| import OAI.Probability.FirstPassage.EndpointConnection |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set Finset |
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| theorem gridConnections_exit_levels (w : Edge → ℝ) (b m₀ K A R H : ℝ) |
| (hK : 0 ≤ K) (hH : 1 ≤ H) (k m J : ℕ) (hJ : 1 ≤ J) |
| (hg : w ∈ gridConnectionsEvent b m₀ K A R k m J) |
| (herr : ∀ j, J ≤ j → j ≤ gridLastLevel m → (K+200)*(j : ℝ)^2*gridWidth k j ≤ H) |
| (x : ℤ) (hlo : -(A+4)*(2 : ℝ)^m ≤ (x : ℝ)) |
| (hhi : (exitColumn x m J : ℝ) ≤ (A+4)*(2 : ℝ)^m) |
| (r : ℝ) (hr : |r| ≤ R*(2 : ℝ)^k) (n : ℕ) (hn : J+n ≤ gridLastLevel m) : |
| ColumnConnection w b r m₀ |
| (∑ i ∈ range (n+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i))) |
| (endpointTube b r H (x-gridMesh m J) (exitColumn x m J+gridMesh m J)) |
| (-endpointColumn (-x) m (J+n)) (exitColumn x m J) := by |
| let T := exitColumn x m J |
| let d := gridMesh m J |
| have hfloor (j : ℕ) : x ≤ -endpointColumn (-x) m j := by |
| have h := (meshFloor_bounds (-x) (gridMesh m j) (gridMesh_pos m j)).1 |
| change meshFloor (-x) (gridMesh m j) ≤ -x at h |
| dsimp [endpointColumn] |
| omega |
| have hT (j : ℕ) (hj : J ≤ j) : -endpointColumn (-x) m j ≤ T := by |
| have hm := endpointColumn_mono (-x) m hj |
| have hd := gridMesh_pos m J |
| dsimp [T,exitColumn,entranceColumn] |
| omega |
| induction n with |
| | zero => |
| have hdiv : gridMesh m J ∣ -endpointColumn (-x) m J := dvd_neg.mpr (meshFloor_dvd (-x) _) |
| have hb := gridConnections_block_connection w b m₀ K A R H hK hH k m J hJ hg J |
| ⟨le_refl _,by simpa using hn⟩ (herr J le_rfl (by simpa using hn)) |
| (-endpointColumn (-x) m J) T (x-d) (T+d) hdiv 1 (by omega) |
| (by simp [T,exitColumn,entranceColumn]; ring) |
| (hlo.trans (by exact_mod_cast hfloor J)) hhi |
| (by dsimp [d]; have hh := hfloor J; omega) (by rfl) r hr |
| simpa [T,d] using hb |
| | succ n ih => |
| have hn' : J+n ≤ gridLastLevel m := by omega |
| have hj : J ≤ J+n+1 := by omega |
| have hjlast : J+n+1 ≤ gridLastLevel m := by omega |
| have hp := ih hn' |
| obtain ⟨N,hN,hstep⟩ := endpointColumn_step (-x) m (J+n) |
| have hdiv : gridMesh m (J+n+1) ∣ -endpointColumn (-x) m (J+n+1) := |
| dvd_neg.mpr (meshFloor_dvd (-x) _) |
| have hdm : gridMesh m (J+n+1) ≤ d := gridMesh_antitone m hj |
| have hbefore := hT (J+n) (by omega) |
| have hstart := hfloor (J+n+1) |
| have hb := gridConnections_block_connection w b m₀ K A R H hK hH k m J hJ hg (J+n+1) |
| ⟨hj,hjlast⟩ (herr _ hj hjlast) (-endpointColumn (-x) m (J+n+1)) |
| (-endpointColumn (-x) m (J+n)) (x-d) (T+d) hdiv N (by omega) (by omega) |
| (hlo.trans (by exact_mod_cast hstart)) |
| ((by exact_mod_cast hbefore : ((-endpointColumn (-x) m (J+n) : ℤ) : ℝ) ≤ (T : ℝ)).trans hhi) (by omega) (by omega) r hr |
| have hc := hb.trans hp |
| rw [add_comm (16*((K+200)*((J+n+1 : ℕ) : ℝ)^2*gridWidth k (J+n+1)))] at hc |
| simpa only [sum_range_succ,Nat.add_assoc] using hc |
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| theorem gridConnections_exit (w : Edge → ℝ) (b m₀ K A R H : ℝ) |
| (hK : 0 ≤ K) (hH : 1 ≤ H) (k m J : ℕ) (hJ : 1 ≤ J) (hlast : J ≤ gridLastLevel m) |
| (hg : w ∈ gridConnectionsEvent b m₀ K A R k m J) |
| (herr : ∀ j, J ≤ j → j ≤ gridLastLevel m → (K+200)*(j : ℝ)^2*gridWidth k j ≤ H) |
| (p : Vertex) (hlo : -(A+4)*(2 : ℝ)^m ≤ (p.1 : ℝ)) |
| (hhi : (exitColumn p.1 m J : ℝ) ≤ (A+4)*(2 : ℝ)^m) (hr : |transverse b p| ≤ R*(2 : ℝ)^k) : |
| ∃ ds, p+displacement ds=endpointVertex b (transverse b p) (exitColumn p.1 m J) ∧ |
| pathCost w p ds ≤ m₀*((exitColumn p.1 m J-p.1 : ℤ) : ℝ)+ |
| ∑ i ∈ range (gridLastLevel m-J+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i)) ∧ |
| PathIn p ds (endpointTube b (transverse b p) H (p.1-gridMesh m J) (exitColumn p.1 m J+gridMesh m J)) := by |
| have hc := gridConnections_exit_levels w b m₀ K A R H hK hH k m J hJ hg herr |
| p.1 hlo hhi (transverse b p) hr (gridLastLevel m-J) (by omega) |
| rw [Nat.add_sub_of_le hlast,endpointColumn_last,neg_neg] at hc |
| simpa only [ColumnConnection,endpointVertex_self] using hc |
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| end PlanarFPP |
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| end OAI |
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