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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 | import Mathlib
import OAI.Geometry.HyperbolicGroups.NormalCell
import OAI.Geometry.HyperbolicGroups.WordPicture
import OAI.Geometry.HyperbolicGroups.IndexedBoundaryPullback
import OAI.Geometry.HyperbolicGroups.WordPictureFilling
namespace OAI
noncomputable section
namespace HyperbolicObstruction.PictureSwitch.CellPicture
open HyperbolicObstruction.WordPictures Equiv Equiv.Perm
attribute [local instance] Classical.propDecidable
variable {S A V B : Type*} [DecidableEq S] [AddCommMonoid B]
variable {T : CellTarget S (A × Bool) V B}
variable (P : CellPicture.{0} T) (hinv : T.rev = invLetter)
variable {R : Set (Word A)} {w : Word A}
/-- Reverse bridge from a literal genus-zero target-cell diagram to an
area-counted word picture. Every local premise is an actual attaching
polygon membership, not a filling or diagram-existence assumption. -/
def wordPicture (xs : List P.Dart) (hx : IsBoundary P.pairing.face xs)
(hw : xs.map (T.edge ∘ P.index) = w)
(hr : ∀ x, x ∉ xs → ∃ ys : List S, IsBoundary T.next ys ∧
P.index x ∈ ys ∧ SymmetricRelator R (ys.map T.edge)) : WordPicture R w where
Dart := P.Dart
pairing := P.pairing
genusZero := P.genusZero
label := T.edge ∘ P.index
paired := fun x => by rw [Function.comp_apply,Function.comp_apply,← hinv]; exact P.respects.2 x
outside := xs
boundary := hx
word := hw
interior := by
intro x hxo
obtain ⟨ys,ht,hy,hR⟩ := hr x hxo
obtain ⟨zs,hz,hzm,hmap⟩ := P.mapped.pullback_boundary x ht hy
refine ⟨zs,hz,hzm,?_⟩
have he : zs.map (T.edge ∘ P.index) = ys.map T.edge := by rw [← hmap,List.map_map]
rw [he]
exact hR
include hinv in
/-- Quantitative extraction of a mapped genus-zero disk. Costs count real
relator moves, including inverse orientations, in the original word alphabet. -/
theorem word_filling (xs : List P.Dart) (hx : IsBoundary P.pairing.face xs)
(hw : xs.map (T.edge ∘ P.index) = w)
(hr : ∀ x, x ∉ xs → ∃ ys : List S, IsBoundary T.next ys ∧
P.index x ∈ ys ∧ SymmetricRelator R (ys.map T.edge)) :
∃ k, Homotopy R w [] k ∧ k ≤ Fintype.card P.Dart / 2 ∧
(k : ℤ) ≤ cycles P.pairing.face :=
(P.wordPicture hinv xs hx hw hr).filling
end HyperbolicObstruction.PictureSwitch.CellPicture
end
end OAI
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