| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.ReferenceResample |
| import OAI.Analysis.HardSphere.Allocation.PhysicalAssembly |
| import OAI.Analysis.HardSphere.Allocation.PhysicalDatum |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge GaussianPacking PathCutoff CellOrbital |
|
|
| lemma nearestLift_eq_of_near {L : Length} (hL : 50<L.val) (c y : Point) |
| (hy : ‖y-c‖≤3) : nearestLift L c (projectPoint L y)=y := by |
| ext j |
| have hj : |y j-c j|≤3 := (norm_le_pi_norm (y-c) j).trans hy |
| have hmem : y j-c j∈Set.Ico (-L.val/2) (-L.val/2+L.val) := by |
| rw [abs_le] at hj |
| constructor <;> linarith |
| change ((AddCircle.equivIco L.val (-L.val/2)) ((y j:AddCircle L.val)-(c j:AddCircle L.val))).val+c j=y j |
| rw [←AddCircle.coe_sub,AddCircle.equivIco_coe_eq hmem] |
| simp |
|
|
| lemma near_lift_unique {L : Length} (hL : 50<L.val) (c y : Point) (x : Torus L) |
| (hp : projectPoint L y=x) (hy : ‖y-c‖≤3) : y=nearestLift L c x := by |
| subst x |
| exact (nearestLift_eq_of_near hL c y hy).symm |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem canonical_certificate_useful_list {ell b Alo A H : ℝ} |
| (hell : 0 < ell) (hb : 0 < b) (hAlo : 0 < Alo) (hA : 0 < A) (hH : 0 < H) : |
| ∃t₀ : ℝ,0 < t₀ ∧ ∀t : ℝ,0 < t → t ≤ t₀ → |
| ∃D₀ : ℝ,1 ≤ D₀ ∧ ∀D α : ℝ,D₀ ≤ D → Alo ≤ α → α ≤ A → |
| ∀(N K m n : ℕ) [Nontrivial (Fin N)] (L : Length),50 < L.val → |
| n=⌈t/(α/D)^2⌉₊ → ∀(w : Time),t*(w:ℝ)=D⁻¹ → |
| ∀(θ : ℝ),θ ≤ ell/6 → |
| ∀(v : Fin 3→Fin K) (c : Point),projectPoint L c=gridCenter L v → |
| ∀(ω : ConfigurationPath N L), |
| Certificate (n+n) (α/D) (D^(1/12:ℝ)) (64*(meanCost+1)) |
| (b*D^(-(1/3):ℝ)/8) (b*D^(-(1/3):ℝ)) ell θ H D (prefixTime midpointTime w) v ω → |
| ∀(label : Fin m→Fin N),Function.Injective label → |
| Finset.univ.image label=eligibleList (n+n) (α/D) (D^(1/12:ℝ)) (64*(meanCost+1)) |
| (b*D^(-(1/3):ℝ)/8) (b*D^(-(1/3):ℝ)) (prefixTime midpointTime w) v ω → |
| ∀(B : Fin 7→Set Point),(∀j,MeasurableSet (B j)) → (∀j,volume (B j)=1) → |
| (∀j u,u∈B j → ‖u-c‖ ≤ 3) → |
| ((Finset.univ.filter (fun i ↦ ¬StrongExcept n t (α/D) (D^(1/12:ℝ)) (64*(meanCost+1)) |
| (b*D^(-(1/3):ℝ)/8) w (nearestLift L c (ω 0 (label i))) (nearestLift L c (ω 1 (label i))) |
| c (fun j ↦ onePathAt j ω) (label i) B)).card:ℝ) ≤ (m:ℝ)/10 := by |
| obtain ⟨cG,MG,hcG,hMG,htail⟩ := wiener_norm_gaussian_tail |
| have htail' (R : ℝ) (hR : 0 ≤ R) : |
| wiener {β | R ≤ ‖β‖} ≤ ENNReal.ofReal MG.toReal*ENNReal.ofReal (Real.exp (-cG*R^2)) := by |
| rw [ENNReal.ofReal_toReal hMG.ne] |
| exact htail R hR |
| obtain ⟨t₀,ht₀,hs⟩ := physical_useful_list hell hb hA hH |
| refine ⟨t₀,ht₀,?_⟩ |
| intro t ht htt |
| obtain ⟨D₁,hD₁,hs⟩ := hs t ht htt (t/Alo^2+1) MG.toReal cG (t/A^2) |
| ENNReal.toReal_nonneg hcG (by positivity) htail' |
| obtain ⟨D₂,hgeom⟩ := eventually_atTop.1 (source_trial_geometry hA hb ht) |
| refine ⟨max D₁ D₂,hD₁.trans (le_max_left _ _),?_⟩ |
| intro D α hDD hαlo hαA N K m n _ L hL hn w hw θ hθ v c hc ω cert label hinj himage B hB hBvol hBv |
| subst n |
| have h1 : D₁ ≤ D := (le_max_left _ _).trans hDD |
| have h2 : D₂ ≤ D := (le_max_right _ _).trans hDD |
| have hα := hAlo.trans_le hαlo |
| rcases hgeom D h2 α hα hαA with |
| ⟨hD,hn,hr1,hsmall,hτ,hnscale,hw0,hw1,hwindow,hlarge,hdrift,hdrift',hwin,hrelax,hiso⟩ |
