English
For a Cauchy filter, define the limit Laurent series as the Hahn series with coefficients given by the limit of the coefficient function.
Русский
Для фильтра Cauchy предел Лаурентового ряда задаётся через гамильтоновоpную серию с коэффициентами, равными пределу функций коэффициентов.
LaTeX
$$$\mathrm{limit} (\mathcal F) := \mathrm{HahnSeries.mk}(\mathrm{coeff}(h\mathcal F))$$$
Lean4
/-- To any Cauchy filter ℱ of `K⸨X⸩`, we can attach a laurent series that is the limit
of the filter. Its `d`-th coefficient is defined as the limit of `Cauchy.coeff hℱ d`, which is
again Cauchy but valued in the discrete space `K`. That sufficiently negative coefficients vanish
follows from `Cauchy.coeff_support_bddBelow` -/
def limit {ℱ : Filter K⸨X⸩} (hℱ : Cauchy ℱ) : K⸨X⸩ :=
HahnSeries.mk (coeff hℱ) <| Set.IsWF.isPWO (coeff_support_bddBelow _).wellFoundedOn_lt
