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fac6777 | 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 | 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
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