| import OAI.Analysis.HyperbolicCones.FormBasic |
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| namespace OAI |
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| /-! Positivity of the Choi |
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| noncomputable section |
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| open scoped Matrix.Norms.L2Operator |
| open Matrix |
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| namespace Paper256 |
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| 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 |
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| 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 _) |
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| 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 |
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| 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 |
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| / |
| 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] |
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| 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 |
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| 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 |
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| 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 |
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| end Paper256 |
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| end |
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| end OAI |
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