| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalCodedMoment |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Groups |
| open Comparison HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| variable {J : Type*} [Fintype J] [DecidableEq J] [MeasurableSpace J] [MeasurableSingletonClass J] |
|
|
| lemma padded_label_encounter (p q : SimplePath x y) (s : DiscreteTorus K→Finset J) |
| (home : J→DiscreteTorus K) |
| (hn : ∀v∈departureSites p∪departureSites q,(s v).Nonempty) |
| (hh : ∀v∈departureSites p∪departureSites q,∀j∈s v,home j=v) |
| (G M : ℝ) (hG : 1≤G) (hM : 0<M) |
| (hm : ∀v∈departureSites p∩departureSites q,M≤2*((s v).card:ℝ)) (R : ℕ) : |
| (∫⁻z,∫⁻w,(ENNReal.ofReal G)^(1+(fieldLabelImage p z∩fieldLabelImage q w).card) |
| ∂Measure.pi (fun v ↦ uniformLabels (s v)) ∂Measure.pi (fun v ↦ uniformLabels (s v)))≤ |
| ENNReal.ofReal (G*Real.exp ((2*(G-1)/M)*(simpleEncounter R p q:ℝ))) := by |
| have G0 : 0≤G := by linarith |
| rw [←lintegral_prod (fun z : (DiscreteTorus K→Option J)×(DiscreteTorus K→Option J) ↦ |
| (ENNReal.ofReal G)^(1+(fieldLabelImage p z.1∩fieldLabelImage q z.2).card)) |
| (measurable_of_countable _).aemeasurable] |
| simp_rw [←ENNReal.ofReal_pow G0] |
| rw [←ofReal_integral_eq_lintegral_ofReal Integrable.of_finite |
| (ae_of_all _ (fun _ ↦ pow_nonneg G0 _))] |
| apply ENNReal.ofReal_le_ofReal |
| apply (padded_label_overlap p q s home hn hh G M hG hM hm).trans |
| apply mul_le_mul_of_nonneg_left _ G0 |
| apply Real.exp_le_exp.mpr |
| apply mul_le_mul_of_nonneg_left |
| · exact_mod_cast shared_departures_le_encounters R p q |
| · positivity |
|
|
| lemma encounter_absorb (G M κ δ : ℝ) (hG : 1≤G) (_hM : 0<M) (hδ : 0<δ) |
| (hκ : 2*(G-1)/M-8*Real.log δ≤κ) (n : ℕ) : |
| ENNReal.ofReal (G*Real.exp ((2*(G-1)/M)*(n:ℝ))) * |
| (ENNReal.ofReal δ)⁻¹^(8*n) ≤ ENNReal.ofReal (G*Real.exp (κ*(n:ℝ))) := by |
| have G0 : 0≤G := by linarith |
| have he : (δ⁻¹)^(8*n)=Real.exp ((-8*Real.log δ)*(n:ℝ)) := by |
| have H := Real.exp_nat_mul (-Real.log δ) (8*n) |
| rw [Real.exp_neg,Real.exp_log hδ] at H |
| exact H.symm.trans (congrArg Real.exp (by push_cast; ring)) |
| rw [←ENNReal.ofReal_inv_of_pos hδ,←ENNReal.ofReal_pow (inv_nonneg.mpr hδ.le), |
| ←ENNReal.ofReal_mul (mul_nonneg G0 (Real.exp_nonneg _)),he] |
| apply ENNReal.ofReal_le_ofReal |
| rw [mul_assoc,←Real.exp_add] |
| apply mul_le_mul_of_nonneg_left _ G0 |
| apply Real.exp_le_exp.mpr |
| nlinarith [Nat.cast_nonneg (α:=ℝ) n] |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| variable {J : Type*} [Fintype J] [DecidableEq J] [MeasurableSpace J] [MeasurableSingletonClass J] |
|
|
| theorem physicalLabel_moment {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (hr : r≤1) (u : Time) |
| (particle : J↪Fin N) (d : PhysicalDatum N K M G L) (hxy : x≠y) |
| (p p' : SimplePath x y) (hM : 0<M) |
| (hsupply : ∀v∈departureSites p∪departureSites p', |
| (M:ℝ)≤2*((branchAccessible hL g m t r R C S u particle v d).card:ℝ)) |
| (hm : ∀w,(Finset.univ.filter (fun j ↦ datumHome hL d (particle j)=w)).card≤M) |
| (hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) |
| (q : ℝ≥0∞) (δ B κ : ℝ) (hδ : 0<δ) (hB : 1≤B) (hBt : routeDensityBound t≤ENNReal.ofReal B) |
| (hκ : 2*(B-1)/(M:ℝ)-8*Real.log δ≤κ) |
| (hq : ∀ j k, j ≠ k → ∀f : physicalSlotAllowed hL g m r R C S u particle d (some k), |
