| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.WordPictureFold |
|
|
| namespace OAI |
|
|
| section |
| section |
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|
|
| noncomputable section |
| namespace HyperbolicObstruction.PictureSwitch |
| open Equiv Equiv.Perm |
| variable {D : Type*} [DecidableEq D] |
| lemma IsBoundary.rotate_append {p : Perm D} {l r : List D} |
| (h : IsBoundary p (l++r)) : IsBoundary p (r++l) := by |
| simpa only [List.rotate_append_length_eq] using h.rotate l.length |
|
|
| lemma IsBoundary.adjacent_rotate {p : Perm D} {xs : List D} (h : IsBoundary p xs) |
| {a b : D} (ha : a ∈ xs) (hab : a ≠ b) (hp : p a = b) : |
| ∃ k < xs.length, ∃ z, xs.rotate k = a::b::z := by |
| obtain ⟨l,r,he⟩ := List.mem_iff_append.mp ha |
| have hb : IsBoundary p ((a::r)++l) := (he ▸ h).rotate_append |
| have hlen : r++l ≠ [] := by |
| intro hn |
| have hc := hb.turn a (by simp) |
| rw [List.cons_append,hn,List.formPerm_singleton,one_apply,hp] at hc |
| exact hab hc.symm |
| obtain ⟨c,z,hz⟩ : ∃ c z, r++l = c::z := List.exists_cons_of_ne_nil hlen |
| have hc := hb.turn a (by simp) |
| rw [List.cons_append,hz,List.formPerm_apply_head _ _ _ (by simpa only [List.cons_append,hz] using hb.nodup),hp] at hc |
| refine ⟨l.length,by simp [he],z,?_⟩ |
| rw [he,List.rotate_append_length_eq,List.cons_append,hz,hc] |
|
|
| end HyperbolicObstruction.PictureSwitch |
|
|
| namespace HyperbolicObstruction.WordPictures |
| namespace Homotopy |
| variable {A : Type*} {R : Set (Word A)} {u v : Word A} {k : ℕ} |
| /-- Moving the boundary basepoint is area-free for a null word; it is not |
| asserted that a cyclic shift represents the same free-group element. -/ |
| lemma rotate_null (h : Homotopy R (v++u) [] k) : Homotopy R (u++v) [] k := by |
| have hf : FreeGroup.mk (u++v) = FreeGroup.mk (u++(v++u)++FreeGroup.invRev u) := by |
| simp only [← FreeGroup.mul_mk,← FreeGroup.inv_mk] |
| group |
| simpa only [Nat.zero_add] using (Homotopy.free hf).trans (h.conjugate u) |
| lemma unrotate_null {w : Word A} (i : ℕ) (hi : i ≤ w.length) |
| (h : Homotopy R (w.rotate i) [] k) : Homotopy R w [] k := by |
| rw [List.rotate_eq_drop_append_take hi] at h |
| simpa only [List.take_append_drop] using h.rotate_null |
| end Homotopy |
|
|
| namespace WordPicture |
| open HyperbolicObstruction.PictureSwitch |
| variable {A : Type*} {R : Set (Word A)} {w : Word A} |
| variable (P : WordPicture R w) |
|
|
| lemma all_outside_rotate (h : ∀ x, x ∈ P.outside) (k : ℕ) : |
| ∀ x, x ∈ (P.rotate k).outside := fun x => List.mem_rotate.mpr (h x) |
|
|
| lemma all_outside_fold {l r : List P.Dart} {a b : P.Dart} |
| (he : P.outside = l++a::b::r) (hc : P.label a = invLetter (P.label b)) |
| (h : ∀ x, x ∈ P.outside) : ∀ x, x ∈ (P.foldAt he hc).outside := by |
| intro x |
| change x ∈ (l++r).attachWith (OffPair a b) _ |
| apply (List.mem_attachWith _ x).mpr |
| have hx := h x.val |
| rw [he] at hx |
| simp only [List.mem_append,List.mem_cons] at hx ⊢ |
| rcases hx with hx | hx |
| · exact Or.inl hx |
| · rcases hx with ha | hb | hx |
| · exact False.elim (x.property.1 ha) |
| · exact False.elim (x.property.2 hb) |
| · exact Or.inr hx |
|
|
| end WordPicture |
| end HyperbolicObstruction.WordPictures |
| end |
| end |
|
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| section |
|
|
| end |
| end |
|
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| end OAI |
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|
