| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.ConditionalLikelihood |
| import OAI.Analysis.HardSphere.Comparison.LocalMoment |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {I C : Type*} [Fintype I] [DecidableEq I] [Fintype C] |
| {X : I → Type*} [∀i,MeasurableSpace (X i)] |
|
|
| omit [Fintype I] [∀i,MeasurableSpace (X i)] in |
| lemma freezeOutside_updateFinset (U s : Finset I) (hs : s⊆U) (x y : ∀i,X i) (v : ∀i:s,X i) : |
| freezeOutside U x (Function.updateFinset y s v)=Function.updateFinset (freezeOutside U x y) s v := by |
| funext i |
| by_cases hi : i∈s |
| · simp [freezeOutside,Function.updateFinset,hi,hs hi] |
| · simp [freezeOutside,Function.updateFinset,hi] |
|
|
| omit [Fintype C] in |
| lemma frozen_partialNormalizer (checks : C → LocalCheck X) (μ : ∀i,Measure (X i)) |
| (U s o : Finset I) (hs : s⊆U) (x y : ∀i,X i) |
| (hx : x∈retainedEvent checks (Finset.univ\U)) : |
| partialNormalizer (frozenChecks checks U x) μ s o y= |
| partialNormalizer checks μ s (o∪(Finset.univ\U)) (freezeOutside U x y) := by |
| apply lintegral_congr |
| intro v |
| rw [frozen_checkWeight checks U (s∪o) x _ hx,freezeOutside_updateFinset U s hs] |
| simp only [Finset.union_assoc] |
|
|
| omit [Fintype I] [∀i,MeasurableSpace (X i)] in |
| lemma freezeOutside_pair {Z : I → Type*} (U : Finset I) (x y : ∀i,X i) (z u : ∀i,Z i) : |
| freezeOutside U (fun i ↦ (x i,z i)) (fun i ↦ (y i,u i))= |
| fun i ↦ (freezeOutside U x y i,freezeOutside U z u i) := by |
| funext i |
| by_cases hi : i∈U <;> simp [freezeOutside,hi] |
|
|
| omit [Fintype C] in |
| lemma frozen_conditionalDensity {Z : I → Type*} [∀i,MeasurableSpace (Z i)] |
| (checks : C → LocalCheck (fun i ↦ X i×Z i)) |
| (κ : ∀i,Kernel (X i) (Z i)) [∀i,IsFiniteKernel (κ i)] |
| (U s o : Finset I) (hs : s⊆U) (x y : ∀i,X i) (z u : ∀i,Z i) |
| (hx : (fun i ↦ (x i,z i))∈retainedEvent checks (Finset.univ\U)) : |
| conditionalDensity (frozenChecks checks U (fun i ↦ (x i,z i))) κ s o y u= |
| conditionalDensity checks κ s (o∪(Finset.univ\U)) (freezeOutside U x y) (freezeOutside U z u) := by |
| unfold conditionalDensity lmarginal |
| have hκ : (fun i:s ↦ κ i (freezeOutside U x y i))=(fun i:s ↦ κ i (y i)) := by |
| funext i;simp only [freezeOutside,ite_eq_left (hs i.property)] |
| rw [hκ] |
| apply lintegral_congr |
| intro v |
| dsimp only |
| rw [frozen_checkWeight checks U (s∪o) _ _ hx,freezeOutside_pair,freezeOutside_updateFinset U s hs] |
| simp only [Finset.union_assoc] |
|
|
| omit [Fintype C] in |
| lemma frozen_conditionalLikelihood {Z : I → Type*} [∀i,MeasurableSpace (Z i)] |
| {R : Type*} [Fintype R] (reference : R → LocalCheck X) |
| (checks : C → LocalCheck (fun i ↦ X i×Z i)) |
| (μ : ∀i,Measure (X i)) (κ : ∀i,Kernel (X i) (Z i)) [∀i,IsFiniteKernel (κ i)] |
| (U s o : Finset I) (hs : s⊆U) (x y : ∀i,X i) (z u : ∀i,Z i) |
| (hx : x∈retainedEvent reference (Finset.univ\U)) |
| (hz : (fun i ↦ (x i,z i))∈retainedEvent checks (Finset.univ\U)) : |
| conditionalLikelihood (frozenChecks reference U x) (frozenChecks checks U (fun i ↦ (x i,z i))) μ κ s o y u= |
| conditionalLikelihood reference checks μ κ s (o∪(Finset.univ\U)) (freezeOutside U x y) (freezeOutside U z u) := by |
| unfold conditionalLikelihood |
| rw [frozen_partialNormalizer reference μ U s o hs x y hx, |
