| import Mathlib | |
| import OAI.Geometry.BallPacking.Rigidity.TwoBallBound | |
| import OAI.Geometry.BallPacking.Packings.ComplementPacking | |
| namespace OAI | |
| noncomputable section | |
| namespace HigherDimensionalBallPacking | |
| theorem HasPacking.strict_pair_bound {n k : ℕ} (hn : 3≤n) | |
| {R : ℝ} {r : Fin k → ℝ} (hR : 0<R) (hr : ∀ i,0<r i) | |
| (hp : HasPacking n k R r) (i j : Fin k) (hij : i≠j) : r i+r j<R := by | |
| by_contra hbad | |
| obtain ⟨a,b,ha,hb,hab,hpab⟩ := hp.unit_pair_counterexample hR hr hij (le_of_not_gt hbad) | |
| exact (not_le_of_gt hab) (normalized_two_ball_bound n hn a b ha hb hpab) | |
| theorem main_theorem (n k : ℕ) (hn : 3 ≤ n) (hk : 1 ≤ k) | |
| (R : ℝ) (r : Fin k → ℝ) (hR : 0 < R) (hr : ∀ i, 0 < r i) : | |
| HasPacking n k R r ↔ PackingInequalities n k R r := by | |
| constructor | |
| · intro hp | |
| exact ⟨hp.volume_lt (by omega) hR hr,hp.strict_pair_bound hn hR hr⟩ | |
| · exact hasPacking_of_inequalities hn hk hR hr | |
| end HigherDimensionalBallPacking | |
| end | |
| end OAI | |