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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | 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
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