English
If D is injective and additive, and q is monic with respect to D, then leadingCoeff_D(p q) = leadingCoeff_D(p) for p monic? Wait: actually the lemma states leadingCoeff D (p q) = leadingCoeff D p when p is monic and q is monic with respect to D.
Русский
При инъективном D и моноических q левая ведущая степень сохраняет ведущий коэффициент, и он равен ведущему коэффициенту p.
LaTeX
$$$$\operatorname{leadingCoeff}_D(p q) = \operatorname{leadingCoeff}_D(p)$$$$
Lean4
theorem pow (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) (hD : D.Injective) (hp : p.Monic D) : (p ^ n).Monic D := by
induction n with
| zero => rw [pow_zero]; exact monic_one hD
| succ n ih => rw [pow_succ']; exact hp.mul hD hadd ih
