File size: 3,887 Bytes
f61e547 | 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 | import OAI.Probability.FirstPassage.EndpointCoarseGeometry
namespace OAI
namespace PlanarFPP
open Set Finset MeasureTheory ProbabilityTheory
lemma endpointBox_finite (b A R : ℝ) (k m : ℕ) : (endpointBox b A R k m).Finite := by
simpa [endpointBox] using finite_tilted_rectangle b 0 0 (A*(2 : ℝ)^m) (R*(2 : ℝ)^k)
noncomputable def endpointBoxVertices (b A R : ℝ) (k m : ℕ) : Finset Vertex :=
(endpointBox_finite b A R k m).toFinset
noncomputable def arbitraryEndpointEvent (b m₀ a A R ε : ℝ) (k m : ℕ) : Set (Edge → ℝ) :=
strictLocalizedEvent (endpointBoxVertices b A R k m) (fun _ _ => True)
(fun _ ds => support b m₀ a (displacement ds)-ε*(2 : ℝ)^k)
lemma mem_arbitraryEndpointEvent (b m₀ a A R ε : ℝ) (k m : ℕ) (w : Edge → ℝ) :
w ∈ arbitraryEndpointEvent b m₀ a A R ε k m ↔
∃ p ds, PathIn p ds (endpointBox b A R k m) ∧ defect w b m₀ a p ds < -ε*(2 : ℝ)^k := by
have hv : (endpointBoxVertices b A R k m : Set Vertex)=endpointBox b A R k m :=
(endpointBox_finite b A R k m).coe_toFinset
simp only [arbitraryEndpointEvent,strictLocalizedEvent,mem_ofPred_eq,true_and,hv,defect]
constructor <;> rintro ⟨p,ds,hin,hc⟩ <;> exact ⟨p,ds,hin,by linarith⟩
lemma measurableSet_arbitraryEndpointEvent (b m₀ a A R ε : ℝ) (k m : ℕ) :
MeasurableSet (arbitraryEndpointEvent b m₀ a A R ε k m) :=
measurableSet_strictLocalizedEvent _ _ _
lemma arbitraryEndpointEvent_depends (b m₀ a A R ε : ℝ) (k m : ℕ) :
DependsOnEdges (arbitraryEndpointEvent b m₀ a A R ε k m)
(internalEdges (endpointBoxVertices b A R k m)) := strictLocalizedEvent_depends _ _ _
noncomputable def coarsePairIndices (A : ℝ) (J : ℕ) : Finset (ℤ×ℤ) :=
((coarseColumnIndices A J) ×ˢ (coarseColumnIndices A J)).filter (fun ij => ij.1 < ij.2)
noncomputable def coarsePairEvent (b m₀ a A R ε : ℝ) (k m J : ℕ) (ij : ℤ×ℤ) : Set (Edge → ℝ) :=
rareFastEvent b m₀ a (coarseRegionFactor A J) (R+1) (ε/2) ((2 : ℝ)^k)
(gridMesh m J*ij.2-gridMesh m J*ij.1) (endpointVertex b 0 (gridMesh m J*ij.1))
theorem arbitraryEndpointEvent_coarse_cover (b m₀ K A R a ε : ℝ)
(hK : 0 ≤ K) (hm : 0 < m₀) (hA : 0 ≤ A) (hR : 0 ≤ R) (hε : 0 < ε)
(k m J : ℕ) (hJ : 1 ≤ J) (hJlast : J ≤ gridLastLevel m) (hlast : gridLastLevel m ≤ k)
(herror : 2*endpointExtensionError K k J+2*|a| ≤ ε*(2 : ℝ)^k/2)
(hscale : (2*R*|a| /m₀)*(2 : ℝ)^k ≤ (gridMesh m J : ℝ)) :
arbitraryEndpointEvent b m₀ a A R ε k m ⊆
(gridConnectionsEvent b m₀ K A R k m J)ᶜ ∪
⋃ ij ∈ coarsePairIndices A J,coarsePairEvent b m₀ a A (R+2*(K+200)) ε k m J ij := by
intro w hw
by_cases hg : w ∈ gridConnectionsEvent b m₀ K A R k m J
· apply Or.inr
obtain ⟨p,ds,hin,hcost⟩ := (mem_arbitraryEndpointEvent ..).mp hw
obtain ⟨X,Y,hXd,hYd,hX,hY,hspan,q,es,hq,he,hpath,hdef⟩ :=
gridConnections_extend_negative_path w b m₀ K A R a ε hK hm hR hε k m J hJ hJlast hlast hg
herror hscale p ds hin hcost
obtain ⟨i,hi,hXi⟩ := coarse_column_index A hA m J X hXd hX
obtain ⟨j,hj,hYj⟩ := coarse_column_index A hA m J Y hYd hY
have hij : i < j := by
have hd := gridMesh_pos m J
have hspan' : gridMesh m J ≤ Y-X := by exact_mod_cast hspan
rw [hXi,hYj] at hspan'
nlinarith
apply mem_iUnion₂.mpr
refine ⟨(i,j),mem_filter.mpr ⟨mem_product.mpr ⟨hi,hj⟩,hij⟩,?_⟩
apply (mem_rareFastEvent ..).mpr
refine ⟨q,es,?_,?_,?_,hdef⟩
· simpa only [endpointVertex] using hq.trans hXi
· simp only [endpointVertex]
rw [he,hYj]
ring
· intro t ht
have hz := endpointBox_into_rareFastRegion b A (R+2*(K+200)) hA k m J X Y hX hspan (hpath t ht)
simpa only [hXi,hYj] using hz
· exact Or.inl hg
end PlanarFPP
end OAI
|