| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.PictureMapFurther |
| import OAI.Geometry.HyperbolicGroups.PermErasePair |
| import OAI.Geometry.HyperbolicGroups.GeneralSwitch |
| import OAI.Geometry.HyperbolicGroups.FaceWalk |
|
|
| namespace OAI |
|
|
| section |
| section |
|
|
|
|
| noncomputable section |
| namespace HyperbolicObstruction.PictureSwitch |
| open Equiv Equiv.Perm |
| attribute [local instance] Classical.propDecidable |
| variable {D : Type*} |
| namespace PictureMap |
| variable {M : PictureMap D} {a b : D} |
|
|
| structure Bigon (a b : D) : Prop where |
| ne : a ≠ b |
| forward : M.face a = b |
| backward : M.face b = a |
|
|
| namespace Bigon |
| variable (h : M.Bigon a b) |
| include h |
|
|
| lemma face_invariant (x : D) : OffPair a b (M.face x) ↔ OffPair a b x := by |
| have h₁ : M.face x = a ↔ x = b := by rw [← h.backward]; exact M.face.apply_eq_iff_eq |
| have h₂ : M.face x = b ↔ x = a := by rw [← h.forward] |
| simp only [OffPair,ne_eq,h₁,h₂,and_comm] |
| |
| lemma switching_vertex : M.vertex.SameCycle a (M.edge b) := by |
| have hh : M.vertex (M.edge b) = a := by rw [vertex,mul_apply,M.edge_edge,h.backward] |
| exact (hh ▸ (SameCycle.rfl (x := M.edge b)).apply_right).symm |
| |
| variable [DecidableEq D] |
|
|
| lemma switched_pair : (M.switch a (M.edge b)).edge a = b := by |
| change swap a (M.edge b) (M.edge (swap a (M.edge b) a)) = b |
| rw [swap_apply_left,M.edge_edge,swap_apply_of_ne_of_ne h.ne.symm (M.edge_ne b).symm] |
| |
| lemma switched_back : (M.switch a (M.edge b)).edge b = a := by |
| have hh := congrArg (M.switch a (M.edge b)).edge h.switched_pair |
| simpa only [edge_edge] using hh.symm |
|
|
| lemma edge_invariant (x : D) : OffPair a b ((M.switch a (M.edge b)).edge x) ↔ OffPair a b x := by |
| have h₁ : (M.switch a (M.edge b)).edge x = a ↔ x = b := by |
| simpa only [h.switched_back] using ((M.switch a (M.edge b)).edge.apply_eq_iff_eq (x := x) (y := b)) |
| have h₂ : (M.switch a (M.edge b)).edge x = b ↔ x = a := by |
| simpa only [h.switched_pair] using ((M.switch a (M.edge b)).edge.apply_eq_iff_eq (x := x) (y := a)) |
| simp only [OffPair,ne_eq,h₁,h₂,and_comm] |
| |
| /-- The zero-chain backtracking bigon is first pinched off by a legal vertex |
| switch and then deleted as a whole component. Its other edge contacts need |
| not initially meet one another. -/ |
| def remove : PictureMap {x // OffPair a b x} := |
| (M.switch a (M.edge b)).restrict (OffPair a b) h.face_invariant h.edge_invariant |
|
|
| lemma remove_face (x : {x // OffPair a b x}) : (h.remove.face x).val = M.face x.val := rfl |
|
|
| lemma remove_genusZero [Fintype D] (hG : M.GenusZero) : h.remove.GenusZero := by |
| exact (M.switch a (M.edge b)).restrict_genusZero (OffPair a b) |
| (M.switch_genusZero_general hG a (M.edge b) h.switching_vertex) |
| h.face_invariant h.edge_invariant |
|
|
| lemma remove_respects {L V : Type*} {label : D → L} {start : L → V} {rev : L → L} |
| (hM : M.Respects label start rev) (hab : label a = rev (label b)) : |
| h.remove.Respects (label ∘ Subtype.val) start rev := by |
| have hs := switch_respects hM (a := a) (b := M.edge b) (hab.trans (hM.2 b).symm) |
| exact ⟨fun x => hs.1 x.val,fun x => hs.2 x.val⟩ |
|
|
| lemma remove_closed {V : Type*} {source target : D → V} |
| (hc : M.FaceClosed source target) : h.remove.FaceClosed (source ∘ Subtype.val) (target ∘ Subtype.val) := |
| fun x => hc x.val |
|
|
| lemma remove_kind {C : Type*} {kind : D → C} (hk : M.FaceKind kind) : |
| h.remove.FaceKind (kind ∘ Subtype.val) := fun x => hk x.val |
|
|
| end Bigon |
| end PictureMap |
| end HyperbolicObstruction.PictureSwitch |
| end |
| end |
|
|
| section |
|
|
| end |
| end |
|
|
| end OAI |
|
|
