| 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] | |
| lemma formPerm_append_join (l : List D) (a b : D) (r : List D) : | |
| (l++a::b::r).formPerm = (l++[a]).formPerm * swap a b * (b::r).formPerm := by | |
| induction l with | |
| | nil => simp only [List.nil_append,List.formPerm_singleton,one_mul,List.formPerm_cons_cons] | |
| | cons x l ih => | |
| cases l with | |
| | nil => simp only [List.cons_append,List.nil_append,List.formPerm_cons_cons,List.formPerm_singleton,mul_one,mul_assoc] | |
| | cons y l => | |
| simp only [List.cons_append,List.formPerm_cons_cons] at ih ⊢ | |
| rw [ih]; simp only [mul_assoc] | |
| namespace IsBoundary | |
| variable {f : Perm D} {l r : List D} {a b : D} | |
| /-- Concatenation is the literal connected-sum splice at the two incoming | |
| corners. The side order, not merely the freely represented word, is retained. -/ | |
| theorem splice_append (hl : IsBoundary f (l++[a])) (hr : IsBoundary f (b::r)) | |
| (hd : ∀ x ∈ l++[a], x ∉ b::r) : | |
| IsBoundary (f * swap a ((b::r).getLast (by simp))) ((l++[a])++(b::r)) := by | |
| let xs := l++[a] | |
| let ys := b::r | |
| let c := ys.getLast (by simp [ys]) | |
| have ha : a ∈ xs := by simp [xs] | |
| have hc : c ∈ ys := List.getLast_mem _ | |
| have hdis : ∀ x ∈ xs, x ∉ ys := hd | |
| have hn : (xs++ys).Nodup := by | |
| rw [List.nodup_append] | |
| exact ⟨hl.nodup,hr.nodup,fun x hx y hy he => hdis x hx (he ▸ hy)⟩ | |
| have hform : (xs++ys).formPerm = xs.formPerm * ys.formPerm * swap a c := by | |
| have hz : ys.formPerm a = a := List.formPerm_apply_of_notMem (hdis a ha) | |
| have hy : ys.formPerm c = b := List.formPerm_apply_getLast b r | |
| have hs : ys.formPerm * swap a c = swap a b * ys.formPerm := by | |
| rw [mul_swap_eq_swap_mul,hz,hy] | |
| dsimp only [xs,ys] | |
| rw [List.append_assoc] | |
| change (l++a::b::r).formPerm = (l++[a]).formPerm * (b::r).formPerm * swap a c | |
| rw [formPerm_append_join] | |
| simp only [mul_assoc] | |
| rw [hs] | |
| have ht : ∀ x ∈ xs++ys, f x = (xs.formPerm*ys.formPerm) x := by | |
| intro x hx | |
| rcases List.mem_append.mp hx with hx | hx | |
| · rw [mul_apply,List.formPerm_apply_of_notMem (hdis x hx),hl.turn x hx] | |
| · have hn : ys.formPerm x ∉ xs := by | |
| intro hh | |
| exact hdis _ hh (List.formPerm_apply_mem_of_mem hx) | |
| rw [mul_apply,List.formPerm_apply_of_notMem hn,hr.turn x hx] | |
| refine ⟨hn,?_⟩ | |
| intro x hx | |
| have hs : swap a c x ∈ xs++ys := by | |
| by_cases he : x = a | |
| · subst x; rw [swap_apply_left]; exact List.mem_append_right xs hc | |
| by_cases hf : x = c | |
| · subst x; rw [swap_apply_right]; exact List.mem_append_left ys ha | |
| · rwa [swap_apply_of_ne_of_ne he hf] | |
| change f (swap a c x) = (xs++ys).formPerm x | |
| rw [ht _ hs,hform]; rfl | |
| end IsBoundary | |
| end HyperbolicObstruction.PictureSwitch | |
| end | |
| end | |
| section | |
| end | |
| end | |
| end OAI | |