| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.SharedRoute |
|
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| namespace OAI |
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| section |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| open HardSphere.Bridge |
| attribute [local instance] Classical.propDecidable |
| variable {E I : Type*} [MeasurableSpace E] [Fintype I] [DecidableEq I] |
| {X Y : I→Type*} [∀i,MeasurableSpace (X i)] [∀i,MeasurableSpace (Y i)] |
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| def routeConfigMap (f : ∀i,X i→Y i) (x : ∀i,X i) : ∀i,Y i := fun i ↦ f i (x i) |
| def routeFactorMap (f : ∀i,X i→Y i) (s : Finset I) (σ : Equiv.Perm I) |
| (p : ∀i:s,X i×X (σ.symm i)) : ∀i:s,Y i×Y (σ.symm i) := |
| fun i ↦ (f i (p i).1,f (σ.symm i) (p i).2) |
| def routePairedMap (f : ∀i,X i→Y i) (s : Finset I) (σ : Equiv.Perm I) |
| (p : (∀i,X i)×(∀i:s,X i×X (σ.symm i))) := |
| (routeConfigMap f p.1,routeFactorMap f s σ p.2) |
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|
| omit [Fintype I] [DecidableEq I] in |
| lemma routeConfigMap_measurable (f : ∀i,X i→Y i) (hf : ∀i,Measurable (f i)) : |
| Measurable (routeConfigMap f) := |
| Measurable.of_eval fun i ↦ (hf i).comp (measurable_pi_apply i) |
| omit [Fintype I] [DecidableEq I] in |
| lemma routeFactorMap_measurable (f : ∀i,X i→Y i) (hf : ∀i,Measurable (f i)) |
| (s : Finset I) (σ : Equiv.Perm I) : Measurable (routeFactorMap f s σ) := |
| Measurable.of_eval fun i ↦ ((hf i).comp (measurable_fst.comp (measurable_pi_apply i))).prodMk |
| ((hf _).comp (measurable_snd.comp (measurable_pi_apply i))) |
| omit [Fintype I] [DecidableEq I] in |
| lemma routePairedMap_measurable (f : ∀i,X i→Y i) (hf : ∀i,Measurable (f i)) |
| (s : Finset I) (σ : Equiv.Perm I) : Measurable (routePairedMap f s σ) := |
| ((routeConfigMap_measurable f hf).comp measurable_fst).prodMk |
| ((routeFactorMap_measurable f hf s σ).comp measurable_snd) |
|
|
| omit [Fintype I] [DecidableEq I] in |
| lemma fiberEquiv_natural (f : ∀i,X i→Y i) {i j : I} (h : i=j) (x : X i) : |
| fiberEquiv h (f i x)=f j (fiberEquiv h x) := by cases h; rfl |
|
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| omit [Fintype I] [∀i,MeasurableSpace (X i)] [∀i,MeasurableSpace (Y i)] in |
| lemma ambientFirst_natural (f : ∀i,X i→Y i) (s : Finset I) (σ : Equiv.Perm I) |
| (p : (∀i,X i)×(∀i:s,X i×X (σ.symm i))) : |
| ambientFirst s σ (routePairedMap f s σ p)=routeConfigMap f (ambientFirst s σ p) := by |
| funext i |
| by_cases hi : i∈s |
| · simp only [ambientFirst,routePairedMap,routeConfigMap,routeFactorMap,Function.updateFinset,dite_eq_left hi] |
| · simp only [ambientFirst,routePairedMap,routeConfigMap,routeFactorMap,Function.updateFinset,dite_eq_right hi] |
|
|
| omit [Fintype I] in |
| lemma ambientSecond_natural (f : ∀i,X i→Y i) (s : Finset I) (σ : Equiv.Perm I) |
| (hσ : ∀i,i∉s→σ i=i) (p : (∀i,X i)×(∀i:s,X i×X (σ.symm i))) : |
| ambientSecond s σ hσ (routePairedMap f s σ p)= |
