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9eeb9c7 | 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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Analysis.HyperbolicCones.FormBasic
namespace OAI
/-! Positivity of the Choi--Lam form, including its principal-minor argument. -/
noncomputable section
open scoped Matrix.Norms.L2Operator
open Matrix
namespace Paper256
theorem cyclic_cubic_nonnegative (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) :
0 ≤ a ^ 2 * b + b ^ 2 * c + c ^ 2 * a - 3 * a * b * c := by
have aux (a b c : ℝ) (hc : 0 ≤ c) (hca : c ≤ a) (hcb : c ≤ b) :
0 ≤ a ^ 2 * b + b ^ 2 * c + c ^ 2 * a - 3 * a * b * c := by
have h := add_nonneg
(mul_nonneg hc (add_nonneg (sq_nonneg (a - b))
(mul_nonneg (sub_nonneg.mpr hca) (sub_nonneg.mpr hcb))))
(mul_nonneg (sq_nonneg (a - c)) (sub_nonneg.mpr hcb))
nlinarith only [h]
rcases le_total a b with hab | hba
· rcases le_total a c with hac | hca
· nlinarith only [aux b c a ha hab hac]
· exact aux a b c hc hca (hca.trans hab)
· rcases le_total b c with hbc | hcb
· nlinarith only [aux c a b hb hbc hba]
· exact aux a b c hc (hcb.trans hba) hcb
theorem zMatrix_det_nonnegative (z : Fin 3 → ℝ) : 0 ≤ Matrix.det (zMatrix z) := by
rw [zMatrix_det]
exact cyclic_cubic_nonnegative _ _ _ (sq_nonneg _) (sq_nonneg _) (sq_nonneg _)
private theorem quadratic_two_nonnegative (a b c x y : ℝ)
(ha : 0 ≤ a) (hc : 0 ≤ c) (hdet : b ^ 2 ≤ a * c) :
0 ≤ a * x ^ 2 + 2 * b * x * y + c * y ^ 2 := by
by_cases ha0 : a = 0
· have hb0 : b = 0 := by nlinarith [sq_nonneg b]
simpa [ha0, hb0] using mul_nonneg hc (sq_nonneg y)
· have hapos : 0 < a := lt_of_le_of_ne ha (Ne.symm ha0)
have h := add_nonneg (sq_nonneg (a * x + b * y))
(mul_nonneg (sub_nonneg.mpr hdet) (sq_nonneg y))
have hm : 0 ≤ a * (a * x ^ 2 + 2 * b * x * y + c * y ^ 2) := by
nlinarith only [h]
exact nonneg_of_mul_nonneg_right hm hapos
private theorem quadratic_three_nonnegative (a b c d e f x y z : ℝ)
(ha : 0 ≤ a) (hd : 0 ≤ d) (hf : 0 ≤ f)
(hab : b ^ 2 ≤ a * d) (hac : c ^ 2 ≤ a * f) (hdf : e ^ 2 ≤ d * f)
(hdet : 0 ≤ a * d * f - a * e ^ 2 - b ^ 2 * f + 2 * b * c * e - c ^ 2 * d) :
0 ≤ a * x ^ 2 + d * y ^ 2 + f * z ^ 2 +
2 * b * x * y + 2 * c * x * z + 2 * e * y * z := by
by_cases ha0 : a = 0
· have hb0 : b = 0 := by nlinarith [sq_nonneg b]
have hc0 : c = 0 := by nlinarith [sq_nonneg c]
have h := quadratic_two_nonnegative d e f y z hd hf hdf
simpa [ha0, hb0, hc0, add_comm, add_left_comm, add_assoc] using h
· have hapos : 0 < a := lt_of_le_of_ne ha (Ne.symm ha0)
have hsmall : (a * e - b * c) ^ 2 ≤ (a * d - b ^ 2) * (a * f - c ^ 2) := by
