| import OAI.Probability.FirstPassage.EndpointExtensions |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set |
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| lemma endpoint_coarse_columns (A : ℝ) (m J : ℕ) (x : ℤ) |
| (hx : |(x : ℝ)| ≤ A*(2 : ℝ)^m) : |
| |(entranceColumn x m J : ℝ)| ≤ (A+2)*(2 : ℝ)^m ∧ |
| |(exitColumn x m J : ℝ)| ≤ (A+2)*(2 : ℝ)^m := by |
| have hl := entranceColumn_span x m J |
| have hr := exitColumn_span x m J |
| have hl' : (gridMesh m J : ℝ) ≤ (x : ℝ)-(entranceColumn x m J : ℝ) ∧ |
| (x : ℝ)-(entranceColumn x m J : ℝ) < 2*(gridMesh m J : ℝ) := by exact_mod_cast hl |
| have hr' : (gridMesh m J : ℝ) ≤ (exitColumn x m J : ℝ)-(x : ℝ) ∧ |
| (exitColumn x m J : ℝ)-(x : ℝ) < 2*(gridMesh m J : ℝ) := by exact_mod_cast hr |
| have hd := gridMesh_le_scale m J |
| have hd0 : (0 : ℝ) < gridMesh m J := by exact_mod_cast gridMesh_pos m J |
| have hx' := abs_le.mp hx |
| constructor <;> apply abs_le.mpr <;> constructor <;> linarith [hl'.1,hl'.2,hr'.1,hr'.2] |
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| lemma endpointBox_mono (b A A' R R' : ℝ) (k m : ℕ) (hA : A ≤ A') (hR : R ≤ R') : |
| endpointBox b A R k m ⊆ endpointBox b A' R' k m := by |
| intro p hp |
| exact ⟨hp.1.trans (mul_le_mul_of_nonneg_right hA (by positivity)), |
| hp.2.trans (mul_le_mul_of_nonneg_right hR (by positivity))⟩ |
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| theorem gridConnections_extend_negative_path (w : Edge → ℝ) (b m₀ K A R a ε : ℝ) |
| (hK : 0 ≤ K) (hm : 0 < m₀) (hR : 0 ≤ R) (hε : 0 < ε) |
| (k m J : ℕ) (hJ : 1 ≤ J) (hJlast : J ≤ gridLastLevel m) (hlast : gridLastLevel m ≤ k) |
| (hg : w ∈ gridConnectionsEvent b m₀ K A R k m J) |
| (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)) |
| (p : Vertex) (ds : List Step) (hin : PathIn p ds (endpointBox b A R k m)) |
| (hcost : defect w b m₀ a p ds < -ε*(2 : ℝ)^k) : |
| ∃ X Y : ℤ, gridMesh m J ∣ X ∧ gridMesh m J ∣ Y ∧ |
| |(X : ℝ)| ≤ (A+2)*(2 : ℝ)^m ∧ |(Y : ℝ)| ≤ (A+2)*(2 : ℝ)^m ∧ |
| (gridMesh m J : ℝ) ≤ ((Y-X : ℤ) : ℝ) ∧ |
| ∃ q es, q.1=X ∧ (q+displacement es).1=Y ∧ |
| PathIn q es (endpointBox b (A+4) (R+2*(K+200)) k m) ∧ |
| defect w b m₀ a q es < -(ε/2)*(2 : ℝ)^k := by |
| let z := p+displacement ds |
| have hp : p ∈ endpointBox b A R k m := by simpa using hin 0 (by omega) |
| have hz : z ∈ endpointBox b A R k m := by simpa [z] using hin ds.length le_rfl |
| obtain ⟨l,_unused,hl,_hu,hld,_hud,hil,_hiu⟩ := |
| gridConnections_endpoint_defects w b m₀ K A R hK k m J hJ hJlast hlast hg p hp |
| obtain ⟨_unused',u,_hl,hu,_hld,hud,_hil,hiu⟩ := |
| gridConnections_endpoint_defects w b m₀ K A R hK k m J hJ hJlast hlast hg z hz |
| let q := endpointVertex b (transverse b p) (entranceColumn p.1 m J) |
| have hrev := negative_defect_reverse_span w hg.1 b m₀ a R ((2 : ℝ)^k) ε hm hR |
| (by positivity) hε.le p z ds rfl hp.2 hz.2 hcost |
| have hspan := extended_columns_span p.1 z.1 m J (2*R*|a| /m₀) ((2 : ℝ)^k) |
| (by convert hrev using 1; ring) hscale |
| refine ⟨entranceColumn p.1 m J,exitColumn z.1 m J,entranceColumn_dvd _ _ _,exitColumn_dvd _ _ _, |
| (endpoint_coarse_columns A m J p.1 hp.1).1,(endpoint_coarse_columns A m J z.1 hz.1).2, |
| hspan,q,l++ds++u,rfl,?_,?_,?_⟩ |
| · rw [displacement_append,displacement_append,← add_assoc,← add_assoc] |
| change ((endpointVertex b (transverse b p) (entranceColumn p.1 m J)+displacement l+ |
| displacement ds)+displacement u).1=exitColumn z.1 m J |
| rw [hl] |
| change (z+displacement u).1=exitColumn z.1 m J |
| rw [hu] |
| rfl |
| · have him : PathIn p ds (endpointBox b (A+4) (R+2*(K+200)) k m) := by |
| intro i hi |
| exact endpointBox_mono b A (A+4) R (R+2*(K+200)) k m (by linarith) (by linarith) (hin i hi) |
| apply pathIn_append _ _ _ _ (pathIn_append _ _ _ _ hil (by simpa [hl] using him)) |
| simpa only [displacement_append,← add_assoc,hl] using hiu |
| · rw [defect_append,defect_append] |
| have he : q+displacement l=p := hl |
| have he' : q+displacement (l++ds)=z := by rw [displacement_append,← add_assoc,he] |
| rw [he,he'] |
| have hb := hld a |
| have hc := hud a |
| change defect w b m₀ a q l ≤ _ at hb |
| linarith |
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| end PlanarFPP |
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| end OAI |
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