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7d9acbd | 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 | import Mathlib
import OAI.Geometry.HyperbolicGroups.BoundaryList
namespace OAI
section
section
noncomputable section
namespace HyperbolicObstruction.PictureSwitch
open Equiv Equiv.Perm SimpleGraph
attribute [local instance] Classical.propDecidable
variable {D V : Type*} [DecidableEq D] {p : Perm D} {G : SimpleGraph V}
variable {xs : List D} (h : IsBoundary p xs) (j : D → G.Dart)
(hc : ∀ x ∈ xs, (j (p x)).fst = (j x).snd)
include h hc in
lemma IsBoundary.graph_chain : (xs.map j).IsChain G.DartAdj := by
rw [List.isChain_iff_getElem]
intro i hi
have hl : i+1 < xs.length := by simpa only [List.length_map] using hi
have hi' : i < xs.length := by omega
simp only [List.getElem_map]
change (j xs[i]).snd = (j xs[i+1]).fst
have he := hc xs[i] (List.getElem_mem hi')
rw [h.turn _ (List.getElem_mem hi'),List.formPerm_apply_getElem _ h.nodup] at he
simpa only [Nat.mod_eq_of_lt hl] using he.symm
include h hc in
/-- Read a literal permutation face as an actual graph closed walk. -/
lemma IsBoundary.graph_closed_walk (hne : xs ≠ []) :
∃ (v : V) (q : G.Walk v v), q.darts = xs.map j := by
have hn : xs.map j ≠ [] := by
intro hh
exact hne (List.map_eq_nil_iff.mp hh)
let q := Walk.ofDarts (xs.map j) hn (h.graph_chain j hc)
have he : ((xs.map j).getLast hn).snd = ((xs.map j).head hn).fst := by
rw [List.getLast_map,List.head_map]
have hh := hc (xs.getLast hne) (List.getLast_mem hne)
rw [h.turn _ (List.getLast_mem hne)] at hh
have hp : xs.formPerm (xs.getLast hne) = xs.head hne := by
cases xs with
| nil => contradiction
| cons a xs => exact List.formPerm_apply_getLast a xs
rw [hp] at hh
exact hh.symm
refine ⟨((xs.map j).head hn).fst,q.copy rfl he,?_⟩
rw [Walk.darts_copy]
exact Walk.darts_ofDarts hn (h.graph_chain j hc)
end HyperbolicObstruction.PictureSwitch
end
end
section
end
end
end OAI
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