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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Probability.FirstPassage.EndpointEntrance
import OAI.Probability.FirstPassage.EndpointExit
import OAI.Probability.FirstPassage.EndpointBounds
namespace OAI
namespace PlanarFPP
open Set
noncomputable def endpointBox (b A R : ℝ) (k m : ℕ) : Set Vertex :=
{z | |(z.1 : ℝ)| ≤ A*(2 : ℝ)^m ∧ |transverse b z| ≤ R*(2 : ℝ)^k}
theorem gridConnections_endpoint_extensions (w : Edge → ℝ) (b m₀ K A R : ℝ)
(hK : 0 ≤ K) (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)
(p : Vertex) (hp : p ∈ endpointBox b A R k m) :
ColumnConnection w b (transverse b p) m₀ (endpointExtensionError K k J)
(endpointBox b (A+4) (R+2*(K+200)) k m) (entranceColumn p.1 m J) p.1 ∧
ColumnConnection w b (transverse b p) m₀ (endpointExtensionError K k J)
(endpointBox b (A+4) (R+2*(K+200)) k m) p.1 (exitColumn p.1 m J) := by
let H := 2*(K+200)*(2 : ℝ)^k
have hH : 1 ≤ H := by
have hs := one_le_pow₀ (by norm_num : (1 : ℝ) ≤ 2) (n := k)
dsimp [H]
nlinarith
have herr (j : ℕ) (_hj : J ≤ j) (hjl : j ≤ gridLastLevel m) :
(K+200)*(j : ℝ)^2*gridWidth k j ≤ H := grid_endpoint_height K hK k j (hjl.trans hlast)
have hd := gridMesh_pos m J
have hdR : (0 : ℝ) < gridMesh m J := by exact_mod_cast hd
have hmesh := gridMesh_le_scale m J
have hL : (0 : ℝ) < 2^m := by positivity
have hs := entranceColumn_span p.1 m J
have ht := exitColumn_span p.1 m J
have hsR : (gridMesh m J : ℝ) ≤ (p.1 : ℝ)-(entranceColumn p.1 m J : ℝ) ∧
(p.1 : ℝ)-(entranceColumn p.1 m J : ℝ) < 2*(gridMesh m J : ℝ) := by exact_mod_cast hs
have htR : (gridMesh m J : ℝ) ≤ (exitColumn p.1 m J : ℝ)-(p.1 : ℝ) ∧
(exitColumn p.1 m J : ℝ)-(p.1 : ℝ) < 2*(gridMesh m J : ℝ) := by exact_mod_cast ht
have hpx := abs_le.mp hp.1
have hleft : -(A+4)*(2 : ℝ)^m ≤ (entranceColumn p.1 m J : ℝ) := by linarith [hsR.2]
have hright : (exitColumn p.1 m J : ℝ) ≤ (A+4)*(2 : ℝ)^m := by linarith [htR.2]
have hpL : -(A+4)*(2 : ℝ)^m ≤ (p.1 : ℝ) := by linarith [hpx.1]
have hpR : (p.1 : ℝ) ≤ (A+4)*(2 : ℝ)^m := by linarith [hpx.2]
have hentr := gridConnections_entrance w b m₀ K A R H hK hH k m J hJ hJlast hg herr p hleft hpR hp.2
have hexit := gridConnections_exit w b m₀ K A R H hK hH k m J hJ hJlast hg herr p hpL hright hp.2
have hc := endpoint_extension_cost_sum K hK k m J hlast hJlast
constructor
· have hconn : ColumnConnection w b (transverse b p) m₀
(∑ i ∈ Finset.range (gridLastLevel m-J+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i)))
(endpointTube b (transverse b p) H (entranceColumn p.1 m J-gridMesh m J) (p.1+gridMesh m J))
(entranceColumn p.1 m J) p.1 := by
simpa only [ColumnConnection,endpointVertex_self] using hentr
apply hconn.mono hc
intro z hz
apply endpoint_localization b A R (K+200) (by linarith) k m p z (gridMesh m J) hmesh hp.1 hp.2
· exact ⟨by have hh := hz.1; omega,
by have hh := hz.2.1; omega⟩
· exact hz.2.2
· have hconn : ColumnConnection w b (transverse b p) m₀
(∑ i ∈ Finset.range (gridLastLevel m-J+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i)))
(endpointTube b (transverse b p) H (p.1-gridMesh m J) (exitColumn p.1 m J+gridMesh m J))
p.1 (exitColumn p.1 m J) := by
simpa only [ColumnConnection,endpointVertex_self] using hexit
apply hconn.mono hc
intro z hz
apply endpoint_localization b A R (K+200) (by linarith) k m p z (gridMesh m J) hmesh hp.1 hp.2
· exact ⟨by have hh := hz.1; omega,
by have hh := hz.2.1; omega⟩
· exact hz.2.2
theorem gridConnections_endpoint_defects (w : Edge → ℝ) (b m₀ K A R : ℝ)
(hK : 0 ≤ K) (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)
(p : Vertex) (hp : p ∈ endpointBox b A R k m) :
∃ ds es,
endpointVertex b (transverse b p) (entranceColumn p.1 m J)+displacement ds=p ∧
p+displacement es=endpointVertex b (transverse b p) (exitColumn p.1 m J) ∧
(∀ a : ℝ, defect w b m₀ a (endpointVertex b (transverse b p) (entranceColumn p.1 m J)) ds ≤
endpointExtensionError K k J+|a|) ∧
(∀ a : ℝ, defect w b m₀ a p es ≤ endpointExtensionError K k J+|a|) ∧
PathIn (endpointVertex b (transverse b p) (entranceColumn p.1 m J)) ds
(endpointBox b (A+4) (R+2*(K+200)) k m) ∧
PathIn p es (endpointBox b (A+4) (R+2*(K+200)) k m) := by
obtain ⟨⟨ds,hd,hc,hin⟩,⟨es,he,hce,hine⟩⟩ :=
gridConnections_endpoint_extensions w b m₀ K A R hK k m J hJ hJlast hlast hg p hp
refine ⟨ds,es,by simpa only [endpointVertex_self] using hd,
by simpa only [endpointVertex_self] using he,?_,?_,hin,by simpa only [endpointVertex_self] using hine⟩
· intro a
exact endpoint_path_defect w b (transverse b p) m₀ a _ _ _ ds hd hc
· intro a
simpa only [endpointVertex_self] using endpoint_path_defect w b (transverse b p) m₀ a _ _ _ es he hce
end PlanarFPP
end OAI
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