deriv_aeval_eq — Mathlib

English

The differential algebra of derivations acts as a Lie algebra on polynomial modules, with derivations acting coefficientwise on polynomial coefficients.

Русский

Алгебра Ли дифференциалов действует на полиномиальные модули по коэффициентам коэффицентов полиномов.

LaTeX

$$$[D_1,D_2](p) = D_1(D_2(p)) - D_2(D_1(p))$ and the actions respect the polynomial structure.$$

Lean4

theorem deriv_aeval_eq (x : R) (p : A[X]) : (aeval x p)′ = aeval x (mapCoeffs p) + aeval x (derivative p) * x′ :=
  by
  convert Derivation.apply_aeval_eq' Differential.deriv _ (Algebra.linearMap A R) ..
  · simp [mapCoeffs]
  · simp [deriv_algebraMap]

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