv.prodAssoc.trans
	((MeasurableEquiv.prodCongr (MeasurableEquiv.refl A) MeasurableEquiv.prodComm).trans
	MeasurableEquiv.prodAssoc.symm)

	@[simp] lemma tripleSwap_apply {A : Type uA2} {B : Type uB2} {C : Type uC2} [MeasurableSpace A]
	[MeasurableSpace B] [MeasurableSpace C] (x : (A × B) × C) :
	tripleSwap x = ((x.1.1,x.2),x.1.2) := rfl

	lemma measurePreserving_tripleSwap {A : Type uA3} {B : Type uB3} {C : Type uC3} [MeasurableSpace A]
	[MeasurableSpace B] [MeasurableSpace C] (μ : Measure A) (ν : Measure B) (ρ : Measure C)
	[SFinite ν] [SFinite ρ] [SFinite μ] :
	MeasurePreserving (tripleSwap : (A × B) × C ≃ᵐ (A × C) × B)
	((μ.prod ν).prod ρ) ((μ.prod ρ).prod ν) := by
	exact ((MeasurePreserving.symm MeasurableEquiv.prodAssoc
	(measurePreserving_prodAssoc μ ρ ν)).comp
	((MeasurePreserving.id μ).prod (Measure.measurePreserving_swap (μ := ν) (ν := ρ)))).comp
	(measurePreserving_prodAssoc μ ν ρ)

	def parameterLayer (f : Z × ℝ → ℝ) : Set ((Z × ℝ) × ℝ) :=
	{z \| 0 < z.1.2 ∧ z.1.2 < f (z.1.1,z.2)}

	lemma measurableSet_parameterLayer {f : Z × ℝ → ℝ} (hf : Measurable f) :
	MeasurableSet (parameterLayer f) :=
	(measurableSet_lt measurable_const measurable_fst.snd).inter
	(measurableSet_lt measurable_fst.snd (hf.comp (measurable_fst.fst.prodMk measurable_snd)))

	noncomputable def parameterLayerMeasure (μ : Measure Z) (f : Z × ℝ → ℝ) :
	Measure ((Z × ℝ) × ℝ) :=
	((μ.prod volume).prod volume).restrict (parameterLayer f)

	instance parameterLayerMeasure_sfinite (μ : Measure Z) [SFinite μ] (f : Z × ℝ → ℝ) :
	SFinite (parameterLayerMeasure μ f) := inferInstanceAs (SFinite (Measure.restrict _ _))

	lemma measurePreserving_layer_tripleSwap (μ : Measure Z) [SFinite μ]
	{f : Z × ℝ → ℝ} (hf : Measurable f) :
	MeasurePreserving (tripleSwap : (Z × ℝ) × ℝ ≃ᵐ (Z × ℝ) × ℝ)
	(layerMeasure (μ.prod volume) f) (parameterLayerMeasure μ f) := by
	exact (measurePreserving_tripleSwap μ volume volume).restrict_preimage
	(measurableSet_parameterLayer hf)

	/-- After permuting the real position and level coordinates, every pair layer
	integral is the iterated nonnegative integral over two transverse/level parameters
	and then the two real line layers. This is valid before proving finiteness. -/
	lemma lintegral_pair_layer_split (μ : Measure Z) [SFinite μ]
	{f g : Z × ℝ → ℝ} (hf : Measurable f) (hg : Measurable g)
	{F : ((Z × ℝ) × ℝ) × ((Z × ℝ) × ℝ) → ℝ≥0∞} (hF : Measurable F) :
	(∫⁻ z, F z ∂(layerMeasure (μ.prod volume) f).prod (layerMeasure (μ.prod volume) g)) =
	∫⁻ p : (Z × ℝ) × (Z × ℝ),
	∫⁻ s : ℝ × ℝ in {s \| (p.1,s) ∈ parameterLayer f} ×ˢ {s \| (p.2,s) ∈ parameterLayer g},
	F (((p.1.1,s.1),p.1.2),((p.2.1,s.2),p.2.2))
	∂volume.prod volume ∂(μ.prod volume).prod (μ.prod volume) := by
	let V := parameterLayer f ×ˢ parameterLayer g
	let G : ((Z × ℝ) × ℝ) × ((Z × ℝ) × ℝ) → ℝ≥0∞ :=
	fun z => F (((z.1.1.1,z.1.2),z.1.1.2),((z.2.1.1,z.2.2),z.2.1.2))
	have hG : Measurable G := by fun_prop
	have hV : MeasurableSet V := (measurableSet_parameterLayer hf).prod (measurableSet_parameterLayer hg)
	calc
	_ = ∫⁻ z, G z ∂(parameterLayerMeasure μ f).prod (parameterLayerMeasure μ g) := by
	exact ((measurePreserving_layer_tripleSwap μ hf).prod
	(measurePreserving_layer_tripleSwap μ hg)).lintegral_comp hG
	_ = ∫⁻ z, V.indicator G z ∂((μ.prod volume).prod volume).prod ((μ.prod volume).prod volume) := by
	rw [parameterLayerMeasure,parameterLayerMeasure,Measure.prod_restrict]
	exact (lintegral_indicator hV G).symm
	_ = ∫⁻ z : ((Z × ℝ) × (Z × ℝ)) × (ℝ × ℝ),
	V.indicator G ((z.1.1,z.2.1),(z.1.2,z.2.2))
	∂((μ.prod volume).prod (μ.prod volume)).prod (volume.prod volume) :=
	lintegral_prod_prod_shuffle _ _ _ _ (hG.indicator hV)
	_ = _ := by
	rw [lintegral_prod _ (by fun_prop)]
	apply lintegral_congr
	intro p
	have hset : MeasurableSet ({s \| (p.1,s) ∈ parameterLayer f} ×ˢ {s \| (p.2,s) ∈ parameterLayer g}) :=
	((measurableSet_parameterLayer hf).preimage measurable_prodMk_left).prod
	((measurableSet_parameterLayer hg).preimage measurable_prodMk_left)
	rw [← lintegral_indicator hset]
	apply lintegral_congr; intro s
	rfl

	end HenonLaneEmden.LayerTonelli
	end LayerSplit

	end OAI