openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileQuotientAlgebra.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileMatrixProduct | |
| namespace OAI | |
| /-! Denominator and quotient estimates used by the free profile certificate. -/ | |
| open Matrix | |
| namespace DefocusingNLS.ProfileCertificate | |
| /-- The tail disk turns a finite matrix gap into a lower bound for the | |
| first component of the matched column. -/ | |
| theorem matched_denominator_lower (A : Matrix (Fin 2) (Fin 2) ℂ) | |
| (r s : ℂ) (ρ : ℝ) (hr : ‖r+s‖ ≤ ρ) : | |
| ‖A 0 1-s*A 0 0‖-ρ*‖A 0 0‖ ≤ ‖A 0 0*r+A 0 1‖ := by | |
| have he : A 0 1-s*A 0 0 = | |
| (A 0 0*r+A 0 1)-A 0 0*(r+s) := by ring | |
| have h := norm_sub_le (A 0 0*r+A 0 1) (A 0 0*(r+s)) | |
| rw [← he, norm_mul] at h | |
| have hm := mul_le_mul_of_nonneg_left hr (norm_nonneg (A 0 0)) | |
| nlinarith | |
| theorem matched_denominator_ne_zero (A : Matrix (Fin 2) (Fin 2) ℂ) | |
| (r s : ℂ) (ρ : ℝ) (hr : ‖r+s‖ ≤ ρ) | |
| (hgap : ρ*‖A 0 0‖ < ‖A 0 1-s*A 0 0‖) : A 0 0*r+A 0 1 ≠ 0 := by | |
| have h := matched_denominator_lower A r s ρ hr | |
| intro hz | |
| rw [hz, norm_zero] at h | |
| linarith | |
| /-- The only change from the zero-tail quotient is a determinant term. -/ | |
| theorem matched_quotient_identity (A : Matrix (Fin 2) (Fin 2) ℂ) (r : ℂ) | |
| (hb : A 0 1 ≠ 0) (hd : A 0 0*r+A 0 1 ≠ 0) : | |
| (A 1 0*r+A 1 1)/(A 0 0*r+A 0 1)-A 1 1/A 0 1 = | |
| -(r*A.det)/(A 0 1*(A 0 0*r+A 0 1)) := by | |
| rw [div_sub_div _ _ hd hb, mul_comm (A 0 0*r+A 0 1) (A 0 1)] | |
| congr 1 | |
| rw [Matrix.det_fin_two] | |
| ring | |
| /-- Quantitative version after lower bounds for both denominators are known. -/ | |
| theorem matched_quotient_bound (A : Matrix (Fin 2) (Fin 2) ℂ) (r : ℂ) | |
| (B D : ℝ) (hB : 0 < B) (hD : 0 < D) | |
| (hb : B ≤ ‖A 0 1‖) (hd : D ≤ ‖A 0 0*r+A 0 1‖) : | |
| ‖(A 1 0*r+A 1 1)/(A 0 0*r+A 0 1)-A 1 1/A 0 1‖ ≤ | |
| ‖r‖*‖A.det‖/(B*D) := by | |
| have hb0 : A 0 1 ≠ 0 := norm_pos_iff.mp (hB.trans_le hb) | |
| have hd0 : A 0 0*r+A 0 1 ≠ 0 := norm_pos_iff.mp (hD.trans_le hd) | |
| rw [matched_quotient_identity A r hb0 hd0, norm_div, norm_neg, norm_mul, norm_mul] | |
| exact div_le_div_of_nonneg_left (mul_nonneg (norm_nonneg _) (norm_nonneg _)) | |
| (mul_pos hB hD) (mul_le_mul hb hd (le_of_lt hD) (norm_nonneg _)) | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |