| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.ConeFrozenTarget |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace HyperbolicObstruction.ConeSkeleton |
| open IncidenceTemplate CoreGeometry LetterPresentation PictureSwitch Equiv Equiv.Perm |
| attribute [local instance] Classical.propDecidable |
| variable (n : ℕ) (w : TableModel.Entry (A := Alphabet n) rowSupport → Bool) |
| (d : (Fin 6 → Alphabet n) → Bool) |
|
|
| /-- The actual oriented macro edge in the indexed attaching polygon. -/ |
| def topDart (a : TopPolygon n w) (i : Fin a.2.2.val.length) : MacroGraph.graph.Dart := |
| ⟨(a.2.2.val.getVert i.val, |
| a.2.2.val.getVert ((finRotate a.2.2.val.length) i).val), |
| (MacroGraph.ofCycle a.2.2.val a.2.2.property).adjacent i⟩ |
|
|
| /-- Forget only the choice of simple circuit surrounding a top step. |
| Frozen fan/decorative cell identities and all their positions are retained. -/ |
| def cellWalkStep : (boundary n w d).Side → WalkStep n w d |
| | ⟨.inl a,b,i⟩ => .inr ⟨.inl a,b,i⟩ |
| | ⟨.inr (.inl a),b,i⟩ => .inl (a.1,if b then topDart n w a i else (topDart n w a i).symm) |
| | ⟨.inr (.inr a),b,i⟩ => .inr ⟨.inr a,b,i⟩ |
|
|
| lemma cellWalkStep_label (s : (boundary n w d).Side) : |
| walkLabel n w d (cellWalkStep n w d s) = (boundary n w d).sideLabel s := by |
| rcases s with ⟨a,b,i⟩ |
| rcases a with a | (a | a) |
| · rfl |
| · cases b with |
| | false => exact macroStep_reverse n w a.1 _ |
| | true => rfl |
| · rfl |
|
|
| lemma cellWalkStep_kind (s : (boundary n w d).Side) : |
| walkKind n w d (cellWalkStep n w d ((boundary n w d).next s)) = |
| walkKind n w d (cellWalkStep n w d s) := by |
| rcases s with ⟨a,b,i⟩ |
| rcases a with a | (a | a) <;> rfl |
|
|
| lemma cellWalkStep_closed (s : (boundary n w d).Side) : |
| walkFront n w d (cellWalkStep n w d ((boundary n w d).next s)) = |
| walkBack n w d (cellWalkStep n w d s) := by |
| rcases s with ⟨a,b,i⟩ |
| rcases a with a | (a | a) |
| · exact congrArg Sum.inr ((frozenBoundary n w d).next_start ⟨.inl a,b,i⟩) |
| · cases b with |
| | false => |
| change Sum.inl (a.1,a.2.2.val.getVert |
| ((finRotate a.2.2.val.length) ((finRotate a.2.2.val.length)⁻¹ i)).val) = |
| Sum.inl (a.1,a.2.2.val.getVert i.val) |
| change Fin a.2.2.val.length at i |
| rw [Perm.inv_def,Equiv.apply_symm_apply] |
| | true => rfl |
| · exact congrArg Sum.inr ((frozenBoundary n w d).next_start ⟨.inr a,b,i⟩) |
|
|
| lemma cellWalkStep_frozen_next (s : (boundary n w d).Side) |
| (h : walkFree n w d (walkKind n w d (cellWalkStep n w d s)) = false) : |
| cellWalkStep n w d ((boundary n w d).next s) = |
| (Equiv.sumCongr (1 : Perm (Top n w × MacroGraph.graph.Dart)) (frozenBoundary n w d).next) |
| (cellWalkStep n w d s) := by |
| rcases s with ⟨a,b,i⟩ |
| rcases a with a | (a | a) |
| · rfl |
| · cases h |
| · rfl |
|
|
| lemma cellWalkStep_frozen_inj {s t : (boundary n w d).Side} |
| (h : walkFree n w d (walkKind n w d (cellWalkStep n w d s)) = false) |
| (he : cellWalkStep n w d s = cellWalkStep n w d t) : s = t := by |
| rcases s with ⟨a,b,i⟩ |
| rcases t with ⟨a',b',i'⟩ |
| rcases a with a | (a | a) |
| · rcases a' with a' | (a' | a') |
| · have hh := Sum.inr.inj he |
| have ha : a = a' := Sum.inl.inj (Sigma.mk.inj hh).1 |
| subst a' |
| have hi := eq_of_heq (Sigma.mk.inj hh).2 |
| cases hi |
| rfl |
| · cases he |
| · have hh := (Sigma.mk.inj (Sum.inr.inj he)).1 |
| cases hh |
| · cases h |
| · rcases a' with a' | (a' | a') |
| · have hh := (Sigma.mk.inj (Sum.inr.inj he)).1 |
| cases hh |
| · cases he |
| · have hh := Sum.inr.inj he |
| have ha : a = a' := Sum.inr.inj (Sigma.mk.inj hh).1 |
| subst a' |
| have hi := eq_of_heq (Sigma.mk.inj hh).2 |
| cases hi |
| rfl |
|
|
| variable {A : Type*} [AddCommGroup A] |
| variable (c : Top n w → MacroGraph.graph.Dart → A) |
| (hc : ∀ a e, c a e.symm = -c a e) (f : FrozenCell n w d → A) |
| variable (weight : Cell n w d → A) |
| universe u |
| variable (P : CellPicture.{u} (target n w d weight)) |
|
|
| /-- Raw mixed pictures are now produced from actual indexed cellular pictures; |
| no simple/reduced/fan-link condition is assumed at this bridge. -/ |
| def cellToWalk : WalkPicture.{u} (walkTarget n w d c hc f) where |
| Dart := P.Dart |
| pairing := P.pairing |
| index := cellWalkStep n w d ∘ P.index |
| closed := by |
| intro x |
| change walkFront n w d (cellWalkStep n w d (P.index (P.pairing.face x))) = _ |
| rw [P.mapped.commute] |
| exact cellWalkStep_closed n w d _ |
| kind := by |
| intro x |
| change walkKind n w d (cellWalkStep n w d (P.index (P.pairing.face x))) = _ |
| rw [P.mapped.commute] |
| exact cellWalkStep_kind n w d _ |
| paired := by |
| intro x |
| change walkLabel n w d (cellWalkStep n w d (P.index (P.pairing.edge x))) = |
| rev n w (walkLabel n w d (cellWalkStep n w d (P.index x))) |
| rw [cellWalkStep_label,cellWalkStep_label] |
| exact P.respects.2 x |
| genusZero := P.genusZero |
|
|
| lemma cellToWalk_frozen : (cellToWalk n w d c hc f weight P).FrozenMapped |
| (walkFrozenTarget n w d c hc f) := by |
| constructor |
| · intro x |
| change cellWalkStep n w d (P.index (P.pairing.face x.val)) = _ |
| exact (congrArg (cellWalkStep n w d) (P.mapped.commute x.val)).trans |
| (cellWalkStep_frozen_next n w d _ x.property) |
| · intro x y hxy he |
| apply Subtype.ext |
| apply P.mapped.injective |
| · exact semiconj_perm_sameCycle (j := fun x : (cellToWalk n w d c hc f weight P).FrozenDart => (x.val : P.Dart)) |
| (t := P.pairing.face) (fun _ => rfl) hxy |
| · exact cellWalkStep_frozen_inj n w d x.property he |
|
|
| end HyperbolicObstruction.ConeSkeleton |
| end |
|
|
| end OAI |
|
|