| have hDp : 0 < D := zero_lt_one.trans_le hD |
| have hr : 0 < α/D := by positivity |
| have hDa : D*(α/D)=α := by field_simp |
| have hbnd := sourceHalf_bounds ht hr hD hAlo |
| (show Alo ≤ D*(α/D) by rw [hDa]; exact hαlo) |
| (show D*(α/D) ≤ A by rw [hDa]; exact hαA) |
| obtain ⟨hm,hmH,hel⟩ := eligible_list_size _ _ _ _ _ _ ell θ H D hDp.le hθ _ v ω cert label hinj himage |
| obtain ⟨y,z,hyp,hzp,hy,hz,hsepY,hsepZ,hj,hreg,hisoY,hisoZ,hfar⟩ := |
| certificate_trial_geometry hL _ _ _ _ _ _ ell θ H D (by positivity) _ v c hc ω cert label hinj hel hiso |
| have hwd : (w:ℝ)=D⁻¹/t := (eq_div_iff ht.ne').mpr (by nlinarith [hw]) |
| have ey : y=(fun i ↦ nearestLift L c (ω 0 (label i))) := by |
| funext i |
| exact near_lift_unique hL c (y i) (ω 0 (label i)) (hyp i) (hy i) |
| have ez : z=(fun i ↦ nearestLift L c (ω 1 (label i))) := by |
| funext i |
| exact near_lift_unique hL c (z i) (ω 1 (label i)) (hzp i) (hz i) |
| have hmain := hs D α h1 hα hαA L N m _ hn hL hm hmH y z c (fun j ↦ onePathAt j ω) label |
| (visitors (gridCenter L v) ω) w B hj hbnd.1 hbnd.2 hr1 hsmall hτ hnscale hw |
| hwindow hlarge hdrift hdrift' (by rw [hwd]; exact hwin) hrelax hy hz hsepY hsepZ |
| hreg hisoY hisoZ hfar hB hBvol hBv |
| simpa only [ey,ez] using hmain |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma labelStartLift_project {N : ℕ} {L : Length} (e : BridgeEndpoints N L) (i : Fin N) : |
| projectPoint L (labelStartLift e i)=e.2 i := |
| congrFun (project_representative N L e.2) i |
|
|
| lemma labelEndLift_project {N : ℕ} {L : Length} (t : ℝ) (e : BridgeEndpoints N L) (i : Fin N) : |
| projectPoint L (labelEndLift t e i)=bridgeEnd N L t e i := by |
| ext j |
| change ((labelStartLift e i j+Real.sqrt (2*t)*e.1 (i,j):ℝ):AddCircle L.val)=_ |
| rw [AddCircle.coe_add] |
| have h := congrFun (labelStartLift_project e i) j |
| change ((labelStartLift e i j:ℝ):AddCircle L.val)=e.2 i j at h |
| rw [h] |
| rfl |
|
|
| lemma exposed_selected_near {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (hK : 1≤K) (h50 : 50<L.val) (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) (p : Fin N→Prop) (b : Fin N→Bool) |
| (hb : ∀i : {i // p i},b i=true) : |
| ∀ᵐw ∂exposedLaw μ (fun _ : Fin N ↦ Bridge.law) p |
| (labelInteraction N L r (2*t)∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b), |
| ∀i : {i // p i},‖labelEndLift (2*t) w.1 i-exposedAnchorLift hL w.1 i‖≤3 := by |
| filter_upwards [recorded_physical_fiber μ r (2*t) |
| (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) |
| (fun i ↦ eligibilityFlag_measurable m r R C S s₀ (2*t) u i) b p, |
| exposed_selected_endpoints hL hK μ m r R C S s₀ (2*t) u p b hb] with w hw he |
| intro i |
| apply near_end_of_positive_confined h50 (2*t) (labelStartLift w.1 i) |
| (labelEndLift (2*t) w.1 i) (exposedAnchorLift hL w.1 i) |
| (exposedAnchorLift_near hL hK w.1 i) |
| · rw [labelEndLift_project,exposedAnchorLift_project] |
| exact (he i).2.1 |
| · have hs : individualFiber p (noisePairAvoid r (2*t) w.1) |
| (fun j β ↦ eligibilityFlag (K:=K) m r R C S s₀ (2*t) u j (w.1,β)) b w.2 i ⊆ |
| {β | ∀s : Time,torusDistance L |
| (projectPoint L (leg (2*t) (labelStartLift w.1 i) (labelEndLift (2*t) w.1 i) β s)) |
