| import OAI.Probability.FirstPassage.EndpointBlocks |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set |
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| def ColumnConnection (w : Edge → ℝ) (b r m₀ E : ℝ) (S : Set Vertex) (X Y : ℤ) : Prop := |
| ∃ ds, endpointVertex b r X+displacement ds=endpointVertex b r Y ∧ |
| pathCost w (endpointVertex b r X) ds ≤ m₀*((Y-X : ℤ) : ℝ)+E ∧ |
| PathIn (endpointVertex b r X) ds S |
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| lemma ColumnConnection.mono {w : Edge → ℝ} {b r m₀ E F : ℝ} {S T : Set Vertex} {X Y : ℤ} |
| (h : ColumnConnection w b r m₀ E S X Y) (hE : E ≤ F) (hS : S ⊆ T) : |
| ColumnConnection w b r m₀ F T X Y := by |
| obtain ⟨ds,hd,hc,hin⟩ := h |
| exact ⟨ds,hd,by linarith,fun i hi => hS (hin i hi)⟩ |
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| lemma ColumnConnection.trans {w : Edge → ℝ} {b r m₀ E F : ℝ} {S : Set Vertex} {X Y Z : ℤ} |
| (h : ColumnConnection w b r m₀ E S X Y) (g : ColumnConnection w b r m₀ F S Y Z) : |
| ColumnConnection w b r m₀ (E+F) S X Z := by |
| obtain ⟨ds,hd,hc,hin⟩ := h |
| obtain ⟨es,he,hce,hine⟩ := g |
| refine ⟨ds++es,?_,?_,?_⟩ |
| · rw [displacement_append,← add_assoc,hd,he] |
| · rw [pathCost_append,hd] |
| push_cast at hc hce ⊢ |
| linarith |
| · exact pathIn_append _ _ _ _ hin (by simpa [hd] using hine) |
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| noncomputable def endpointTube (b r H : ℝ) (B T : ℤ) : Set Vertex := |
| {z | B ≤ z.1 ∧ z.1 ≤ T ∧ |transverse b z-r| ≤ H} |
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| lemma gridConnections_block_connection (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) |
| (j : ℕ) (hj : J ≤ j ∧ j ≤ gridLastLevel m) |
| (he : (K+200)*(j : ℝ)^2*gridWidth k j ≤ H) |
| (X Y B T : ℤ) (hX : gridMesh m j ∣ X) (n : ℕ) (hn : n ≤ 16) |
| (hXY : X+(n : ℤ)*gridMesh m j=Y) |
| (hlo : -(A+4)*(2 : ℝ)^m ≤ (X : ℝ)) |
| (hhi : (Y : ℝ) ≤ (A+4)*(2 : ℝ)^m) |
| (hB : B ≤ X-gridMesh m j) (hT : Y+gridMesh m j ≤ T) |
| (r : ℝ) (hr : |r| ≤ R*(2 : ℝ)^k) : |
| ColumnConnection w b r m₀ (16*((K+200)*(j : ℝ)^2*gridWidth k j)) |
| (endpointTube b r H B T) X Y := by |
| obtain ⟨ds,hd,hc,hin⟩ := gridConnections_block w b m₀ K A R H hK hH k m J hJ hg j hj he |
| X hX n hlo (by simpa [hXY] using hhi) r hr |
| have hn' : (n : ℝ) ≤ 16 := by exact_mod_cast hn |
| have herr : (n : ℝ)*((K+200)*(j : ℝ)^2*gridWidth k j) ≤ |
| 16*((K+200)*(j : ℝ)^2*gridWidth k j) := by |
| apply mul_le_mul_of_nonneg_right hn' |
| exact mul_nonneg (mul_nonneg (by linarith) (sq_nonneg _)) (by exact_mod_cast (gridWidth_pos k j).le) |
| refine ⟨ds,by simpa [hXY] using hd,?_,?_⟩ |
| · have heq : (n : ℤ)*gridMesh m j=Y-X := by omega |
| rw [heq] at hc |
| linarith |
| · intro i hi |
| have hz := hin i hi |
| rw [hXY] at hz |
| exact ⟨hB.trans hz.1,hz.2.1.trans hT,hz.2.2⟩ |
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| end PlanarFPP |
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| end OAI |
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