| import Mathlib | |
| import OAI.Geometry.HyperbolicGroups.BoundaryList | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace HyperbolicObstruction.PictureSwitch | |
| open Equiv Equiv.Perm | |
| attribute [local instance] Classical.propDecidable | |
| variable {D : Type*} [DecidableEq D] [Fintype D] | |
| /-- Every finite face orbit has a literal nonempty boundary enumeration, | |
| including monogons (which Perm.toList alone omits). -/ | |
| theorem exists_boundary (p : Perm D) (a : D) : | |
| ∃ xs, IsBoundary p xs ∧ a ∈ xs := by | |
| by_cases ha : p a = a | |
| · refine ⟨[a],⟨by simp,?_⟩,by simp⟩ | |
| intro x hx | |
| have he : x = a := by simpa only [List.mem_singleton] using hx | |
| subst x | |
| simpa only [List.formPerm_singleton,one_apply] using ha | |
| · have has : a ∈ p.support := mem_support.mpr ha | |
| refine ⟨p.toList a,⟨p.nodup_toList a,?_⟩,mem_toList_iff.mpr ⟨SameCycle.rfl,has⟩⟩ | |
| intro x hx | |
| rw [formPerm_toList] | |
| rw [cycleOf_apply,ite_eq_left (mem_toList_iff.mp hx).1] | |
| lemma IsBoundary.length_eq_card {p : Perm D} {xs : List D} (h : IsBoundary p xs) | |
| {a : D} (ha : a ∈ xs) : | |
| xs.length = Fintype.card {x : D // p.SameCycle a x} := by | |
| rw [Fintype.card_of_subtype xs.toFinset (by | |
| intro x | |
| rw [List.mem_toFinset] | |
| exact ⟨h.sameCycle ha,h.mem_of_sameCycle ha⟩),List.toFinset_card_of_nodup h.nodup] | |
| end HyperbolicObstruction.PictureSwitch | |
| end | |
| end | |
| section | |
| end | |
| end | |
| end OAI | |