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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | 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
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