| routeConfigMap f (ambientSecond s σ hσ p) := by |
| funext i |
| by_cases hi : i∈s |
| · simp only [ambientSecond,routePairedMap,routeConfigMap,Function.updateFinset,dite_eq_left hi, |
| reverseSelected_apply,routeFactorMap] |
| exact fiberEquiv_natural f _ _ |
| · simp only [ambientSecond,routePairedMap,routeConfigMap,Function.updateFinset,dite_eq_right hi] |
|
|
| omit [Fintype I] in |
| lemma ambientSharedKernel_apply_measure (s : Finset I) (σ : Equiv.Perm I) |
| (hσ : ∀i,i∉s→σ i=i) (ξ : Kernel E (∀i,X i)) [IsFiniteKernel ξ] |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) (e : E) : |
| ambientSharedKernel s σ hσ ξ ρ A e=(ξ e).compProd |
| ((ambientRefreshKernel s σ hσ ρ A).comap (Prod.mk e) measurable_prodMk_left) := by |
| ext C hC |
| rw [ambientSharedKernel,Kernel.compProd_apply hC,Measure.compProd_apply hC] |
| rfl |
|
|
| omit [Fintype I] in |
| lemma ambientRefresh_map (f : ∀i,X i→Y i) (hf : ∀i,Measurable (f i)) |
| (s : Finset I) (σ : Equiv.Perm I) (hσ : ∀i,i∉s→σ i=i) |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,Y i))) (hA : MeasurableSet A) (e : E) (x : ∀i,X i) : |
| (ambientRefreshKernel s σ hσ ρ |
| ((fun p : E×(∀i,X i) ↦ (p.1,routeConfigMap f p.2)) ⁻¹' A) (e,x)).map |
| (routeFactorMap f s σ)= |
| ambientRefreshKernel s σ hσ (fun i ↦ (ρ i).map (Prod.map (f i) (f (σ.symm i)))) |
| A (e,routeConfigMap f x) := by |
| have hP : Measurable (fun p : E×(∀i,X i) ↦ (p.1,routeConfigMap f p.2)) := |
| measurable_fst.prodMk ((routeConfigMap_measurable f hf).comp measurable_snd) |
| have hB : MeasurableSet {p : ∀i:s,Y i×Y (σ.symm i) | |
| (e,ambientFirst s σ (routeConfigMap f x,p))∈A ∧ |
| (e,ambientSecond s σ hσ (routeConfigMap f x,p))∈A} := |
| (ambientRefreshEvent_measurable s σ hσ A hA).preimage |
| (measurable_prodMk_left (x:=(e,routeConfigMap f x))) |
| rw [ambientRefreshKernel_apply _ _ _ _ _ (hA.preimage hP), |
| ambientRefreshKernel_apply _ _ _ _ _ hA] |
| have he : {p : ∀i:s,X i×X (σ.symm i) | |
| (e,ambientFirst s σ (x,p))∈((fun q : E×(∀i,X i) ↦ (q.1,routeConfigMap f q.2)) ⁻¹' A) ∧ |
| (e,ambientSecond s σ hσ (x,p))∈((fun q : E×(∀i,X i) ↦ (q.1,routeConfigMap f q.2)) ⁻¹' A)} = |
| (routeFactorMap f s σ) ⁻¹' {p | (e,ambientFirst s σ (routeConfigMap f x,p))∈A ∧ |
| (e,ambientSecond s σ hσ (routeConfigMap f x,p))∈A} := by |
| ext p |
| simp only [mem_ofPred_eq,mem_preimage] |
| change _ ↔ (e,ambientFirst s σ (routePairedMap f s σ (x,p)))∈A ∧ |
| (e,ambientSecond s σ hσ (routePairedMap f s σ (x,p)))∈A |
| rw [ambientFirst_natural,ambientSecond_natural] |
| rw [he,conditioned_map_preimage _ _ (routeFactorMap_measurable f hf s σ) hB] |
| congr 1 |
| have hmap := Measure.pi_map_pi (μ:=fun i:s ↦ ρ i e) |
| (fun i:s ↦ ((hf i).prodMap (hf (σ.symm i))).aemeasurable) |
| simp only [Kernel.map_apply _ ((hf _).prodMap (hf _))] |
| exact hmap |
|
|
| end HardSphere.Comparison |
| end |
| end |
|
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| end OAI |
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