| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.Allocation |
| import OAI.Analysis.HardSphere.PathBounds.InteractingVisits |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.FreeVisits |
| open HardSphere.Wiener HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital |
|
|
| def visitorCount {n K : ℕ} (L : Length) (E : Set (ConfigurationPath 1 L)) |
| (v : DiscreteTorus K) (ω : ConfigurationPath n L) : ℕ := by |
| classical |
| exact (Finset.univ.filter (fun i : Fin n ↦ onePathAt i ω∈E ∧ pathVisitsBall L 12 (siteCenter L v) i ω)).card |
|
|
| lemma visitorCount_sum {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| (E : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath n L) : |
| (∑v∈A,visitorCount L E v ω)=defectiveTotalVisits L A E ω := by |
| classical |
| simp only [visitorCount,Finset.card_filter] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| by_cases hi : onePathAt i ω∈E |
| · simp only [hi,true_and,pathVisitedSites,ite_eq_left,Finset.card_filter] |
| · simp only [hi,false_and,ite_false,Finset.sum_const_zero,pathVisitedSites] |
|
|
| lemma visitorCount_measurable {n K : ℕ} (L : Length) {E : Set (ConfigurationPath 1 L)} |
| (hE : MeasurableSet E) (v : DiscreteTorus K) : Measurable (visitorCount (n:=n) L E v) := by |
| classical |
| have he : visitorCount (n:=n) L E v=(fun ω ↦ ∑i : Fin n, if onePathAt i ω∈E ∧ pathVisitsBall L 12 (siteCenter L v) i ω then (1:ℕ) else 0) := by |
| funext ω |
| simp only [visitorCount,Finset.card_filter] |
| rw [he] |
| apply Finset.measurable_fun_sum |
| intro i _ |
| have hi : MeasurableSet {ω : ConfigurationPath n L | onePathAt i ω∈E} := hE.preimage (onePathAt_continuous i).measurable |
| exact measurable_const.ite (hi.inter (pathVisitsBall_isClosed L 12 (siteCenter L v) i).measurableSet) measurable_const |
|
|
| lemma defectiveTotalVisits_measurable {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| {E : Set (ConfigurationPath 1 L)} (hE : MeasurableSet E) : Measurable (defectiveTotalVisits (n:=n) L A E) := by |
| have he : defectiveTotalVisits (n:=n) L A E=fun ω ↦ ∑v∈A,visitorCount L E v ω := by |
| funext ω; exact (visitorCount_sum L A E ω).symm |
| rw [he] |
| exact Finset.measurable_fun_sum _ (fun v _ ↦ visitorCount_measurable L hE v) |
|
|
| lemma joint_visitor_sum {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| (E : Set (ConfigurationPath 1 L)) {H : ℝ} {ω : ConfigurationPath n L} |
| (h : ∀v∈A,H≤(visitorCount L E v ω:ℝ)) : |
| H*(A.card:ℝ)≤(defectiveTotalVisits L A E ω:ℝ) := by |
| rw [←visitorCount_sum L A E ω,Nat.cast_sum] |
| have hs := Finset.sum_le_sum h |
| simpa only [Finset.sum_const,nsmul_eq_mul,mul_comm] using hs |
|
|
| lemma joint_visitors_markov {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| {E : Set (ConfigurationPath 1 L)} (hE : MeasurableSet E) |
| (μ : Measure (ConfigurationPath n L)) {η H Q : ℝ} (hη : 0≤η) |
| (hmgf : (∫⁻ω,ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L A E ω:ℝ))) ∂μ)≤ |
| ENNReal.ofReal (Real.exp (Q*(A.card:ℝ)))) : |
| μ {ω | ∀v∈A,H≤(visitorCount L E v ω:ℝ)}≤ENNReal.ofReal (Real.exp ((Q-η*H)*(A.card:ℝ))) := by |
| let F := fun ω : ConfigurationPath n L ↦ ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L A E ω:ℝ))) |
