| import OAI.Probability.FirstPassage.EndpointCoarseGeometry |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set Finset MeasureTheory ProbabilityTheory |
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| 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) |
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| noncomputable def endpointBoxVertices (b A R : ℝ) (k m : ℕ) : Finset Vertex := |
| (endpointBox_finite b A R k m).toFinset |
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| 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) |
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| 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⟩ |
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| 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 _ _ _ |
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| 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 _ _ _ |
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| noncomputable def coarsePairIndices (A : ℝ) (J : ℕ) : Finset (ℤ×ℤ) := |
| ((coarseColumnIndices A J) ×ˢ (coarseColumnIndices A J)).filter (fun ij => ij.1 < ij.2) |
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| 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)) |
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| 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 |
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| end PlanarFPP |
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| end OAI |
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