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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 | 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
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