| import OAI.Probability.FirstPassage.EndpointGeometry |
| import OAI.Probability.FirstPassage.GridConnections |
|
|
| namespace OAI |
|
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| namespace PlanarFPP |
| open Set Finset |
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| theorem gridConnections_block (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 : ℤ) (hX : gridMesh m j ∣ X) (n : ℕ) |
| (hlo : -(A+4)*(2 : ℝ)^m ≤ (X : ℝ)) |
| (hhi : ((X+(n : ℤ)*gridMesh m j : ℤ) : ℝ) ≤ (A+4)*(2 : ℝ)^m) |
| (r : ℝ) (hr : |r| ≤ R*(2 : ℝ)^k) : |
| ∃ ds, endpointVertex b r X+displacement ds= |
| endpointVertex b r (X+(n : ℤ)*gridMesh m j) ∧ |
| pathCost w (endpointVertex b r X) ds ≤ |
| m₀*((n : ℤ)*gridMesh m j : ℤ)+(n : ℝ)*((K+200)*(j : ℝ)^2*gridWidth k j) ∧ |
| PathIn (endpointVertex b r X) ds {z | X-gridMesh m j ≤ z.1 ∧ |
| z.1 ≤ X+(n : ℤ)*gridMesh m j+gridMesh m j ∧ |transverse b z-r| ≤ H} := by |
| let d := gridMesh m j |
| let p (i : ℕ) := endpointVertex b r (X+(i : ℤ)*d) |
| let S : Set Vertex := {z | X-d ≤ z.1 ∧ z.1 ≤ X+(n : ℤ)*d+d ∧ |transverse b z-r| ≤ H} |
| let c : ℝ := m₀*d+(K+200)*(j : ℝ)^2*gridWidth k j |
| have hd : 0 < d := gridMesh_pos m j |
| have hp0 : p 0 ∈ S := by |
| dsimp [p,S,endpointVertex] |
| refine ⟨by omega,?_,?_⟩ |
| · have hn : (0 : ℤ) ≤ n := by positivity |
| nlinarith |
| · simpa [endpointVertex] using (endpointVertex_close b r X).le.trans hH |
| have step (i : ℕ) (hi : i < n) : |
| ∃ ds, p i+displacement ds=p (i+1) ∧ pathCost w (p i) ds ≤ c ∧ PathIn (p i) ds S := by |
| have hdiv : d ∣ X+(i : ℤ)*d := dvd_add hX (dvd_mul_left _ _) |
| obtain ⟨q,hq⟩ := hdiv |
| have hq' : gridMesh m j*q=X+(i : ℤ)*d := hq.symm |
| have hicol : (X : ℝ) ≤ ((X+(i : ℤ)*d : ℤ) : ℝ) := by |
| exact_mod_cast (show X ≤ X+(i : ℤ)*d by have h : 0 ≤ (i : ℤ)*d := mul_nonneg (by positivity) hd.le; omega) |
| have hiend : X+(i : ℤ)*d+d ≤ X+(n : ℤ)*d := by |
| have hi' : (i : ℤ)+1 ≤ n := by omega |
| nlinarith |
| have hir : ((X+(i : ℤ)*d+d : ℤ) : ℝ) ≤ (A+4)*(2 : ℝ)^m := by |
| exact (by exact_mod_cast hiend : ((X+(i : ℤ)*d+d : ℤ) : ℝ) ≤ ((X+(n : ℤ)*d : ℤ) : ℝ)).trans hhi |
| obtain ⟨ds,hend,hcost,hin⟩ := gridConnectionsEvent_paths w b m₀ K A R hK k m J hJ hg j hj q |
| ⟨by simpa [hq'] using hlo.trans hicol,by simpa [hq'] using hir⟩ r hr (p i) (p (i+1)) |
| ⟨by simpa [p,endpointVertex] using hq'.symm,(endpointVertex_close b r _).le⟩ |
| ⟨by simp [p,endpointVertex,hq',Nat.cast_add,Nat.cast_one]; ring,(endpointVertex_close b r _).le⟩ |
| refine ⟨ds,hend,hcost,?_⟩ |
| intro t ht |
| have hz := hin t ht |
| rw [hq'] at hz |
| change X+(i : ℤ)*d-d ≤ (p i+displacement (ds.take t)).1 ∧ |
| (p i+displacement (ds.take t)).1 ≤ X+(i : ℤ)*d+2*d ∧ |
| |transverse b (p i+displacement (ds.take t))-r| ≤ (K+200)*(j : ℝ)^2*gridWidth k j at hz |
| change X-d ≤ (p i+displacement (ds.take t)).1 ∧ |
| (p i+displacement (ds.take t)).1 ≤ X+(n : ℤ)*d+d ∧ _ |
| refine ⟨?_,?_,hz.2.2.trans he⟩ |
| · have hi0 : (0 : ℤ) ≤ i := by positivity |
| nlinarith [hz.1] |
| · nlinarith [hz.2.1] |
| obtain ⟨ds,hend,hcost,hin⟩ := exists_path_chain w p S (fun _ => c) n hp0 step |
| refine ⟨ds,by simpa [p] using hend,?_,by simpa [p,S] using hin⟩ |
| have heq : (∑ _i ∈ range n,c) = |
| m₀*((n : ℤ)*gridMesh m j : ℤ)+(n : ℝ)*((K+200)*(j : ℝ)^2*gridWidth k j) := by |
| simp [c,d] |
| ring |
| rw [heq] at hcost |
| simpa [p] using hcost |
|
|
| end PlanarFPP |
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| end OAI |
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