openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileTaylorBound.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileStateInterpretation | |
| namespace OAI | |
| /-! The computed K=34 majorant bounds the actual parameter-dependent product. -/ | |
| open Matrix | |
| namespace DefocusingNLS.ProfileCertificate | |
| private theorem prefix_remainder (ns : List ℕ) (b z : ℝ) | |
| (hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) : | |
| let s := ns.foldr step initial | |
| Dominates ((radius : ℝ)^2 • rationalMatrix s.error) | |
| ((ns.map (profileFactor ((centerB : ℝ)+b) ((centerZ : ℝ)+z))).prod- | |
| linearModel (complexMatrix s.value) (complexMatrix s.derivB) (complexMatrix s.derivZ) b z) := by | |
| induction ns with | |
| | nil => | |
| dsimp only [List.foldr_nil] | |
| intro i j | |
| fin_cases i <;> fin_cases j <;> | |
| simp [initial, linearModel, complexMatrix, rationalMatrix, Matrix2.zero, | |
| Matrix2.one, Matrix2.toMatrix] | |
| | cons n ns ih => | |
| dsimp only at ih ⊢ | |
| rw [List.foldr_cons, List.map_cons, List.prod_cons, profileFactor_affine, | |
| state_value_step, state_derivB_step, state_derivZ_step, state_error_step, | |
| Matrix.smul_mul] | |
| exact taylor_step_dominates _ _ _ _ _ _ _ _ _ _ _ _ _ b z (radius : ℝ) | |
| (((n : ℝ)+1)⁻¹) (by positivity) hb hz | |
| (absoluteMatrix_dominates (factor n)) (absoluteMatrix_dominates parameterB) | |
| (absoluteMatrix_dominates parameterZ) (absoluteMatrix_dominates _) | |
| (absoluteMatrix_dominates _) ih | |
| /-- Parameter-uniform Taylor remainder with the precise certificate matrix E. -/ | |
| theorem profileProduct_remainder (b z : ℝ) | |
| (hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) : | |
| Dominates ((radius : ℝ)^2 • rationalMatrix result.error) | |
| (profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34- | |
| linearModel (complexMatrix result.value) (complexMatrix result.derivB) | |
| (complexMatrix result.derivZ) b z) := by | |
| simpa only [profileFactor_product, result] using prefix_remainder (List.range 34) b z hb hz | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |