| import Mathlib |
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| namespace OAI |
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|
| noncomputable section |
| namespace CKSArea |
| noncomputable section |
| open Matrix |
| open scoped Matrix |
| abbrev Mat2 := Matrix (Fin 2) (Fin 2) ℝ |
|
|
| def density (g : Mat2) : ℝ := Real.sqrt g.det |
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|
| lemma real_symmetric {n : Type*} {g : Matrix n n ℝ} (h : g.IsHermitian) (i j : n) : |
| g j i = g i j := by |
| simpa only [Matrix.conjTranspose_apply,star_trivial] using congrFun (congrFun h i) j |
|
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| lemma mixed_det_nonneg {g h : Mat2} (hg : g.PosDef) (hh : h.PosSemidef) : |
| 0 ≤ g 0 0*h 1 1+g 1 1*h 0 0-2*g 0 1*h 0 1 := by |
| have ha : 0 < g 0 0 := hg.diag_pos |
| have he : 0 ≤ h 0 0 := hh.diag_nonneg |
| have hd : 0 < g 0 0*g 1 1-(g 0 1)^2 := by |
| have hh' := hg.det_pos |
| rw [Matrix.det_fin_two,real_symmetric hg.isHermitian 0 1] at hh' |
| nlinarith |
| have hp := hh.dotProduct_mulVec_nonneg ![-g 0 1,g 0 0] |
| simp only [Matrix.mulVec, dotProduct, Fin.sum_univ_two, star_trivial, |
| Matrix.cons_val_zero,Matrix.cons_val_one, |
| Matrix.cons_val_fin_one] at hp |
| rw [real_symmetric hh.isHermitian 0 1] at hp |
| have hn : 0 ≤ g 0 0*(g 0 0*h 1 1+g 1 1*h 0 0-2*g 0 1*h 0 1) := by |
| nlinarith [mul_nonneg hd.le he] |
| exact nonneg_of_mul_nonneg_right hn ha |
|
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| lemma det_lower_bound {g p : Mat2} {c : ℝ} (hg : g.PosDef) (hc : 0 ≤ c) |
| (h : (p-c • g).PosSemidef) : c^2*g.det ≤ p.det := by |
| let e := p-c • g |
| have he : e.PosSemidef := h |
| have hp : p = c • g+e := by dsimp [e]; abel |
| have hmix := mixed_det_nonneg hg he |
| have hedet : 0 ≤ e.det := he.det_nonneg |
| have hg10 := real_symmetric hg.isHermitian 0 1 |
| have he10 := real_symmetric he.isHermitian 0 1 |
| rw [hp,Matrix.det_fin_two,Matrix.det_fin_two] |
| simp only [Matrix.add_apply,Matrix.smul_apply,smul_eq_mul] |
| rw [hg10,he10] |
| rw [Matrix.det_fin_two,he10] at hedet |
| nlinarith [mul_nonneg hc hmix] |
|
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| theorem density_lower_bound {g p : Mat2} {c : ℝ} (hg : g.PosDef) (hc : 0 ≤ c) |
| (h : (p-c • g).PosSemidef) : c*density g ≤ density p := by |
| have hh := Real.sqrt_le_sqrt (det_lower_bound hg hc h) |
| rw [Real.sqrt_mul (sq_nonneg c),Real.sqrt_sq_eq_abs,abs_of_nonneg hc] at hh |
| exact hh |
|
|
| end |
| end CKSArea |
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| end |
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| end OAI |
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|
