English
In the zeros-poles extraction setting, the trailing coefficient at x equals the update product in D.
Русский
В настройке извлечения нулей и полюсов завершающий коэффициент в x равен произведению обновления D.
LaTeX
$$$\mathrm{meromorphicTrailingCoeffAt}\left(f, x\right) = \prod^{\mathrm{fin}}_{u} (x - u)^{\mathrm{update}(D, x, 0, u)}$$$
Lean4
/-- Formulation of `MeromorphicAt.frequently_zero_iff_eventuallyEq_zero` as an identity principle: If
`f` and `g` are meromorphic at `x`, then `f` and `g` agree eventually in a punctured neighborhood of
`x` iff they agree at points arbitrarily close to (but different from) `x`.
-/
theorem frequently_eq_iff_eventuallyEq (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
(∃ᶠ z in 𝓝[≠] x, f z = g z) ↔ f =ᶠ[𝓝[≠] x] g :=
by
rw [eventuallyEq_iff_sub, ← (hf.sub hg).frequently_zero_iff_eventuallyEq_zero]
simp_rw [Pi.sub_apply, sub_eq_zero]
