| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.ProductNormalize |
| import OAI.Analysis.HardSphere.Geometry.FrozenChecks |
|
|
| 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_update_self (U : Finset I) (x : ∀i,X i) (u : ∀i:U,X i) : |
| freezeOutside U x (Function.updateFinset x U u)=Function.updateFinset x U u := by |
| funext i |
| by_cases hi : i∈U <;> simp [freezeOutside,Function.updateFinset,hi] |
|
|
| omit [Fintype C] in |
| lemma frozen_local_mean (checks : C→LocalCheck X) (μ : ∀i,Measure (X i)) |
| (U : Finset I) (x : ∀i,X i) (hx : x∈retainedEvent checks (Finset.univ\U)) |
| (F : (∀i,X i)→ℝ≥0∞) : |
| lmarginal μ U (fun y ↦ checkWeight (frozenChecks checks U x) U y*F (freezeOutside U x y)) x / |
| checkNormalizer (frozenChecks checks U x) μ U x= |
| lmarginal μ U (fun y ↦ checkWeight checks Finset.univ y*F y) x / |
| lmarginal μ U (checkWeight checks Finset.univ) x := by |
| have hU : U∪(Finset.univ\U)=Finset.univ := Finset.union_sdiff_of_subset (Finset.subset_univ _) |
| unfold checkNormalizer lmarginal |
| congr 1 <;> apply lintegral_congr <;> intro u <;> (try dsimp only) <;> |
| rw [frozen_checkWeight checks U U _ _ hx,hU,freezeOutside_update_self] |
|
|
| lemma local_mean_integrates (checks : C→LocalCheck X) (μ : ∀i,Measure (X i)) |
| [∀i,IsProbabilityMeasure (μ i)] (U : Finset I) (F : (∀i,X i)→ℝ≥0∞) |
| (hF : Measurable F) (A : ℝ≥0∞) (hA : A≠⊤) |
| (hlocal : ∀x,x∈retainedEvent checks (Finset.univ\U) → |
| lmarginal μ U (fun y ↦ checkWeight checks Finset.univ y*F y) x / |
| lmarginal μ U (checkWeight checks Finset.univ) x≤A) : |
| (∫⁻x,F x ∂conditionedMeasure (Measure.pi μ) (retainedEvent checks Finset.univ))≤A := by |
| let w := checkWeight checks Finset.univ |
| have hw : Measurable w := measurable_checkWeight checks _ |
| have hraw : (∫⁻x,w x*F x ∂Measure.pi μ)≤A*∫⁻x,w x ∂Measure.pi μ := by |
| rw [←lintegral_const_mul _ hw] |
| apply lintegral_le_of_lmarginal_le U (f:=fun y ↦ w y*F y) (g:=fun y ↦ A*w y) (hw.mul hF) (measurable_const.mul hw) |
| intro x |
| have hden : lmarginal μ U w x≠⊤ := by |
| have hd : lmarginal μ U w x≤1 := by |
| calc |
| _≤lmarginal μ U (fun _ ↦ 1) x := lmarginal_mono (checkWeight_le_one checks _) x |
| _=1 := by simp [lmarginal] |
| exact ne_top_of_le_ne_top ENNReal.one_ne_top hd |
| by_cases hx : x∈retainedEvent checks (Finset.univ\U) |
| · have h := (ENNReal.div_le_iff_le_mul (Or.inr hA) (Or.inl hden)).mp (hlocal x hx) |
| change _≤∫⁻u,A*w (Function.updateFinset x U u) ∂Measure.pi (fun i:U ↦ μ i) |
| rw [lintegral_const_mul _ (show Measurable (fun u : ∀i:U,X i ↦ w (Function.updateFinset x U u)) from hw.comp measurable_updateFinset)] |
| exact h |
| · have hz : lmarginal μ U (fun y ↦ w y*F y) x=0 := by |
| unfold lmarginal |
| apply lintegral_eq_zero_of_ae_eq_zero |
| apply ae_of_all |
| intro u |
| dsimp only |
| have hout : checkWeight checks (Finset.univ\U) (Function.updateFinset x U u)=0 := by |
| rw [checkWeight_depends checks (Finset.univ\U) (x:=Function.updateFinset x U u) (y:=x) |
| (fun i hi ↦ by simp [Function.updateFinset,(Finset.mem_sdiff.mp hi).2])] |
| exact Set.indicator_of_notMem hx _ |
| have hfull : w (Function.updateFinset x U u)=0 := le_antisymm |
| ((checkWeight_antitone checks (Finset.subset_univ (Finset.univ\U)) _).trans_eq hout) bot_le |
| simp only [hfull,zero_mul,Pi.zero_apply] |
| rw [hz] |
| exact bot_le |
| let Z := (Measure.pi μ) (retainedEvent checks Finset.univ) |
| have hZ : (∫⁻x,w x ∂Measure.pi μ)=Z := by |
| exact (lintegral_indicator_const (measurableSet_retainedEvent checks _) (1:ℝ≥0∞)).trans (one_mul _) |
| have hint : (∫⁻x,F x ∂conditionedMeasure (Measure.pi μ) (retainedEvent checks Finset.univ))= |
| Z⁻¹*∫⁻x,w x*F x ∂Measure.pi μ := by |
| rw [conditionedMeasure,lintegral_smul_measure,←lintegral_indicator (measurableSet_retainedEvent checks _)] |
| congr 1 |
| apply lintegral_congr |
| intro x |
| by_cases hx : x∈retainedEvent checks Finset.univ <;> simp [w,checkWeight,hx] |
| rw [hint] |
| calc |
| _≤Z⁻¹*(A*Z) := mul_le_mul' (le_refl _) (hraw.trans_eq (congrArg (A*·) hZ)) |
| _≤A := by |
| by_cases hz : Z=0 |
| · simp [hz] |
| · have ht : Z≠⊤ := measure_ne_top _ _ |
| calc |
| _=A*(Z⁻¹*Z) := by ac_rfl |
| _≤A := by rw [ENNReal.inv_mul_cancel hz ht,mul_one] |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