| frozen_partialNormalizer checks _ U s o hs _ _ hz, |
| freezeOutside_pair, |
| frozen_conditionalDensity checks κ U s o hs x y z u hz] |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {I R C D : Type*} [Fintype I] [DecidableEq I] [Fintype R] [Fintype C] [Fintype D] |
| {X Z W : I→Type*} [∀i,MeasurableSpace (X i)] [∀i,MeasurableSpace (Z i)] |
| [∀i,MeasurableSpace (W i)] |
|
|
| omit [Fintype I] [∀i,MeasurableSpace (X i)] in |
| lemma freezeOutside_self (U : Finset I) (x : ∀i,X i) : freezeOutside U x x=x := by |
| funext i |
| by_cases hi : i∈U <;> simp [freezeOutside,hi] |
|
|
| lemma local_complement (s o U : Finset I) (hso : Disjoint s o) (hU : s∪o=U) : |
| o∪(Finset.univ\U)=Finset.univ\s := by |
| ext i |
| have hd := Finset.disjoint_left.mp hso |
| simp only [Finset.mem_union,Finset.mem_sdiff,Finset.mem_univ,true_and] |
| rw [←hU] |
| simp only [Finset.mem_union] |
| constructor |
| · rintro (ho|hnot) |
| · exact fun hs ↦ hd hs ho |
| · exact fun hs ↦ hnot (Or.inl hs) |
| · intro hs |
| by_cases ho : i∈o |
| · exact Or.inl ho |
| · exact Or.inr (not_or.mpr ⟨hs,ho⟩) |
|
|
| theorem frozen_pair_mean_integrates |
| (reference : R→LocalCheck X) (checks : C→LocalCheck (fun i ↦ X i×Z i)) |
| (other : D→LocalCheck (fun i ↦ X i×W i)) |
| (μ : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (μ i)] |
| (κ : ∀i,Kernel (X i) (Z i)) [∀i,IsFiniteKernel (κ i)] |
| (η : ∀i,Kernel (X i) (W i)) [∀i,IsFiniteKernel (η i)] |
| (s t os ot U : Finset I) (hs : s⊆U) (ht : t⊆U) |
| (hso : Disjoint s os) (hto : Disjoint t ot) (hsU : s∪os=U) (htU : t∪ot=U) |
| (z : ∀i,Z i) (w : ∀i,W i) |
| (hS : ∀x,x∈retainedEvent reference (Finset.univ\U) → |
| (fun i ↦ (x i,z i))∈retainedEvent checks (Finset.univ\U)) |
| (hT : ∀x,x∈retainedEvent reference (Finset.univ\U) → |
| (fun i ↦ (x i,w i))∈retainedEvent other (Finset.univ\U)) |
| (A : ℝ≥0∞) (hA : A≠⊤) |
| (hlocal : ∀x,x∈retainedEvent reference (Finset.univ\U) → |
| let Ref := frozenChecks reference U x |
| let ChS := frozenChecks checks U (fun i ↦ (x i,z i)) |
| let ChT := frozenChecks other U (fun i ↦ (x i,w i)) |
| lmarginal μ U (fun y ↦ checkWeight Ref U y * |
| conditionalLikelihood Ref ChS μ κ s os y z * |
| conditionalLikelihood Ref ChT μ η t ot y w) x / |
| checkNormalizer Ref μ U x≤A) : |
| (∫⁻x,conditionalLikelihood reference checks μ κ s (Finset.univ\s) x z * |
| conditionalLikelihood reference other μ η t (Finset.univ\t) x w |
| ∂conditionedMeasure (Measure.pi μ) (retainedEvent reference Finset.univ))≤A := by |
| apply local_mean_integrates reference μ U _ |
| ((measurable_conditionalLikelihood reference checks μ κ _ _ z).mul |
| (measurable_conditionalLikelihood reference other μ η _ _ w)) A hA |
| intro x hx |
| rw [←frozen_local_mean reference μ U x hx] |
| have h := hlocal x hx |
| dsimp only at h |
| convert h using 2 |
| apply congrArg (fun f ↦ lmarginal μ U f x) |
| funext y |
| rw [frozen_conditionalLikelihood reference checks μ κ U s os hs x y z z hx (hS x hx), |
| frozen_conditionalLikelihood reference other μ η U t ot ht x y w w hx (hT x hx), |
| freezeOutside_self,freezeOutside_self,local_complement s os U hso hsU, |
| local_complement t ot U hto htU] |
| exact (mul_assoc _ _ _).symm |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
