| import OAI.Combinatorics.MatchingPSD.SequentialTests |
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| namespace OAI |
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| universe u_1 |
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| namespace PerfectMatchingPSD |
| open Matrix |
| open scoped MatrixOrder |
| variable {R : Type u_1} [Fintype R] [DecidableEq R] |
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| 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 |
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| 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] |
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| 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] |
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| 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 |
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| end PerfectMatchingPSD |
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| end OAI |
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