| 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_entrance_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 ≤ (entranceColumn x m J : ℝ)) |
| (hhi : (x : 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 (entranceColumn x m J-gridMesh m J) (x+gridMesh m J)) |
| (entranceColumn x m J) (endpointColumn x m (J+n)) := by |
| let B := entranceColumn x m J |
| let d := gridMesh m J |
| have hfloor (j : ℕ) : endpointColumn x m j ≤ x := |
| (meshFloor_bounds x _ (gridMesh_pos m j)).1 |
| have hfloorR (j : ℕ) : (endpointColumn x m j : ℝ) ≤ (x : ℝ) := by exact_mod_cast hfloor j |
| have hB (j : ℕ) (hj : J ≤ j) : B ≤ endpointColumn x m j := by |
| have hm := endpointColumn_mono x m hj |
| have hd := gridMesh_pos m J |
| dsimp [B,entranceColumn] |
| omega |
| induction n with |
| | zero => |
| 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)) B |
| (endpointColumn x m J) (B-d) (x+d) (entranceColumn_dvd x m J) 1 (by omega) |
| (by simp [B,entranceColumn]) hlo |
| ((hfloorR J).trans hhi) (by rfl) (by dsimp [d]; have hh := hfloor J; omega) r hr |
| simpa [B,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) := |
| dvd_trans (gridMesh_divides m (J+n)) (meshFloor_dvd x _) |
| have hdm : gridMesh m (J+n+1) ≤ d := gridMesh_antitone m hj |
| have hbefore := hB (J+n) (by omega) |
| 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)) |
| (endpointColumn x m (J+n+1)) (B-d) (x+d) hdiv N (by omega) hstep |
| (hlo.trans (by exact_mod_cast hbefore)) |
| ((hfloorR (J+n+1)).trans hhi) |
| (by omega) (by have hh := hfloor (J+n+1); omega) r hr |
| have hc := hp.trans hb |
| simpa only [sum_range_succ,Nat.add_assoc] using hc |
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| theorem gridConnections_entrance (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 ≤ (entranceColumn p.1 m J : ℝ)) |
| (hhi : (p.1 : ℝ) ≤ (A+4)*(2 : ℝ)^m) (hr : |transverse b p| ≤ R*(2 : ℝ)^k) : |
| ∃ ds, endpointVertex b (transverse b p) (entranceColumn p.1 m J)+displacement ds=p ∧ |
| pathCost w (endpointVertex b (transverse b p) (entranceColumn p.1 m J)) ds ≤ |
| m₀*((p.1-entranceColumn p.1 m J : ℤ) : ℝ)+ |
| ∑ i ∈ range (gridLastLevel m-J+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i)) ∧ |
| PathIn (endpointVertex b (transverse b p) (entranceColumn p.1 m J)) ds |
| (endpointTube b (transverse b p) H (entranceColumn p.1 m J-gridMesh m J) (p.1+gridMesh m J)) := by |
| have hc := gridConnections_entrance_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] 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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