| import OAI.Probability.FirstPassage.Paths | |
| import Mathlib.Data.List.Count | |
| import Lean.Elab.Tactic.Omega | |
| namespace OAI | |
| /-! The closedness count in crossings.tex:cr:oriented-paths. A contour need not | |
| be simple at vertices; its zero displacement alone gives the half-length count. -/ | |
| namespace PlanarFPP | |
| theorem displacement_counts (ds : List Step) : | |
| (displacement ds).1=(ds.count Step.east : ℤ)-ds.count Step.west ∧ | |
| (displacement ds).2=(ds.count Step.north : ℤ)-ds.count Step.south ∧ | |
| ds.length=ds.count Step.east+ds.count Step.west+ds.count Step.north+ds.count Step.south := by | |
| induction ds with | |
| | nil => simp | |
| | cons d ds ih => | |
| cases d <;> simp_all [displacement_cons,Step.delta] <;> omega | |
| /-- Exactly half the edges of a closed square-lattice walk point upward or | |
| leftward, as required before selecting bad sites in the source contour proof. -/ | |
| theorem closed_walk_up_left_count (ds : List Step) (h : displacement ds=0) : | |
| ds.length=2*(ds.count Step.north+ds.count Step.west) := by | |
| obtain ⟨hx,hy,hl⟩ := displacement_counts ds | |
| rw [h] at hx hy | |
| simp only [Prod.fst_zero,Prod.snd_zero] at hx hy | |
| omega | |
| end PlanarFPP | |
| end OAI | |