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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileMatrixMajorant
namespace OAI
/-! The entrywise two-parameter Taylor step underlying the K=34 certificate. -/
open Matrix
namespace DefocusingNLS.ProfileCertificate
abbrev CMatrix := Matrix (Fin 2) (Fin 2) ℂ
abbrev RealMatrix := Matrix (Fin 2) (Fin 2) ℝ
noncomputable def linearModel (U X Y : CMatrix) (b z : ℝ) : CMatrix := U+b • X+z • Y
theorem linearModel_step_identity (F G H U X Y A : CMatrix) (b z r : ℝ) :
r • ((F+b • G+z • H)*A) -
linearModel (r • (F*U)) (r • (G*U+F*X)) (r • (H*U+F*Y)) b z =
r • ((F+(b • G+z • H))*(A-linearModel U X Y b z) +
(b • G+z • H)*(b • X+z • Y)) := by
simp only [linearModel, Matrix.mul_add, Matrix.add_mul, Matrix.mul_sub,
Matrix.smul_mul, Matrix.mul_smul, smul_add, smul_sub, smul_smul]
module
/-- The real matrix E is exactly the manuscript's second-order majorant. -/
theorem taylor_step_dominates (F G H U X Y A : CMatrix)
(Fbound Gbound Hbound Xbound Ybound E : RealMatrix)
(b z δ r : ℝ) (hr : 0 ≤ r) (hb : |b| ≤ δ) (hz : |z| ≤ δ)
(hF : Dominates Fbound F) (hG : Dominates Gbound G)
(hH : Dominates Hbound H) (hX : Dominates Xbound X) (hY : Dominates Ybound Y)
(hE : Dominates (δ^2 • E) (A-linearModel U X Y b z)) :
Dominates
(δ^2 • (r • ((Fbound+δ • (Gbound+Hbound))*E +
(Gbound+Hbound)*(Xbound+Ybound))))
(r • ((F+b • G+z • H)*A) -
linearModel (r • (F*U)) (r • (G*U+F*X)) (r • (H*U+F*Y)) b z) := by
rw [linearModel_step_identity]
have hP : Dominates (δ • (Gbound+Hbound)) (b • G+z • H) := by
simpa only [smul_add] using (hG.smul_le b δ hb).add (hH.smul_le z δ hz)
have hV : Dominates (δ • (Xbound+Ybound)) (b • X+z • Y) := by
simpa only [smul_add] using (hX.smul_le b δ hb).add (hY.smul_le z δ hz)
have h := ((hF.add hP).mul hE).add (hP.mul hV)
have hs := h.smul r
rw [abs_of_nonneg hr] at hs
convert! hs 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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