import Mathlib

namespace OAI

namespace BorsukNine

abbrev Vector4 := EuclideanSpace ℝ (Fin 4)
abbrev Matrix4 := EuclideanSpace ℝ (Fin 4 × Fin 4)

def projector (u : Vector4) : Matrix4 :=
  WithLp.toLp 2 (fun ij => u ij.1 * u ij.2)

def projectorSet : Set Matrix4 :=
  projector '' {u : Vector4 | ‖u‖ = 1}

def traceOneSymmetric : Set Matrix4 :=
  {A | (∀ i j : Fin 4, A (i, j) = A (j, i)) ∧ ∑ i : Fin 4, A (i, i) = 1}

def HasTenSmallCover : Prop :=
  ∃ C : Fin 10 → Set Matrix4,
    (∀ i, C i ⊆ projectorSet) ∧
    projectorSet ⊆ ⋃ i, C i ∧
    ∀ i, Metric.diam (C i) < Real.sqrt 2

end BorsukNine

end OAI