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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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import OAI.Combinatorics.MatchingPSD.NormalizedFactors
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6 u_7
namespace PerfectMatchingPSD
open MatchingSmoothing Matrix
open scoped BigOperators
attribute [local instance] Classical.propDecidable
noncomputable def colorCut {n : ℕ} {V : Type u_1} [Fintype V]
(e : V ≃ Fin n) (z : V → Bool) (hz : Odd (Fintype.card {v // z v = true})) : OddCut n := by
classical
refine ⟨Finset.univ.filter (fun v => z (e.symm v) = true), ?_⟩
have he := Fintype.card_congr (Equiv.subtypeEquivOfSubtype' (p := fun v => z v = true) e)
have ho : Odd (Fintype.card {v : Fin n // z (e.symm v) = true}) := by
rw [← he]
exact hz
simpa only [Fintype.card_subtype] using ho
lemma colorCut_slack {n : ℕ} {V : Type u_2} [Fintype V]
(e : V ≃ Fin n) (z : V → Bool) (hz : Odd (Fintype.card {v // z v = true}))
(M : Pairing V) :
slack (Sum.inr (colorCut e z hz)) (fromPairing (Pairing.transport e M)) =
tableCoordinate z (true,false) M - 1 := by
classical
rw [slack, crossingCount_tableCoordinate, tableCoordinate_transport]
congr 2
funext v
simp [colorCut]
lemma true_card_comp_equiv {V : Type u_3} {W : Type u_4} [Fintype V] [Fintype W]
(z : V → Bool) (e : W ≃ V) :
Fintype.card {w // z (e w) = true} = Fintype.card {v // z v = true} :=
Fintype.card_congr (Equiv.subtypeEquiv e (fun _ => Iff.rfl))
lemma true_card_sum_false {V : Type u_5} {W : Type u_6} [Fintype V] [Fintype W] (z : V → Bool) :
Fintype.card {v : V ⊕ W // Sum.elim z (fun _ => false) v = true} =
Fintype.card {v : V // z v = true} := by
classical
let e : {v : V ⊕ W // Sum.elim z (fun _ => false) v = true} ≃ {v : V // z v = true} :=
{ toFun := fun v => match v with
| ⟨Sum.inl a,h⟩ => ⟨a,h⟩
| ⟨Sum.inr _,h⟩ => nomatch h
invFun := fun v => ⟨Sum.inl v.1,v.2⟩
left_inv := by
rintro ⟨v,h⟩
cases v with
| inl v => rfl
| inr v => exact False.elim (Bool.false_ne_true h)
right_inv := by intro v; rfl }
exact Fintype.card_congr e
lemma tableCoordinate_false {V : Type u_7} [Fintype V] (M : Pairing V) :
tableCoordinate (fun _ : V => false) (true,false) M = 0 := by
simp [tableCoordinate]
lemma global_factors_of_factorization {n r s q : ℕ} (hn : Even n) (hnpos : 1 ≤ n)
(hfac : HasFactorization n r) {W : Type} [Fintype W]
(e : (GlobalVertices s q ⊕ W) ≃ Fin n) (MW : Pairing W) :
∃ (Q : Equiv.Perm (GlobalVertices s q) → (Fin q → Bool) → Matrix (Fin r) (Fin r) ℝ)
(B : Pairing (GlobalVertices s q) → Matrix (Fin r) (Fin r) ℝ),
(∀ a y, (Q a y).PosSemidef) ∧ (∀ M, (B M).PosSemidef) ∧
(∀ a y, (Q a y).trace ≤ (n:ℝ)+1) ∧ (∀ M, (B M).trace ≤ (r:ℝ)) ∧
∀ a y M, (Q a y * B (Pairing.relabel a M)).trace =
tableCoordinate (globalCutColor y) (true,false) M - 1 := by
classical
obtain ⟨A,B,hA,hB,hpair,htrA,htrB⟩ := normalized_factorization hn hnpos hfac
let z (a : Equiv.Perm (GlobalVertices s q)) (y : Fin q → Bool) : GlobalVertices s q ⊕ W → Bool :=
Sum.elim (globalCutColor y ∘ a.symm) (fun _ => false)
have hz (a : Equiv.Perm (GlobalVertices s q)) (y : Fin q → Bool) :
Odd (Fintype.card {v // z a y v = true}) := by
rw [true_card_sum_false]
change Odd (Fintype.card {v // globalCutColor y (a.symm v) = true})
rw [true_card_comp_equiv]
exact globalCutColor_odd s q y
let U (a : Equiv.Perm (GlobalVertices s q)) (y : Fin q → Bool) := colorCut e (z a y) (hz a y)
refine ⟨fun a y => A (Sum.inr (U a y)),
fun M => B (fromPairing (Pairing.transport e (Pairing.sum M MW))),
fun a y => hA _, fun M => hB _, fun a y => htrA _, fun M => htrB _, ?_⟩
intro a y M
rw [hpair, colorCut_slack, tableCoordinate_sum, tableCoordinate_false, add_zero]
congr 1
rw [← Pairing.transport_perm, tableCoordinate_transport]
congr 1
ext v
simp
end PerfectMatchingPSD
end OAI
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