| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.CellAffinity |
| import OAI.Analysis.HardSphere.Allocation.AllocatedLinearity |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
|
|
| lemma measureAffinity_map_equiv {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (e : X≃ᵐY) (P Q : Measure X) [IsFiniteMeasure P] [IsFiniteMeasure Q] : |
| measureAffinity (P.map e) (Q.map e)=measureAffinity P Q := by |
| unfold measureAffinity affinity |
| rw [←Measure.map_add _ _ e.measurable,e.measurableEmbedding.integral_map] |
| apply integral_congr_ae |
| filter_upwards [e.measurableEmbedding.rnDeriv_map P (P+Q),e.measurableEmbedding.rnDeriv_map Q (P+Q)] with x hx hy |
| rw [hx,hy] |
|
|
| lemma actual_common_gain {X : Type*} [MeasurableSpace X] (σ P Q : Measure X) |
| [IsFiniteMeasure σ] [IsFiniteMeasure P] [IsFiniteMeasure Q] |
| {B : ℝ} (hB : 1 ≤ B) (hm : (1/2:ℝ≥0∞) ≤ σ univ) |
| (hp : σ≪P) (hq : σ≪Q) |
| (hip : Integrable (fun z ↦ (σ.rnDeriv P z).toReal^2) P) |
| (hiq : Integrable (fun z ↦ (σ.rnDeriv Q z).toReal^2) Q) |
| (hmp : (∫z,(σ.rnDeriv P z).toReal^2 ∂P) ≤ 2*B) |
| (hmq : (∫z,(σ.rnDeriv Q z).toReal^2 ∂Q) ≤ 2*B) : |
| 1/(8*B) ≤ measureAffinity P Q := by |
| have hm' : (1/2:ℝ) ≤ σ.real univ := by |
| have H := ENNReal.toReal_mono (measure_ne_top σ univ) hm |
| norm_num [Measure.real] at H ⊢ |
| exact H |
| have h := common_measure_affinity hp hq (by linarith : 0 < σ.real univ) hip hiq hmp hmq |
| rw [show (2*B)*(2*B)=(2*B)^2 by ring,Real.sqrt_sq (by linarith : 0 ≤ 2*B)] at h |
| refine le_trans ?_ h |
| apply (div_le_div_iff₀ (by linarith : 0 < 8*B) (by linarith : 0 < 2*B)).mpr |
| have hm2 : (1/4:ℝ) ≤ (σ.real univ)^2 := by nlinarith |
| nlinarith |
|
|
| end HardSphere.Comparison |
|
|
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
|
|
| lemma physical_cell_posts_sum {N K : ℕ} [NeZero K] {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1 ≤ K) (b : Bool) |
| (P : Measure (Torus L×Configuration N L)) [SFinite P] : |
| (∑tag : Fin 3→Fin K,∑extra : Fin 3→Fin K, |
| physicalCellVolumePost L extra tag b ∘ₘ P)= |
| (P.prod volume).map (cellObservation b) := by |
| simp_rw [physicalCellVolumePost,←Measure.map_comp _ _ (cellObservation_measurable b), |
| cellExtraKernel_comp _ volume _ _ (centeredCell_measurable L _) (centeredCell_measurable L _)] |
| simp_rw [←Measure.map_finset_sum' (cellObservation_measurable b).aemeasurable] |
| congr 1 |
| have he (tag extra : Fin 3→Fin K) : |
| {p : (Torus L×Configuration N L)×Torus L | |
| p.1.1∈centeredCell L (gridCenter L tag) ∧ p.2∈centeredCell L (gridCenter L extra)}= |
| {p | (gridHome hL p.1.1,gridHome hL p.2)=(tag,extra)} := by |
| ext p |
| simp only [Set.mem_ofPred_eq,Prod.mk.injEq,gridHome_eq_iff hL hK] |
| simp_rw [he] |
| rw [←Fintype.sum_prod_type (fun ij : (Fin 3→Fin K)×(Fin 3→Fin K) ↦ |
| (P.prod volume).restrict {p | (gridHome hL p.1.1,gridHome hL p.2)=ij})] |
| apply sum_restrict_partition |
| intro i |
| exact (((gridHome_measurable hL).comp (measurable_fst.comp measurable_fst)).prodMk |
| ((gridHome_measurable hL).comp measurable_snd)) (measurableSet_singleton i) |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