nlinarith only [mul_nonneg ha hdet]
have hs := quadratic_two_nonnegative (a * d - b ^ 2) (a * e - b * c)
(a * f - c ^ 2) y z (sub_nonneg.mpr hab) (sub_nonneg.mpr hac) hsmall
have hm : 0 ≤ a * (a * x ^ 2 + d * y ^ 2 + f * z ^ 2 +
2 * b * x * y + 2 * c * x * z + 2 * e * y * z) := by
nlinarith only [sq_nonneg (a * x + b * y + c * z), hs]
exact nonneg_of_mul_nonneg_right hm hapos
/-- For a symmetric three-by-three matrix, its principal minors certify positivity. -/
theorem posSemidef_three_of_principal_minors (M : Mat 3 ℝ) (hM : M.IsHermitian)
(hdiag : ∀ i, 0 ≤ M i i)
(h01 : M 0 1 ^ 2 ≤ M 0 0 * M 1 1)
(h02 : M 0 2 ^ 2 ≤ M 0 0 * M 2 2)
(h12 : M 1 2 ^ 2 ≤ M 1 1 * M 2 2) (hdet : 0 ≤ Matrix.det M) :
M.PosSemidef := by
have h10 : M 1 0 = M 0 1 := by simpa using hM.apply 0 1
have h20 : M 2 0 = M 0 2 := by simpa using hM.apply 0 2
have h21 : M 2 1 = M 1 2 := by simpa using hM.apply 1 2
have hd : 0 ≤ M 0 0 * M 1 1 * M 2 2 - M 0 0 * M 1 2 ^ 2 -
M 0 1 ^ 2 * M 2 2 + 2 * M 0 1 * M 0 2 * M 1 2 - M 0 2 ^ 2 * M 1 1 := by
rw [Matrix.det_fin_three, h10, h20, h21] at hdet
nlinarith only [hdet]
refine Matrix.PosSemidef.of_dotProduct_mulVec_nonneg hM fun x => ?_
have h := quadratic_three_nonnegative (M 0 0) (M 0 1) (M 0 2)
(M 1 1) (M 1 2) (M 2 2) (x 0) (x 1) (x 2)
(hdiag 0) (hdiag 1) (hdiag 2) h01 h02 h12 hd
simp [dotProduct, Matrix.mulVec, Fin.sum_univ_succ, h10, h20, h21]
nlinarith only [h]
theorem zMatrix_posSemidef (z : Fin 3 → ℝ) : (zMatrix z).PosSemidef := by
apply posSemidef_three_of_principal_minors _ (zMatrix_isHermitian z)
(zMatrix_diag_nonnegative z)
· have h := zMatrix_principal_minor_two_nonnegative z 0
simp [Matrix.det_fin_two, Matrix.submatrix, zMatrix, Fin.reduceAdd] at h ⊢
nlinarith only [h]
· have h := zMatrix_principal_minor_two_nonnegative z 2
simp [Matrix.det_fin_two, Matrix.submatrix, zMatrix, Fin.reduceAdd, Fin.reduceEq] at h ⊢
nlinarith only [h]
· have h := zMatrix_principal_minor_two_nonnegative z 1
simp [Matrix.det_fin_two, Matrix.submatrix, zMatrix, Fin.reduceAdd, Fin.reduceEq] at h ⊢
nlinarith only [h]
· exact zMatrix_det_nonnegative z
theorem choiLam_nonnegative (z y : Fin 3 → ℝ) : 0 ≤ choiLam z y := by
have h := (zMatrix_posSemidef z).dotProduct_mulVec_nonneg y
simpa only [star_trivial, zMatrix_quadratic] using h
theorem qMatrix_posSemidef (y : Fin 3 → ℝ) : (qMatrix y).PosSemidef := by
refine Matrix.PosSemidef.of_dotProduct_mulVec_nonneg (qMatrix_isHermitian y) fun z => ?_
simpa only [star_trivial, qMatrix_quadratic] using choiLam_nonnegative z y
end Paper256
end
end OAI
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