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8e4d533 | 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 | 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
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