File size: 3,735 Bytes
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 | import OAI.Probability.FirstPassage.EndpointGeometry
import OAI.Probability.FirstPassage.GridConnections
namespace OAI
namespace PlanarFPP
open Set Finset
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
end OAI
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