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 packing_necessity (n k : ℕ) (hn : 3 ≤ n) (hk : 1 ≤ k)
    (R : ℝ) (r : Fin k → ℝ) (hR : 0 < R) (hr : ∀ i, 0 < r i)
    (hp : HasPacking n k R r) : PackingInequalities n k R r := by
  sorry

end HigherDimensionalBallPacking

end OAI