| (projectPoint L (exposedAnchorLift hL w.1 i))≤10} := by |
| intro β hβ |
| have hf := hβ.1 |
| rw [hb i] at hf |
| have ho : ownEligible (K:=K) m r R C S s₀ (2*t) u i (w.1,β) := by |
| simpa only [eligibilityFlag,decide_eq_true_eq] using hf |
| rw [ownEligible_at hL hK m r R C S s₀ (2*t) u i w.1 β _ (he i)] at ho |
| intro s |
| rw [exposedAnchorLift_project] |
| exact ho.2 s |
| exact ne_zero_of_lt (lt_of_lt_of_le (pos_iff_ne_zero.mpr (hw.2.1 i)) (measure_mono hs)) |
|
|
| def canonicalShift {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (e : BridgeEndpoints N L) (i : Fin N) : Point := |
| anchorPoint (gridHome hL (e.2 i))-exposedAnchorLift hL e i |
|
|
| lemma canonicalShift_period {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (e : BridgeEndpoints N L) (i : Fin N) : projectPoint L (canonicalShift hL e i)=0 := by |
| ext j |
| change (((anchorPoint (gridHome hL (e.2 i)) j-exposedAnchorLift hL e i j):ℝ):AddCircle L.val)=0 |
| rw [AddCircle.coe_sub] |
| exact sub_eq_zero.mpr (congrFun ((anchorPoint_project L _).trans (exposedAnchorLift_project hL e i).symm) j) |
|
|
| lemma canonicalShift_anchor {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (e : BridgeEndpoints N L) (i : Fin N) : |
| exposedAnchorLift hL e i+canonicalShift hL e i=anchorPoint (gridHome hL (e.2 i)) := by |
| unfold canonicalShift |
| abel |
|
|
| lemma canonicalShift_start {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (hK : 1≤K) (h50 : 50<L.val) (e : BridgeEndpoints N L) (i : Fin N) : |
| labelStartLift e i+canonicalShift hL e i= |
| nearestLift L (anchorPoint (gridHome hL (e.2 i))) (e.2 i) := by |
| apply near_lift_unique h50 |
| · rw [projectPoint_shift _ (canonicalShift_period hL e i),labelStartLift_project] |
| · have he : labelStartLift e i+canonicalShift hL e i-anchorPoint (gridHome hL (e.2 i))= |
| labelStartLift e i-exposedAnchorLift hL e i := by unfold canonicalShift; abel |
| rw [he] |
| exact exposedAnchorLift_near hL hK e i |
|
|
| lemma canonicalShift_end {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (h50 : 50<L.val) (e : BridgeEndpoints N L) (i : Fin N) (t : ℝ) |
| (hz : ‖labelEndLift (2*t) e i-exposedAnchorLift hL e i‖≤3) : |
| labelEndLift (2*t) e i+canonicalShift hL e i= |
| nearestLift L (anchorPoint (gridHome hL (e.2 i))) (bridgeEnd N L (2*t) e i) := by |
| apply near_lift_unique h50 |
| · rw [projectPoint_shift _ (canonicalShift_period hL e i),labelEndLift_project] |
| · have he : labelEndLift (2*t) e i+canonicalShift hL e i-anchorPoint (gridHome hL (e.2 i))= |
| labelEndLift (2*t) e i-exposedAnchorLift hL e i := by unfold canonicalShift; abel |
| rwa [he] |
|
|
| lemma physicalMutual_shift_preimage {I : Type*} {L : Length} (r : ℝ) |
| (c : I→Point) (hc : ∀i,projectPoint L (c i)=0) : |
| (fun f : I→C(Time,Point) ↦ fun i ↦ Bridge.shiftPath (c i) (f i)) ⁻¹' |
| physicalMutual (L:=L) r=physicalMutual (L:=L) r := by |
| ext f |
| simp only [physicalMutual,pairEvent,mem_preimage,mem_ofPred_eq,project_shift _ (hc _)] |
| have he (i j : I) : TrialCollision (L:=L) r (projectCurve L (f j)) (Bridge.shiftPath (c i) (f i)) ↔ |
| TrialCollision (L:=L) r (projectCurve L (f j)) (f i) := by |
| simp only [TrialCollision,Bridge.shiftPath,ContinuousMap.add_apply,ContinuousMap.const_apply, |
