File size: 1,791 Bytes
e5a97c6 | 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 45 46 47 48 49 50 51 52 53 | import Mathlib
namespace OAI
namespace PerfectMatchingPSD
abbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2}
def IsPerfectMatching {n : ℕ} (M : Finset (Edge n)) : Prop :=
∀ v : Fin n, (M.filter fun e => e.1.1 = v ∨ e.1.2 = v).card = 1
abbrev PerfectMatching (n : ℕ) :=
{M : Finset (Edge n) // IsPerfectMatching M}
abbrev OddCut (n : ℕ) := {U : Finset (Fin n) // Odd U.card}
abbrev Row (n : ℕ) := Edge n ⊕ OddCut n
def Crosses {n : ℕ} (U : Finset (Fin n)) (e : Edge n) : Prop :=
(e.1.1 ∈ U ∧ e.1.2 ∉ U) ∨ (e.1.1 ∉ U ∧ e.1.2 ∈ U)
noncomputable def crossingCount {n : ℕ} (U : Finset (Fin n))
(M : PerfectMatching n) : ℕ := by
classical
exact (M.1.filter (Crosses U)).card
noncomputable def slack {n : ℕ} (i : Row n) (M : PerfectMatching n) : ℝ := by
classical
exact match i with
| Sum.inl e => if e ∈ M.1 then 1 else 0
| Sum.inr U => (crossingCount U.1 M : ℝ) - 1
/-- Real PSD factors of order `r`, without equivariance or rank-one restrictions. -/
def HasFactorization (n r : ℕ) : Prop :=
∃ (A : Row n → Matrix (Fin r) (Fin r) ℝ)
(B : PerfectMatching n → Matrix (Fin r) (Fin r) ℝ),
(∀ i, (A i).PosSemidef) ∧ (∀ M, (B M).PosSemidef) ∧
∀ i M, slack i M = Matrix.trace (A i * B M)
/-- The least positive order of a real PSD factorization. -/
noncomputable def psdRank (n : ℕ) : ℕ :=
sInf {r : ℕ | 0 < r ∧ HasFactorization n r}
/-- Every positive real power is eventually smaller than the PSD rank,
uniformly over all even vertex counts. The exponentiation is real `rpow`. -/
def MainClaim : Prop :=
∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧
∀ n : ℕ, n₀ ≤ n → Even n → (n : ℝ) ^ C < (psdRank n : ℝ)
end PerfectMatchingPSD
end OAI
|