| import OAI.NumberTheory.Ostmann.Tree.CycleCut |
| import OAI.NumberTheory.Ostmann.Tree.CycleWalk |
|
|
| namespace OAI |
|
|
| namespace Ostmann.Tree |
| open scoped BigOperators |
|
|
| def quartetCut (d : ℕ) : Density.Cut (d+2) d := Density.Cut.ofLE (by omega) |
|
|
| theorem exists_quartet_cycle {d b : ℕ} (P : LeafPartition (d+2) b) |
| (hb : 4*b ≤ 3*2^(d+2)) : |
| Nonempty (BalancedSelection P.label (quartetCut d).label) := by |
| apply exists_balanced_selection |
| have hr : 2^(d+2)=4*2^d := by rw [pow_add]; ring |
| simp only [Leaves,Fintype.card_fin,Fintype.card_fun,Fintype.card_bool] |
| omega |
|
|
| theorem quartet_cycle_project {F : Type*} [Field F] {d b : ℕ} |
| (P : LeafPartition (d+2) b) |
| (c : BalancedSelection P.label (quartetCut d).label) |
| (z : Fˣ) (M : Leaves (d+2) → Fˣ) : |
| (quartetCut d).project (c.act z M) = (quartetCut d).project M := by |
| classical |
| funext v |
| rw [Density.Cut.project_eq_label_product,Density.Cut.project_eq_label_product] |
| exact c.right_product z M v |
|
|
| theorem quartet_cycle_components {F : Type*} [Field F] {d b : ℕ} |
| (P : LeafPartition (d+2) b) |
| (c : BalancedSelection P.label (quartetCut d).label) |
| (z : Fˣ) (M : Leaves (d+2) → Fˣ) : |
| P.componentProduct (c.act z M) = P.componentProduct M := by |
| classical |
| funext v |
| exact c.left_product z M v |
|
|
| end Ostmann.Tree |
|
|
| end OAI |
|
|