| have hc : Measurable (fun k : ℕ ↦ (k:ℝ)) := measurable_of_countable _ |
| have hF : Measurable F := ((measurable_const.mul (hc.comp (defectiveTotalVisits_measurable L A hE))).exp).ennreal_ofReal |
| have hp : ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ))))≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr (Real.exp_pos _)) |
| have hm := meas_ge_le_lintegral_div (μ:=μ) hF.aemeasurable hp ENNReal.ofReal_ne_top |
| calc |
| _ ≤ μ {ω | ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ))))≤F ω} := by |
| apply measure_mono |
| intro ω hω |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonneg_left (joint_visitor_sum L A E hω) hη |
| _ ≤ _ := hm |
| _ ≤ ENNReal.ofReal (Real.exp (Q*(A.card:ℝ)))/ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ)))) := ENNReal.div_le_div_right hmgf _ |
| _ = _ := by |
| rw [←ENNReal.ofReal_div_of_pos (Real.exp_pos _),←Real.exp_sub] |
| congr 2 |
| ring |
|
|
| end HardSphere.FreeVisits |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Groups |
| open Wiener CellOrbital PathCutoff FreeVisits |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma centeredCell_distance {L : Length} {c x : Torus L} (hx : x∈centeredCell L c) : |
| torusDistance L x c≤1 := by |
| have hd (j : Fin 3) : dist (x j) (c j)≤(1/2:ℝ) := by |
| rw [dist_eq_norm,←centered_rep_abs] |
| exact abs_le.mpr ⟨by linarith [(hx j).1], (hx j).2.le⟩ |
| unfold torusDistance |
| apply Real.sqrt_le_one.mpr |
| calc |
| _ ≤ ∑ _j : Fin 3,(1/4:ℝ) := Finset.sum_le_sum (fun j _ ↦ by nlinarith [dist_nonneg (x:=x j) (y:=c j),hd j]) |
| _ ≤ 1 := by norm_num |
|
|
| @[simp] lemma intervalTime_zero_one (s : Time) : intervalTime 0 1 s=s := by |
| apply Subtype.ext |
| simp [intervalTime] |
|
|
| lemma smallDiameter_distance {L : Length} (ω : ConfigurationPath 1 L) |
| (h : intervalDiameter 0 1 ω≤1) (s q : Time) : torusDistance L (ω s 0) (ω q 0)≤1 := by |
| have hh := (intervalDistance_le 0 1 s q ω).trans h |
| simpa only [intervalTime_zero_one] using hh |
|
|
| def longVisitors {N : ℕ} {L : Length} (c : Torus L) (ω : ConfigurationPath N L) : Finset (Fin N) := |
| Finset.univ.filter (fun i ↦ pathVisitsBall L 12 c i ω ∧ 1 < intervalDiameter 0 1 (onePathAt i ω)) |
|
|
| def visitors {N : ℕ} {L : Length} (c : Torus L) (ω : ConfigurationPath N L) : Finset (Fin N) := |
| Finset.univ.filter (fun i ↦ pathVisitsBall L 12 c i ω) |
|
|
| structure Certificate {N K : ℕ} [Nontrivial (Fin N)] {L : Length} |
| (m : ℕ) (r R C S s₀ ell θ H D : ℝ) (u : Time) (v : Fin 3→Fin K) |
| (ω : ConfigurationPath N L) : Prop where |
| low : ell*D≤(centeredCount L (gridCenter L v) (ω 0):ℝ) |
| closeY : ((localCloseLabels L s₀ (gridCenter L v) (ω 0)).card:ℝ)<θ*D |
| closeZ : ((localCloseLabels L s₀ (gridCenter L v) (ω 1)).card:ℝ)<θ*D |
| visit : ((visitors (gridCenter L v) ω).card:ℝ)≤H*D |
| regular : ∀i,pathVisitsBall L 12 (gridCenter L v) i ω→Regular m r R C u S (onePathAt i ω) |
| long : ((longVisitors (gridCenter L v) ω).card:ℝ)<θ*D |
|
|
| lemma eligible_of_not_discarded {N K : ℕ} [Nontrivial (Fin N)] {L : Length} |
| (m : ℕ) (r R C S s₀ : ℝ) (u : Time) (v : Fin 3→Fin K) (ω : ConfigurationPath N L) |
