File size: 3,624 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 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 | import OAI.Combinatorics.MatchingPSD.SequentialTests
namespace OAI
universe u_1
namespace PerfectMatchingPSD
open Matrix
open scoped MatrixOrder
variable {R : Type u_1} [Fintype R] [DecidableEq R]
lemma two_div_success {μ ε : ℝ} (hμ : 2 ≤ μ) (hε : 0 ≤ ε) (hub : μ ≤ 2+ε) :
1-ε ≤ 2/μ := by
rw [le_div_iff₀ (by linarith : 0 < μ)]
have hh : ε ≤ ε*μ := by nlinarith
nlinarith
lemma normalized_gauge_pair (G Q T : Matrix R R ℝ) (hu : IsUnit G) :
(G*Q*G*(G⁻¹*T*G⁻¹)).trace = (Q*T).trace := by
have hunit := (Matrix.isUnit_iff_isUnit_det G).mp hu
have hl : G⁻¹*G=1 := Matrix.nonsing_inv_mul G hunit
have hr : G*G⁻¹=1 := Matrix.mul_nonsing_inv G hunit
have he : G*Q*G*(G⁻¹*T*G⁻¹) = G*(Q*T)*G⁻¹ := by
calc
_ = G*Q*(G*G⁻¹)*T*G⁻¹ := by simp only [mul_assoc]
_ = _ := by rw [hr, mul_one]; simp only [mul_assoc]
rw [he, Matrix.trace_mul_cycle, hl, one_mul]
lemma inverse_sqrt_congruence (D : Matrix R R ℝ) (hD : D.PosDef) :
(CFC.sqrt D)⁻¹ * D * (CFC.sqrt D)⁻¹ = 1 := by
let G := CFC.sqrt D
have hGG : G*G=D := CFC.sqrt_mul_sqrt_self D hD.posSemidef.nonneg
have hu : IsUnit G := (IsUnit.mul_iff.mp (hGG ▸ hD.isUnit)).1
have hl : G⁻¹*G=1 := Matrix.nonsing_inv_mul G ((Matrix.isUnit_iff_isUnit_det G).mp hu)
have hr : G*G⁻¹=1 := Matrix.mul_nonsing_inv G ((Matrix.isUnit_iff_isUnit_det G).mp hu)
change G⁻¹*D*G⁻¹=1
rw [← hGG, ← mul_assoc, hl, one_mul, hr]
lemma states_of_domination {q : ℕ}
(Q : (Fin q → Bool) → Matrix R R ℝ) (T : Fin q → Bool → Matrix R R ℝ)
(D : Matrix R R ℝ) (hD : D.PosDef)
(hQ : ∀ y, (Q y).PosSemidef) (hT : ∀ i b, (T i b).PosSemidef)
(hdom : ∀ i, T i false + T i true ≤ D)
(ε : ℝ) (hε : 0 ≤ ε)
(htr : ∀ y, 2 ≤ (Q y*D).trace ∧ (Q y*D).trace ≤ 2+ε)
(hbit : ∀ y i, (Q y * T i (y i)).trace = 2) :
∃ (ρ : (Fin q → Bool) → Matrix R R ℝ) (F : Fin q → Bool → Matrix R R ℝ),
(∀ y, (ρ y).PosSemidef) ∧ (∀ y, (ρ y).trace = 1) ∧
(∀ i b, 0 ≤ F i b) ∧ (∀ i, F i false + F i true ≤ 1) ∧
∀ y i, 1-ε ≤ (ρ y*F i (y i)).trace := by
let G := CFC.sqrt D
have hG : G.PosSemidef := (CFC.sqrt_nonneg D).posSemidef
have hGG : G*G=D := CFC.sqrt_mul_sqrt_self D hD.posSemidef.nonneg
have hu : IsUnit G := (IsUnit.mul_iff.mp (hGG ▸ hD.isUnit)).1
have hGi : G⁻¹.PosSemidef := hG.inv
let μ := fun y => (Q y*D).trace
have hμ (y : Fin q → Bool) : 0 < μ y := lt_of_lt_of_le (by norm_num) (htr y).1
refine ⟨fun y => (μ y)⁻¹ • (G*Q y*G), fun i b => G⁻¹*T i b*G⁻¹,
?_, ?_, ?_, ?_, ?_⟩
· intro y
apply Matrix.PosSemidef.smul _ (le_of_lt (inv_pos.mpr (hμ y)))
simpa only [hG.isHermitian.eq] using (hQ y).conjTranspose_mul_mul_same G
· intro y
rw [Matrix.trace_smul, Matrix.trace_mul_cycle, hGG, Matrix.trace_mul_comm]
change (μ y)⁻¹ * μ y = 1
exact inv_mul_cancel₀ (ne_of_gt (hμ y))
· intro i b
apply Matrix.PosSemidef.nonneg
simpa only [hGi.isHermitian.eq] using (hT i b).conjTranspose_mul_mul_same G⁻¹
· intro i
have hg := congruence_mono (hdom i) G⁻¹
have hGit : (G⁻¹)ᵀ=G⁻¹ := hGi.isHermitian.eq
rw [hGit, inverse_sqrt_congruence D hD, mul_add, add_mul] at hg
exact hg
· intro y i
rw [smul_mul_assoc, Matrix.trace_smul, normalized_gauge_pair G (Q y) (T i (y i)) hu,
hbit y i]
change 1-ε ≤ (μ y)⁻¹*2
rw [mul_comm, ← div_eq_mul_inv]
exact two_div_success (htr y).1 hε (htr y).2
end PerfectMatchingPSD
end OAI
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