powerSeriesFamily_hsum_zero — Mathlib

English

From any function with bounded-below support, construct a Hahn series with these coefficients.

Русский

Из любой функции с ограниченной снизу опорой строится Hahn-ряд с этими коэффициентами.

LaTeX

$$ofSuppBddBelow f hf$$

Lean4

theorem powerSeriesFamily_hsum_zero (f : PowerSeries R) :
    (powerSeriesFamily 0 f).hsum = f.constantCoeff • (1 : HahnSeries Γ V) :=
  by
  ext g
  by_cases hg : g = 0
  · simp only [hg, coeff_hsum]
    rw [finsum_eq_single _ 0 (fun n hn ↦ by simp [hn])]
    simp
  · rw [coeff_hsum, finsum_eq_zero_of_forall_eq_zero fun n ↦ (by by_cases hn : n = 0 <;> simp [hg, hn])]
    simp [hg]

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