| projectPoint_shift _ (hc i)] |
| simp only [he] |
|
|
| lemma canonical_reference_transport {N K : ℕ} {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) |
| (m : ℕ) (r R C S t : ℝ) (u : Time) (e : BridgeEndpoints N L) |
| (p : Fin N→Prop) (w : {i // ¬p i}→Bridge.Noise) |
| (hz : ∀i : {i // p i},‖labelEndLift (2*t) e i-exposedAnchorLift hL e i‖≤3) : |
| (conditionedMeasure (Measure.pi fun i : {i // p i} ↦ |
| individualReference t (labelStartLift e i) (labelEndLift (2*t) e i) |
| (permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i) |
| (exposedAnchorLift hL e i) (frozenPaths (2*t) e p w))) (physicalMutual (L:=L) r)).map |
| (fun (f : {i // p i} → C(Time,Point)) (i : {i // p i}) ↦ Bridge.shiftPath (canonicalShift hL e i) (f i)) = |
| conditionedMeasure (Measure.pi fun i : {i // p i} ↦ |
| individualReference t |
| (nearestLift L (anchorPoint (gridHome hL (e.2 i))) (e.2 i)) |
| (nearestLift L (anchorPoint (gridHome hL (e.2 i))) (bridgeEnd N L (2*t) e i)) |
| (permittedPath m r R C S u |
| (nearestLift L (anchorPoint (gridHome hL (e.2 i))) (e.2 i)) |
| (nearestLift L (anchorPoint (gridHome hL (e.2 i))) (bridgeEnd N L (2*t) e i)) |
| (anchorPoint (gridHome hL (e.2 i))) (frozenPaths (2*t) e p w))) |
| (physicalMutual (L:=L) r) := by |
| let μ := fun i : {i // p i} ↦ individualReference t (labelStartLift e i) (labelEndLift (2*t) e i) |
| (permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i) |
| (exposedAnchorLift hL e i) (frozenPaths (2*t) e p w)) |
| have (i : {i // p i}) : IsFiniteMeasure (μ i) := conditioned_finite _ _ |
| have H := conditioned_pi_map μ (fun i ↦ Bridge.shiftPath (canonicalShift hL e i)) |
| (fun i ↦ (Bridge.shiftPath_continuous _).measurable) (physicalMutual_measurable (I:={i // p i}) (L:=L) r) |
| rw [physicalMutual_shift_preimage (I:={i // p i}) r |
| (fun i ↦ canonicalShift hL e i) (fun i ↦ canonicalShift_period hL e i)] at H |
| rw [H] |
| congr 1 |
| apply congrArg Measure.pi |
| funext i |
| have h := individualReference_shift t (labelStartLift e i) (labelEndLift (2*t) e i) (canonicalShift hL e i) |
| (permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i) |
| (exposedAnchorLift hL e i) (frozenPaths (2*t) e p w)) |
| (permittedPath m r R C S u (labelStartLift e i+canonicalShift hL e i) |
| (labelEndLift (2*t) e i+canonicalShift hL e i) |
| (exposedAnchorLift hL e i+canonicalShift hL e i) (frozenPaths (2*t) e p w)) |
| (permittedPath_measurable ..) |
| (permittedPath_shift m r R C S u _ _ _ _ (canonicalShift_period hL e i) _) |
| simpa only [canonicalShift_start hL hK h50,canonicalShift_end hL h50 e i t (hz i), |
| canonicalShift_anchor] using h |
|
|
| theorem actual_allocated_canonical_reference {N K M G : ℕ} {L : Length} |
| (hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val) |
| (μ : Measure (BridgeEndpoints N L)) [SFinite μ] |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) |
| (g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M)) : |
| let p := fun i ↦ i∈groupUnion g a |
| ∀ᵐw ∂exposedLaw (allocationBase μ (gridAllocationKernel hL (G*M)) b a) |
| (fun _ : Fin N ↦ Bridge.law) p |
| (labelInteraction N L r (2*t)∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b), |
