| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.GridAllocation |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison CellOrbital |
| attribute [local instance] Classical.propDecidable |
|
|
| def GroupWeak {N : ℕ} (m G : ℕ) (g : Fin G) (B : Finset (Fin N)) |
| (a : Allocation (G*(2*m)) (Fin N)) : Prop := |
| ∃f,a=some f ∧ m≤(Finset.univ.filter (fun k ↦ f (groupPosition (2*m) G g k)∈B)).card |
|
|
| lemma allocation_group_weak {N : ℕ} (E B : Finset (Fin N)) (m G : ℕ) (g : Fin G) |
| (hE : G*(2*m)≤E.card) (hB : 10*(E∩B).card≤E.card) : |
| (allocationPMF E (G*(2*m))).toMeasure {a | GroupWeak m G g B a}≤ |
| ENNReal.ofReal ((2/5:ℝ)^m) := by |
| let f : Allocation (G*(2*m)) (Fin N)→Allocation (2*m) (Fin N) := |
| Option.map (fun a ↦ (groupPosition (2*m) G g).trans a) |
| have hf : Measurable f := measurable_of_countable _ |
| have hmap : ((allocationPMF E (G*(2*m))).toMeasure).map f=(allocationPMF E (2*m)).toMeasure := by |
| rw [PMF.toMeasure_map _ _ hf,allocated_group_uniform E _ _ hE] |
| have hsub : {a | GroupWeak m G g B a}⊆f ⁻¹' {a | ∃b,a=some b ∧ m≤(Finset.univ.filter (fun k ↦ b k∈B)).card} := by |
| rintro a ⟨b,rfl,hb⟩ |
| exact ⟨_,rfl,hb⟩ |
| have hm : 2*m≤E.card := (show 2*m≤G*(2*m) by |
| have hg : 1≤G := Nat.succ_le_iff.mpr (Nat.zero_le g.val |>.trans_lt g.isLt) |
| simpa using Nat.mul_le_mul_right (2*m) hg).trans hE |
| apply (measure_mono hsub).trans |
| rw [←Measure.map_apply hf (by exact MeasurableSet.of_discrete),hmap] |
| exact allocation_half_bad E B m hm hB |
|
|
| lemma grid_group_joint_weak {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (p : BridgeEndpoints N L×(Fin N→Bool)) (m G : ℕ) (g : Fin G) |
| (B : (Fin 3→Fin K)→Finset (Fin N)) (A : Finset (Fin 3→Fin K)) |
| (hE : ∀v∈A,G*(2*m)≤(recordedList hL v p).card) |
| (hB : ∀v∈A,10*((recordedList hL v p)∩B v).card≤(recordedList hL v p).card) : |
| gridAllocationKernel hL (G*(2*m)) p {a | ∀v∈A,GroupWeak m G g (B v) (a v)}≤ |
| (ENNReal.ofReal ((2/5:ℝ)^m))^A.card := by |
| rw [gridAllocationKernel_apply] |
| rw [show (Measure.pi (fun v ↦ (allocationPMF (recordedList hL v p) (G*(2*m))).toMeasure)) |
| {a | ∀v∈A,GroupWeak m G g (B v) (a v)} = |
| ∏v∈A,(allocationPMF (recordedList hL v p) (G*(2*m))).toMeasure {a | GroupWeak m G g (B v) a} from |
| pi_failure_sets _ (fun v ↦ {a | GroupWeak m G g (B v) a}) A] |
| calc |
| _ ≤ ∏v∈A,ENNReal.ofReal ((2/5:ℝ)^m) := Finset.prod_le_prod (fun v hv ↦ |
| allocation_group_weak _ _ m G g (hE v hv) (hB v hv)) |
| _ = _ := by simp |
|
|
| end HardSphere.Groups |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Groups |
| open Wiener Bridge GaussianPacking PathCutoff Comparison CellOrbital |
|
|
| lemma endpointEligible_unique {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| {v w : Fin 3→Fin K} {s₀ : ℝ} {Y Z : Configuration N L} {i : Fin N} |
| (hv : EndpointEligible v s₀ Y Z i) (hw : EndpointEligible w s₀ Y Z i) : v=w := by |
| by_contra h |
| exact Set.disjoint_left.mp (centeredCell_disjoint hL hK h) hv.1 hw.1 |
|
|
| lemma ownEligible_at {N K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| (m : ℕ) (r R C S s₀ t : ℝ) (u : Time) |
| (i : Fin N) (e : BridgeEndpoints N L) (β : Bridge.Noise) (v : Fin 3→Fin K) |
| (he : EndpointEligible v s₀ e.2 (bridgeEnd N L t e) i) : |
| ownEligible (K:=K) m r R C S s₀ t u i (e,β) ↔ |
| Regular m r R C u S (projectCurve L (leg t (labelStartLift e i) (labelEndLift t e i) β)) ∧ |
| Confined (gridCenter L v) (projectCurve L (leg t (labelStartLift e i) (labelEndLift t e i) β)) := by |
| have hp : ownPath i t (e,β)=projectCurve L (leg t (labelStartLift e i) (labelEndLift t e i) β) := by |
| exact labelBridgePath_lift t e (fun _ ↦ β) i |
| unfold ownEligible |
| rw [hp] |
| constructor |
| · rintro ⟨w,hw,hreg,hconf⟩ |
| have hh := endpointEligible_unique hL hK he hw |
| subst w |
| exact ⟨hreg,hconf⟩ |
| · rintro ⟨hreg,hconf⟩ |
| exact ⟨v,he,hreg,hconf⟩ |
|
|
| lemma near_anchor_lift {L : Length} (hL : 1≤L.val) (y : Point) (v : Torus L) |
| (hy : projectPoint L y∈centeredCell L v) : |
| ∃c : Point,projectPoint L c=v ∧ ‖y-c‖≤1/2 := by |
| have hm := (centeredCell_image hL v (projectPoint L y)).mp hy |
| choose s hs he using hm |
| refine ⟨fun j ↦ y j-s j,?_,?_⟩ |
| · funext j |
| change ((y j-s j:ℝ):AddCircle L.val)=v j |
| rw [AddCircle.coe_sub,he j] |
| change ((y j:ℝ):AddCircle L.val)-(((y j:ℝ):AddCircle L.val)-v j)=v j |
| abel |
| · apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤1/2)).mpr |
| intro j |
| change |y j-(y j-s j)|≤1/2 |
| rw [sub_sub_cancel] |
| exact abs_le.mpr ⟨by linarith [(hs j).1],(hs j).2.le⟩ |
|
|
| end HardSphere.Groups |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
