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dae740b | 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 | import Mathlib
namespace OAI
namespace HigherDimensionalBallPacking
open scoped ContDiff
abbrev Phase (n : ℕ) := Fin n → ℂ
noncomputable def capacity {n : ℕ} (z : Phase n) : ℝ :=
Real.pi * ∑ j, Complex.normSq (z j)
def closedBall (n : ℕ) (r : ℝ) : Set (Phase n) := {z | capacity z ≤ r}
def openBall (n : ℕ) (r : ℝ) : Set (Phase n) := {z | capacity z < r}
noncomputable def standardForm {n : ℕ} (u v : Phase n) : ℝ :=
∑ j, ((u j).re * (v j).im - (u j).im * (v j).re)
def SymplecticOn {n : ℕ} (U : Set (Phase n)) (f : Phase n → Phase n) : Prop :=
IsOpen U ∧ ContDiffOn ℝ ∞ f U ∧
Topology.IsEmbedding (fun x : U => f x) ∧
∀ x ∈ U, ∀ u v : Phase n,
standardForm (fderiv ℝ f x u) (fderiv ℝ f x v) = standardForm u v
def HasPacking (n k : ℕ) (R : ℝ) (r : Fin k → ℝ) : Prop :=
∃ (U : Fin k → Set (Phase n)) (f : Fin k → Phase n → Phase n),
(∀ i, closedBall n (r i) ⊆ U i ∧ SymplecticOn (U i) (f i) ∧
Set.MapsTo (f i) (closedBall n (r i)) (openBall n R)) ∧
(∀ i j, i ≠ j →
Disjoint (f i '' closedBall n (r i)) (f j '' closedBall n (r j)))
def PackingInequalities (n k : ℕ) (R : ℝ) (r : Fin k → ℝ) : Prop :=
(∑ i, r i ^ n) < R ^ n ∧ ∀ i j, i ≠ j → r i + r j < R
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
sorry
end HigherDimensionalBallPacking
end OAI
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