| import Mathlib | |
| namespace OAI | |
| namespace EuclideanSteinitzBergstrom | |
| abbrev Vector (d : ℕ) := EuclideanSpace ℝ (Fin d) | |
| def SignedPrefixBound (C : ℝ) : Prop := | |
| ∀ (d N : ℕ), 1 ≤ d → 1 ≤ N → | |
| ∀ (v : Fin N → Vector d), (∀ i, ‖v i‖ ≤ 1) → | |
| ∃ ε : Fin N → ℝ, (∀ i, ε i = -1 ∨ ε i = 1) ∧ | |
| ∀ k : ℕ, k ≤ N → | |
| ‖∑ i ∈ Finset.univ.filter (fun i : Fin N => i.val < k), ε i • v i‖ ≤ | |
| C * Real.sqrt d | |
| def OrderingPrefixBound (C : ℝ) : Prop := | |
| ∀ (d N : ℕ), 1 ≤ d → 1 ≤ N → | |
| ∀ (v : Fin N → Vector d), (∀ i, ‖v i‖ ≤ 1) → (∑ i, v i) = 0 → | |
| ∃ π : Equiv.Perm (Fin N), ∀ k : ℕ, k ≤ N → | |
| ‖∑ i ∈ Finset.univ.filter (fun i : Fin N => i.val < k), v (π i)‖ ≤ | |
| C * Real.sqrt d | |
| theorem main : ∃ C : ℝ, SignedPrefixBound C ∧ OrderingPrefixBound C := by | |
| sorry | |
| end EuclideanSteinitzBergstrom | |
| end OAI | |