| ((fiberNormalize (Kernel.const _ (Measure.pi fun _ : {i // p i} ↦ Bridge.law)) |
| (blockEvent p (labelInteraction N L r (2*t)∩ |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b))) w).map |
| (fun β (i : {i // p i}) ↦ Bridge.shiftPath (canonicalShift hL w.1 i) |
| (leg (2*t) (labelStartLift w.1 i) (labelEndLift (2*t) w.1 i) (β i))) = |
| conditionedMeasure (Measure.pi fun i : {i // p i} ↦ |
| individualReference t |
| (nearestLift L (anchorPoint (gridHome hL (w.1.2 i))) (w.1.2 i)) |
| (nearestLift L (anchorPoint (gridHome hL (w.1.2 i))) (bridgeEnd N L (2*t) w.1 i)) |
| (permittedPath m r R C S u |
| (nearestLift L (anchorPoint (gridHome hL (w.1.2 i))) (w.1.2 i)) |
| (nearestLift L (anchorPoint (gridHome hL (w.1.2 i))) (bridgeEnd N L (2*t) w.1 i)) |
| (anchorPoint (gridHome hL (w.1.2 i))) (frozenPaths (2*t) w.1 p w.2))) |
| (physicalMutual (L:=L) r) := by |
| dsimp only |
| let p := fun i ↦ i∈groupUnion g a |
| let base := allocationBase μ (gridAllocationKernel hL (G*M)) b a |
| by_cases hb : ∀i : {i // p i},b i=true |
| · have hn := exposed_selected_near hL hK h50 base m r R C S s₀ t u p b hb |
| have hd : (fun i ↦ Classical.propDecidable (p i)) = |
| (fun i ↦ Finset.decidableMem i (groupUnion g a)) := Subsingleton.elim _ _ |
| rw [hd] at hn |
| filter_upwards [actual_allocated_physical_reference hL hK h50 μ m r R C S s₀ t u g b a, |
| hn] with w hw hz |
| have H := canonical_reference_transport hL hK h50 m r R C S t u w.1 p w.2 hz |
| rw [←hw] at H |
| have hm : Measurable (fun β : {i // p i}→Bridge.Noise ↦ fun i : {i // p i} ↦ |
| leg (2*t) (labelStartLift w.1 i) (labelEndLift (2*t) w.1 i) (β i)) := |
| Measurable.of_eval (fun i ↦ (leg_continuous (2*t) _ _).measurable.comp (measurable_pi_apply i)) |
| have hc : Measurable (fun f : {i // p i}→C(Time,Point) ↦ fun i : {i // p i} ↦ |
| Bridge.shiftPath (canonicalShift hL w.1 i) (f i)) := by fun_prop |
| rw [Measure.map_map hc hm] at H |
| have hC : MeasurableSet (blockEvent p (labelInteraction N L r (2*t)∩ |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b)) := |
| ((labelInteraction_measurable N L r (2*t)).inter |
| (flagEvent_measurable _ (fun i ↦ eligibilityFlag_measurable m r R C S s₀ (2*t) u i) b)).preimage |
| (blockGlue_measurable p) |
| have he : fiberNormalize (Kernel.const _ (Measure.pi fun _ : {i // p i} ↦ Bridge.law)) |
| (blockEvent p (labelInteraction N L r (2*t)∩ |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b)) w = |
| conditionedMeasure (Measure.pi fun _ : {i // p i} ↦ Bridge.law) |
| {β | blockGlue p (w,β) ∈ labelInteraction N L r (2*t)∩ |
| flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b} := by |
| change (fiberRestrict _ _ w univ)⁻¹ • fiberRestrict _ _ w=_ |
| rw [fiberRestrict_apply _ hC] |
| simp only [Kernel.const_apply,Measure.restrict_apply_univ,conditionedMeasure] |
| rfl |
| rw [he] |
| convert H using 1 <;> congr 10 |
| all_goals try exact Subsingleton.elim _ _ |
| all_goals rw [hd] |
| · filter_upwards [allocated_selected_flags hL μ g b a |
| (labelInteraction N L r (2*t)∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b)] with w hw |
| exact (hb (fun i ↦ hw i i.property)).elim |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
