openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileMatrixMajorant.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileMatrixProduct | |
| namespace OAI | |
| /-! Entrywise majorants for the finite free-profile Taylor remainder. -/ | |
| open Matrix | |
| namespace DefocusingNLS.ProfileCertificate | |
| /-- Entrywise complex modulus bounds, retaining the two-term matrix product. -/ | |
| def Dominates (E : Matrix (Fin 2) (Fin 2) ℝ) | |
| (A : Matrix (Fin 2) (Fin 2) ℂ) : Prop := ∀ i j, ‖A i j‖ ≤ E i j | |
| namespace Dominates | |
| theorem nonneg {E : Matrix (Fin 2) (Fin 2) ℝ} {A : Matrix (Fin 2) (Fin 2) ℂ} | |
| (h : Dominates E A) (i j : Fin 2) : 0 ≤ E i j := (norm_nonneg _).trans (h i j) | |
| theorem add {E F : Matrix (Fin 2) (Fin 2) ℝ} {A B : Matrix (Fin 2) (Fin 2) ℂ} | |
| (hA : Dominates E A) (hB : Dominates F B) : Dominates (E+F) (A+B) := by | |
| intro i j | |
| exact (norm_add_le _ _).trans (add_le_add (hA i j) (hB i j)) | |
| theorem mul {E F : Matrix (Fin 2) (Fin 2) ℝ} {A B : Matrix (Fin 2) (Fin 2) ℂ} | |
| (hA : Dominates E A) (hB : Dominates F B) : Dominates (E*F) (A*B) := by | |
| intro i j | |
| simp only [Matrix.mul_apply, Fin.sum_univ_two] | |
| apply (norm_add_le _ _).trans | |
| rw [norm_mul, norm_mul] | |
| exact add_le_add | |
| (mul_le_mul (hA i 0) (hB 0 j) (norm_nonneg _) (hA.nonneg i 0)) | |
| (mul_le_mul (hA i 1) (hB 1 j) (norm_nonneg _) (hA.nonneg i 1)) | |
| theorem smul {E : Matrix (Fin 2) (Fin 2) ℝ} {A : Matrix (Fin 2) (Fin 2) ℂ} | |
| (hA : Dominates E A) (r : ℝ) : Dominates (|r| • E) (r • A) := by | |
| intro i j | |
| simp only [Matrix.smul_apply, norm_smul, Real.norm_eq_abs, smul_eq_mul] | |
| exact mul_le_mul_of_nonneg_left (hA i j) (abs_nonneg r) | |
| theorem mono {E F : Matrix (Fin 2) (Fin 2) ℝ} {A : Matrix (Fin 2) (Fin 2) ℂ} | |
| (hA : Dominates E A) (hEF : ∀ i j, E i j ≤ F i j) : Dominates F A := by | |
| intro i j | |
| exact (hA i j).trans (hEF i j) | |
| theorem smul_le {E : Matrix (Fin 2) (Fin 2) ℝ} {A : Matrix (Fin 2) (Fin 2) ℂ} | |
| (hA : Dominates E A) (r δ : ℝ) (hr : |r| ≤ δ) : Dominates (δ • E) (r • A) := by | |
| apply (hA.smul r).mono | |
| intro i j | |
| exact mul_le_mul_of_nonneg_right hr (hA.nonneg i j) | |
| end Dominates | |
| noncomputable def rationalMatrix (E : RMatrix) : Matrix (Fin 2) (Fin 2) ℝ := | |
| (Matrix2.toMatrix E).map (Rat.castHom ℝ) | |
| noncomputable def absoluteMatrix (A : QMatrix) : Matrix (Fin 2) (Fin 2) ℝ := | |
| rationalMatrix (Matrix2.map RationalComplex.absOne A) | |
| theorem absoluteMatrix_dominates (A : QMatrix) : Dominates (absoluteMatrix A) (complexMatrix A) := by | |
| intro i j | |
| fin_cases i <;> fin_cases j <;> exact RationalComplex.norm_toComplex_le_absOne _ | |
| /-- The exact algebraic remainder identity. -/ | |
| theorem product_remainder (F P U V R : Matrix (Fin 2) (Fin 2) ℂ) : | |
| (F+P)*(U+V+R)-(F*U+F*V+P*U) = (F+P)*R+P*V := by | |
| noncomm_ring | |
| /-- Propagate one Taylor remainder without losing the entrywise product structure. -/ | |
| theorem product_remainder_dominates | |
| (F P U V R : Matrix (Fin 2) (Fin 2) ℂ) | |
| (Fbound N Vbound E : Matrix (Fin 2) (Fin 2) ℝ) (δ : ℝ) | |
| (hF : Dominates Fbound F) (hP : Dominates (δ • N) P) | |
| (hV : Dominates (δ • Vbound) V) (hR : Dominates (δ^2 • E) R) : | |
| Dominates (δ^2 • ((Fbound+δ • N)*E+N*Vbound)) | |
| ((F+P)*(U+V+R)-(F*U+F*V+P*U)) := by | |
| rw [product_remainder] | |
| have h := ((hF.add hP).mul hR).add (hP.mul hV) | |
| convert! h using 1 | |
| ext i j | |
| simp only [Matrix.add_apply, Matrix.smul_apply, Matrix.mul_apply, | |
| Fin.sum_univ_two, smul_eq_mul] | |
| ring | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |