| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.ConeFullNormal |
| import OAI.Geometry.HyperbolicGroups.ActualFragment |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| namespace HyperbolicObstruction.ConeSkeleton |
| open IncidenceTemplate CoreGeometry LetterPresentation PictureSwitch Equiv Equiv.Perm |
| open Occurrences.Shape |
| attribute [local instance] Classical.propDecidable |
| universe u |
| variable (n : ℕ) (w : TableModel.Entry (A := Alphabet n) rowSupport → Bool) |
| (d : (Fin 6 → Alphabet n) → Bool) |
| 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 (P : WalkPicture.{u} (walkTarget n w d c hc f)) |
|
|
| lemma all_area : actualArea n w d c hc f P (fun _ => true) = Fintype.card P.Face := by |
| apply Fintype.card_congr |
| exact { toFun := Subtype.val |
| invFun := fun x => ⟨x,rfl⟩ |
| left_inv := fun _ => rfl |
| right_inv := fun _ => rfl } |
|
|
| lemma area_le_darts (select : P.Face → Bool) : |
| actualArea n w d c hc f P select ≤ Fintype.card P.Dart := by |
| have h1 : Fintype.card (SelectedFace n w d c hc f P select) ≤ Fintype.card P.Face := |
| Fintype.card_le_of_injective _ Subtype.val_injective |
| exact h1.trans (Fintype.card_le_of_surjective P.faceClass (fun x => ⟨x.out,P.faceClass_out x⟩)) |
|
|
| lemma all_boundary_zero : actualBoundary n w d c hc f P (fun _ => true) = 0 := by |
| have : IsEmpty (ActualRawBoundary n w d c hc f P (fun _ => true)) := |
| ⟨fun x => x.property.2.2 rfl⟩ |
| have : IsEmpty (RadialBoundarySide n w d c hc f P (fun _ => true)) := |
| ⟨fun x => x.property.2.2 rfl⟩ |
| simp only [actualBoundary,Fintype.card_of_isEmpty,Nat.add_zero] |
|
|
| /-- A bounded closed normal picture has no faces. This uses the genuine |
| rank event for its size, not an abstract incidence or link assumption. -/ |
| theorem small_normal_empty {M : ℕ} {α β δ : ℝ} |
| (ha : 0 < α) (hbeta : 7*α < β) (hd : 0 < δ) |
| (hgood : Occurrences.GoodUpTo M α β δ (Sum.elim w d)) |
| (hs : P.Simple) (hm : P.FrozenMapped (walkFrozenTarget n w d c hc f)) |
| (hred : P.FullyReduced) (hrad : P.RadialInjective (isApex n w)) (hb : P.NoBigon) |
| (hM : 8 * actualArea n w d c hc f P (fun _ => true) ≤ M) : IsEmpty P.Dart := by |
| have hh := actual_fragment_bound n w d c hc f P (fun _ => true) hs hm hred hrad ha.le hgood hb hM |
| rw [all_boundary_zero,all_area] at hh |
| simp only [← Nat.card_eq_fintype_card,Nat.cast_zero] at hh |
| have hk := defectConstant_pos ha hbeta hd |
| have hz : Fintype.card P.Face = 0 := by |
| rw [← Nat.card_eq_fintype_card] |
| have hz : (Nat.card P.Face : ℝ) ≤ 0 := by nlinarith |
| exact_mod_cast (le_antisymm hz (Nat.cast_nonneg (α := ℝ) (Nat.card P.Face))) |
| have : IsEmpty P.Face := Fintype.card_eq_zero_iff.mp hz |
| exact ⟨fun x => isEmptyElim (P.faceClass x)⟩ |
|
|
| /-- Bounded spherical-chain vanishing: the checked finite surgeries produce |
| normal form and preserve the literal chain. No topological realization is |
| asserted by this combinatorial statement. -/ |
| theorem small_closed_chain_zero {M : ℕ} {α β δ : ℝ} |
| (ha : 0 < α) (hbeta : 7*α < β) (hd : 0 < δ) |
| (hgood : Occurrences.GoodUpTo M α β δ (Sum.elim w d)) |
| (hm : P.FrozenMapped (walkFrozenTarget n w d c hc f)) |
| (hM : 8 * Fintype.card P.Dart ≤ M) : P.chain = 0 := by |
| obtain ⟨Q,⟨hPQ⟩,hmQ,hsQ,hbQ,hrQ,hradQ,hchain,_⟩ := normalizedPicture n w d c hc f P hm |
| have hcard : Fintype.card Q.Dart ≤ Fintype.card P.Dart := |
| Fintype.card_le_of_injective hPQ.map hPQ.injective |
| have hA := area_le_darts n w d c hc f Q (fun _ => true) |
| have : IsEmpty Q.Dart := small_normal_empty n w d c hc f Q ha hbeta hd hgood hsQ hmQ hrQ hradQ hbQ (by omega) |
| rw [← hchain] |
| simp only [WalkPicture.chain,Finset.univ_eq_empty,Finset.sum_empty] |
|
|
| end HyperbolicObstruction.ConeSkeleton |
| end |
|
|
| end OAI |
|
|
