| import OAI.Probability.FirstPassage.ShapeDeficit |
| import OAI.Probability.FirstPassage.HighDensityOriented |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Filter |
| open scoped Topology |
|
|
| def crossingChain (initialPath : List Step) (trial connector : ℕ → List Step) : ℕ → List Step |
| | 0 => initialPath ++ trial 0 |
| | n+1 => crossingChain initialPath trial connector n ++ (connector n ++ trial (n+1)) |
|
|
| theorem crossingChain_endpoint (initialPath : List Step) (trial connector : ℕ → List Step) |
| (a z : ℕ → Vertex) (hp : displacement initialPath = a 0) |
| (ht : ∀ n, a n+displacement (trial n)=z n) |
| (hc : ∀ n, z n+displacement (connector n)=a (n+1)) (n : ℕ) : |
| displacement (crossingChain initialPath trial connector n)=z n := by |
| induction n with |
| | zero => simpa only [crossingChain,displacement_append,hp] using ht 0 |
| | succ n ih => |
| simp only [crossingChain,displacement_append,ih,← add_assoc,hc,ht] |
|
|
| theorem crossingChain_defect (w : Edge → ℝ) (b m a e : ℝ) |
| (initialPath : List Step) (trial connector : ℕ → List Step) (p z : ℕ → Vertex) |
| (hp : displacement initialPath=p 0) |
| (ht : ∀ n, p n+displacement (trial n)=z n) |
| (hc : ∀ n, z n+displacement (connector n)=p (n+1)) |
| (htrial : ∀ n, defect w b m a (p n) (trial n) ≤ -e) |
| (hconnector : ∀ n, defect w b m a (z n) (connector n) ≤ e/2) (n : ℕ) : |
| defect w b m a 0 (crossingChain initialPath trial connector n) ≤ |
| defect w b m a 0 initialPath-((n : ℝ)+1)*e+(n : ℝ)*e/2 := by |
| induction n with |
| | zero => |
| simp only [crossingChain,defect_append,zero_add,hp] |
| have hh := htrial 0 |
| norm_num |
| linarith |
| | succ n ih => |
| rw [crossingChain,defect_append,crossingChain_endpoint initialPath trial connector p z hp ht hc, |
| zero_add,defect_append,hc] |
| have h1 := htrial (n+1) |
| have h2 := hconnector n |
| push_cast |
| linarith |
|
|
| theorem oriented_path_coordinates (v : ℕ → Vertex) (h0 : v 0=0) |
| (hs : ∀ n, v (n+1)=v n+(1,0) ∨ v (n+1)=v n+(0,1)) (n : ℕ) : |
| (v n).1+(v n).2=(n : ℤ) ∧ 0 ≤ (v n).1 ∧ 0 ≤ (v n).2 ∧ (v n).2 ≤ n := by |
| induction n with |
| | zero => simp [h0] |
| | succ n ih => |
| rcases hs n with he | he <;> rw [he] <;> simp only [Prod.fst_add,Prod.snd_add] <;> |
| push_cast <;> omega |
|
|
| theorem UniformShape.no_crossing_chain (w : Edge → ℝ) (hw : ∀ e, 0 ≤ w e) |
| (μ : Plane → ℝ) (hu : UniformShape w μ) (b m a e B : ℝ) (he : 0 < e) (hB : 0 < B) |
| (hf : ∀ v, m*v 0+a*(v 1-b*v 0) ≤ μ v) |
| (initialPath : List Step) (trial connector : ℕ → List Step) (p z : ℕ → Vertex) |
| (hp : displacement initialPath=p 0) |
| (ht : ∀ n, p n+displacement (trial n)=z n) |
| (hc : ∀ n, z n+displacement (connector n)=p (n+1)) |
| (htrial : ∀ n, defect w b m a (p n) (trial n) ≤ -e) |
| (hconnector : ∀ n, defect w b m a (z n) (connector n) ≤ e/2) |
| (hescape : Tendsto (fun n => ‖latticeEmbed (z n)‖) atTop atTop) |
| (hbound : ∀ n, ‖latticeEmbed (z n)‖ ≤ B*((n : ℝ)+1)) : False := by |
| apply hu.no_linear_deficit w hw μ (fun v => m*v 0+a*(v 1-b*v 0)) hf z |
| (crossingChain initialPath trial connector) (defect w b m a 0 initialPath) (e/2) B |
| (by linarith) hB hescape hbound |
| (crossingChain_endpoint initialPath trial connector p z hp ht hc) |
| intro n |
| have hh := crossingChain_defect w b m a e initialPath trial connector p z hp ht hc htrial hconnector n |
| rw [defect,crossingChain_endpoint initialPath trial connector p z hp ht hc] at hh |
| simp only [support,transverse] at hh |
| change pathCost w 0 (crossingChain initialPath trial connector n) ≤ |
| m*(z n).1+a*((z n).2-b*(z n).1)+defect w b m a 0 initialPath-e/2*n |
| linarith |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
