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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 54 55 56 57 58 59 60 | import Mathlib
import OAI.Combinatorics.MatchingPSD.AffineCertificate
import OAI.Combinatorics.MatchingPSD.AffineSlacks
import OAI.Combinatorics.MatchingPSD.TightMatchings
import OAI.Combinatorics.MatchingPSD.NormalizedFactors
namespace OAI
namespace PerfectMatchingPSD
open MatchingSmoothing Finset Matrix
open scoped BigOperators
theorem affineLift_hasFactorization {n r : ℕ} (hn : 4 ≤ n) (he : Even n)
(h : HasAffineLift n r) : HasFactorization n r := by
classical
obtain ⟨L, T, hT⟩ := h
have hvertex (M : PerfectMatching n) : ∃ X : Matrix (Fin r) (Fin r) ℝ,
X ∈ L ∧ X.PosSemidef ∧ T X = matchingPoint M := by
have hm : matchingPoint M ∈ matchingPolytope n :=
subset_convexHull ℝ _ (Set.mem_range_self M)
rw [← hT] at hm
obtain ⟨X, hX, heq⟩ := hm
exact ⟨X, hX.1, hX.2, heq⟩
choose B hBL hB hBT using hvertex
have hrow (i : Row n) : ∃ A : Matrix (Fin r) (Fin r) ℝ,
A.PosSemidef ∧ ∀ M, (A * B M).trace = slack i M := by
obtain ⟨M₀, hzero⟩ := row_has_tight_matching hn he i
have hnonneg (X : Matrix (Fin r) (Fin r) ℝ) (hXL : X ∈ L)
(hXP : X.PosSemidef) : 0 ≤ ((slackAffine i).comp T) X := by
apply slackAffine_nonneg
rw [← hT]
exact ⟨X, ⟨hXL, hXP⟩, rfl⟩
have hz : ((slackAffine i).comp T) (B M₀) = 0 := by
simpa only [AffineMap.comp_apply, hBT, slackAffine_vertex] using hzero
obtain ⟨A, hA, hAB⟩ := AffinePSD.affine_slack_certificate L B hB hBL
((slackAffine i).comp T) hnonneg M₀ hz
refine ⟨A, hA, fun M => ?_⟩
simpa only [AffineMap.comp_apply, hBT, slackAffine_vertex] using hAB M
choose A hA hAB using hrow
exact ⟨A, B, hA, hB, fun i M => (hAB i M).symm⟩
lemma factorization_order_pos {n r : ℕ} (hn : 1 ≤ n) (he : Even n)
(h : HasFactorization n r) : 0 < r := by
classical
by_contra hr
have hr0 : r = 0 := by omega
subst r
obtain ⟨M⟩ := perfectMatching_nonempty he
let v : Fin n := ⟨0, by omega⟩
have hc := M.2 v
obtain ⟨e, heq⟩ := Finset.card_eq_one.mp hc
have heM : e ∈ M.1 := (Finset.mem_filter.mp (heq.symm ▸ Finset.mem_singleton_self e)).1
obtain ⟨A, B, hA, hB, hf⟩ := h
have hz := hf (Sum.inl e) M
simp [slack, heM, Matrix.trace] at hz
end PerfectMatchingPSD
end OAI
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