| physicalCollisionReference hL g m t r R C S u particle d j |
| {p | TrialCollision r (projectCurve L f.val) p.val}≤q) |
| (hsmall : ENNReal.ofReal δ+(4*(49^3*(M+1)):ℕ)*(routeDensityBound t*q)≤1) (b : Bool) : |
| (∫⁻z,∫⁻z',∫⁻w,physicalRouteLikelihood hL g m t r R C S u particle hxy b (d,(p,z)) w * |
| physicalRouteLikelihood hL g m t r R C S u particle hxy b (d,(p',z')) w |
| ∂physicalPrimaryReference hL g m t r R C S u particle (siteIndex (if b then y else x)) d |
| ∂physicalLabels hL g m t r R C S u particle d |
| ∂physicalLabels hL g m t r R C S u particle d)≤ |
| ENNReal.ofReal (B*Real.exp (κ*(simpleEncounter 30 p p':ℝ))) := by |
| let s := fun v ↦ branchAccessible hL g m t r R C S u particle v d |
| have hM' : (0:ℝ)<M := by exact_mod_cast hM |
| have hn : ∀v∈departureSites p∪departureSites p',(s v).Nonempty := by |
| intro v hv |
| apply Finset.card_pos.mp |
| have H := hsupply v hv |
| have : (0:ℝ)<(s v).card := by dsimp [s]; linarith |
| exact_mod_cast this |
| have hhome : ∀v∈departureSites p∪departureSites p',∀j∈s v, |
| indexSite (datumHome hL d (particle j))=v := by |
| intro v _ j hj |
| rw [(branchAccessible_home hL g m t r R C S u particle v d j hj).1] |
| exact indexSite_siteIndex v |
| have ha : ∀i : Fin p.val.length,(s (pathVertexEmbedding p i.castSucc)).Nonempty := by |
| intro i |
| exact hn _ (Finset.mem_union_left _ (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)) |
| have ha' : ∀i : Fin p'.val.length,(s (pathVertexEmbedding p' i.castSucc)).Nonempty := by |
| intro i |
| exact hn _ (Finset.mem_union_right _ (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)) |
| have Hav := padded_label_encounter p p' s (fun j ↦ indexSite (datumHome hL d (particle j))) |
| hn hhome B M hB hM' (fun v hv ↦ hsupply v (Finset.mem_union_left _ (Finset.mem_inter.mp hv).1)) 30 |
| have hlabels : physicalLabels hL g m t r R C S u particle d=Measure.pi (fun v ↦ uniformLabels (s v)) := by |
| rw [physicalLabels, finiteKernelPi_apply] |
| rfl |
| calc |
| _ ≤ ∫⁻z,∫⁻z',(ENNReal.ofReal B)^(1+(fieldLabelImage p z∩fieldLabelImage p' z').card)* |
| (ENNReal.ofReal δ)⁻¹^(8*simpleEncounter 30 p p') |
| ∂physicalLabels hL g m t r R C S u particle d |
| ∂physicalLabels hL g m t r R C S u particle d := by |
| apply lintegral_mono_ae |
| filter_upwards [physicalLabels_supported hL g m t r R C S u particle d] with z hz |
| apply lintegral_mono_ae |
| filter_upwards [physicalLabels_supported hL g m t r R C S u particle d] with z' hz' |
| apply (physicalCoded_moment hL g m t r R C S ht hr u particle d hxy p p' z z' hz hz' |
| ha ha' hm hp q (ENNReal.ofReal δ) (ENNReal.ofReal_pos.mpr hδ) hq hsmall b).trans |
| exact mul_le_mul' (pow_le_pow_left' hBt _) le_rfl |
| _ = (∫⁻z,∫⁻z',(ENNReal.ofReal B)^(1+(fieldLabelImage p z∩fieldLabelImage p' z').card) |
| ∂physicalLabels hL g m t r R C S u particle d |
| ∂physicalLabels hL g m t r R C S u particle d) * |
| (ENNReal.ofReal δ)⁻¹^(8*simpleEncounter 30 p p') := by |
| simp_rw [lintegral_mul_const _ (measurable_of_countable _)] |
| _ ≤ ENNReal.ofReal (B*Real.exp ((2*(B-1)/M)*(simpleEncounter 30 p p':ℝ))) * |
| (ENNReal.ofReal δ)⁻¹^(8*simpleEncounter 30 p p') := by |
| apply mul_le_mul' _ le_rfl |
| rwa [hlabels] |
| _ ≤ _ := encounter_absorb B M κ δ hB hM' hδ hκ _ |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
