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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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Mathlib
import OAI.Geometry.HyperbolicGroups.ConeTopCollapse
import OAI.Geometry.HyperbolicGroups.ConePresentationGroup
import OAI.Geometry.HyperbolicGroups.WordSubstitution
namespace OAI
noncomputable section
namespace HyperbolicObstruction.ConeSkeleton
open IncidenceTemplate CoreGeometry LetterPresentation
variable (n : ℕ) (w : TableModel.Entry (A := Alphabet n) rowSupport → Bool)
(d : (Fin 6 → Alphabet n) → Bool)
lemma square_collapsedBoundary (s : Square n w) :
collapsedBoundary n w d (.inr (.inr (.inl s))) =
literalSquare n (extendRow n (.inr (.inr s.1)) s.2.val) s.1 := by
rfl
lemma six_collapsedBoundary (s : Six n d) :
collapsedBoundary n w d (.inr (.inr (.inr s))) = List.ofFn s.val := by
rfl
/-- Uniform factor 30 for every actual bounded normal-form cell. The
fan paths, macro walk, exceptional loose edge and both decoration families
have all been expanded literally into the finite simple-cycle presentation. -/
theorem cell_collapse_filling (a : Cell n w d) :
∃ k ≤ 30, WordPictures.Homotopy (SimpleWordRelator n w d)
(collapsedBoundary n w d a) [] k := by
rcases a with f | (a | (s | s))
· obtain ⟨k,hk,hh⟩ := fan_collapse_filling n w d f
exact ⟨k,by omega,hh⟩
· exact top_collapse_filling n w d a
· rw [square_collapsedBoundary]
have hp : w (TableModel.row rowSupport (extendRow n (.inr (.inr s.1)) s.2.val)
(.inr (.inr s.1))) = true := by
simpa only [row_extend] using s.2.property
refine ⟨1,by decide,?_⟩
simpa only [List.nil_append,List.append_nil] using
WordPictures.Homotopy.move (WordPictures.Move.face [] [] _
(SimpleWordRelator.square _ s.1 hp))
· rw [six_collapsedBoundary]
refine ⟨1,by decide,?_⟩
simpa only [List.nil_append,List.append_nil] using
WordPictures.Homotopy.move (WordPictures.Move.face [] [] _
(SimpleWordRelator.six s.val s.property))
lemma collapse_letter_word (z : List (Alphabet n)) :
(z.map (letterDart n w)).flatMap (collapse n w) = z := by
induction z with
| nil => rfl
| cons a z ih => simp only [List.map_cons,List.flatMap_cons,collapse_letter,List.singleton_append,ih]
/-- Quantitative replacement of a literal polygon word derivation. This is
an algebraic substitution theorem, not an assumed bound on disk production. -/
theorem polygon_filling_to_simple {z : List (Alphabet n)} {k : ℕ}
(h : WordPictures.Homotopy (Set.range (boundary n w d).attachingWord)
(z.map (letterDart n w)) [] k) :
∃ a ≤ 30*k, WordPictures.Homotopy (SimpleWordRelator n w d) z [] a := by
have hs : ∀ r ∈ Set.range (boundary n w d).attachingWord,
∃ a ≤ 30, WordPictures.Homotopy (SimpleWordRelator n w d) (r.flatMap (collapse n w)) [] a := by
rintro r ⟨c,rfl⟩
exact cell_collapse_filling n w d c
obtain ⟨a,ha,hh⟩ := h.substitute (collapse n w) (fun e => collapse_rev n w e) hs
exact ⟨a,ha,by simpa only [collapse_letter_word,List.flatMap_nil] using hh⟩
/-- The cone-polygon comparison creates no new relations among original
letters. Combined with the opposite implication, it is an exact null-word
equivalence in the original marked presentation, not a surjection claim. -/
theorem nullWord_polygon_iff (z : List (Alphabet n)) :
PresentedGroup.mk (FreeGroup.mk '' Set.range (boundary n w d).attachingWord)
(FreeGroup.mk (z.map (letterDart n w))) = 1 ↔
PresentedGroup.mk (Relator n w d) (FreeGroup.mk z) = 1 := by
constructor
· intro hz
obtain ⟨k,hk⟩ := WordPictures.existence (z.map (letterDart n w)) hz
obtain ⟨a,ha,hh⟩ := polygon_filling_to_simple n w d hk
have he := WordPictures.sound hh
apply PresentedGroup.mk_eq_one_iff.mpr
have hm := PresentedGroup.mk_eq_one_iff.mp he
rw [simple_presentation_normalClosure] at hm
exact hm
· exact nullWord_to_polygon n w d z
end HyperbolicObstruction.ConeSkeleton
end
end OAI
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