| import OAI.Probability.FirstPassage.BoundaryWinding |
| import Mathlib.GroupTheory.Perm.Cycle.Concrete |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Finset Equiv |
|
|
| theorem exists_cycle_nonzero_sum {α : Type*} [Fintype α] [DecidableEq α] |
| (p : Equiv.Perm α) (hp : ∀ x, p x ≠ x) (w : α → ℤ) |
| (hw : ∑ x, w x ≠ 0) : |
| ∃ x, ∑ y ∈ (p.toList x).toFinset, w y ≠ 0 := by |
| classical |
| let := Equiv.Perm.SameCycle.setoid p |
| let Q := Quotient (Equiv.Perm.SameCycle.setoid p) |
| let : Fintype Q := Fintype.ofFinite Q |
| let c : α → Q := Quotient.mk (Equiv.Perm.SameCycle.setoid p) |
| have hs := sum_fiberwise (univ : Finset α) c w |
| by_contra hn |
| push Not at hn |
| have hz (q : Q) : ∑ y ∈ univ.filter (fun y => c y = q), w y = 0 := by |
| induction q using Quotient.inductionOn with |
| | h x => |
| change (∑ y ∈ univ.filter (fun y => c y = c x), w y) = 0 |
| have he : univ.filter (fun y => c y = c x) = (p.toList x).toFinset := by |
| ext y |
| simp only [mem_filter, mem_univ, true_and, List.mem_toFinset, |
| Equiv.Perm.mem_toList_iff, Equiv.Perm.mem_support] |
| constructor |
| · intro hy |
| exact ⟨(Quotient.exact hy : p.SameCycle y x).symm, hp x⟩ |
| · intro hy |
| exact Quotient.sound hy.1.symm |
| rw [he] |
| exact hn x |
| simp_rw [hz] at hs |
| exact hw (by simpa using hs.symm) |
|
|
| theorem exists_boundary_cycle (S : Finset Vertex) (h0 : (0 : Vertex) ∈ S) : |
| ∃ x : BoundarySide S, |
| let es := (boundaryPerm S).toList x |
| es.Nodup ∧ 0 < es.length ∧ |
| (∀ e ∈ es, ((boundaryPerm S) e).val.tail = e.val.head) ∧ |
| (∀ e (he : e ∈ es), (es.next e he).val.tail = e.val.head) ∧ |
| (∑ e ∈ es.toFinset, e.val.rayCross) ≠ 0 := by |
| classical |
| have hw : ∑ e : BoundarySide S, e.val.rayCross ≠ 0 := by |
| have hh := boundarySides_rayCross_sum S |
| rw [ite_eq_left h0] at hh |
| rw [Finset.sum_coe_sort] |
| omega |
| obtain ⟨x,hx⟩ := exists_cycle_nonzero_sum (boundaryPerm S) |
| (boundaryPerm_ne_self S) (fun e => e.val.rayCross) hw |
| refine ⟨x, Equiv.Perm.nodup_toList _ _, ?_, ?_, ?_, hx⟩ |
| · exact Equiv.Perm.length_toList_pos_of_mem_support _ _ |
| (Equiv.Perm.mem_support.mpr (boundaryPerm_ne_self S x)) |
| · intro e he |
| exact boundaryPerm_tail S e |
| · intro e he |
| rw [Equiv.Perm.next_toList_eq_apply] |
| exact boundaryPerm_tail S e |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