| (hreg : ∀i,pathVisitsBall L 12 (gridCenter L v) i ω→Regular m r R C u S (onePathAt i ω)) |
| (i : Fin N) (hy : ω 0 i∈centeredCell L (gridCenter L v)) |
| (hY : i∉localCloseLabels L s₀ (gridCenter L v) (ω 0)) |
| (hZ : i∉localCloseLabels L s₀ (gridCenter L v) (ω 1)) |
| (hshort : i∉longVisitors (gridCenter L v) ω) : Eligible m r R C S s₀ u v i ω := by |
| have hyc := centeredCell_distance hy |
| have hv : pathVisitsBall L 12 (gridCenter L v) i ω := ⟨0,by linarith⟩ |
| have hsmall : intervalDiameter 0 1 (onePathAt i ω)≤1 := by |
| by_contra! h |
| exact hshort (Finset.mem_filter.mpr ⟨Finset.mem_univ _,hv,h⟩) |
| have hd (s : Time) : torusDistance L (ω s i) (ω 0 i)≤1 := smallDiameter_distance (onePathAt i ω) hsmall s 0 |
| have hc (s : Time) : torusDistance L (ω s i) (gridCenter L v)≤2 := by |
| have hh := torusDistance_triangle' L (ω s i) (ω 0 i) (gridCenter L v) |
| linarith [hd s] |
| have hNY : s₀<nearestParticleDistance L (ω 0) i := by |
| by_contra! hh |
| apply hY |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,by linarith,hh⟩ |
| have hNZ : s₀<nearestParticleDistance L (ω 1) i := by |
| by_contra! hh |
| apply hZ |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,by linarith [hc 1],hh⟩ |
| refine ⟨⟨hy,by linarith [hc 1],?_⟩,hreg i hv,?_⟩ |
| · intro j hji |
| exact ⟨hNY.trans_le (nearestParticleDistance_le L (ω 0) i j hji), |
| hNZ.trans_le (nearestParticleDistance_le L (ω 1) i j hji)⟩ |
| · intro s |
| exact (hc s).trans (by norm_num) |
|
|
| lemma certificate_eligible_count {N K : ℕ} [Nontrivial (Fin N)] {L : Length} |
| (m : ℕ) (r R C S s₀ ell θ H D : ℝ) (u : Time) (v : Fin 3→Fin K) |
| (ω : ConfigurationPath N L) (h : Certificate m r R C S s₀ ell θ H D u v ω) : |
| (ell-3*θ)*D≤((eligibleList m r R C S s₀ u v ω).card:ℝ) := by |
| let E := eligibleList m r R C S s₀ u v ω |
| let A := localCloseLabels L s₀ (gridCenter L v) (ω 0) |
| let B := localCloseLabels L s₀ (gridCenter L v) (ω 1) |
| let T := longVisitors (gridCenter L v) ω |
| have hsub : Finset.univ.filter (fun i ↦ ω 0 i∈centeredCell L (gridCenter L v))⊆((E∪A)∪B)∪T := by |
| intro i hi |
| by_cases he : i∈E |
| · simp [he] |
| by_cases ha : i∈A |
| · simp [ha] |
| by_cases hb : i∈B |
| · simp [hb] |
| by_cases ht : i∈T |
| · simp [ht] |
| exfalso |
| exact he (Finset.mem_filter.mpr ⟨Finset.mem_univ _,eligible_of_not_discarded m r R C S s₀ u v ω |
| h.regular i (Finset.mem_filter.mp hi).2 ha hb ht⟩) |
| have hcard : centeredCount L (gridCenter L v) (ω 0)≤E.card+A.card+B.card+T.card := by |
| apply (Finset.card_le_card hsub).trans |
| exact (Finset.card_union_le _ _).trans (Nat.add_le_add_right |
| ((Finset.card_union_le _ _).trans (Nat.add_le_add_right (Finset.card_union_le _ _) _)) _) |
| have hc : (centeredCount L (gridCenter L v) (ω 0):ℝ)≤(E.card:ℝ)+(A.card:ℝ)+(B.card:ℝ)+(T.card:ℝ) := by exact_mod_cast hcard |
| have h1 := h.low |
| have h2 := h.closeY |
| have h3 := h.closeZ |
| have h4 := h.long |
| change _≤(E.card:ℝ) |
| change (A.card:ℝ)<θ*D at h2 |
| change (B.card:ℝ)<θ*D at h3 |
| change (T.card:ℝ)<θ*D at h4 |
| nlinarith |
|
|
| end HardSphere.Groups |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
