allRoots_nodup — Mathlib

English

The shortAddLongRoot decomposes as the sum of shortRoot and longRoot with coefficients 1 and 1: shortAddLongRoot = shortRoot + longRoot.

Русский

shortAddLongRoot раскладывается как сумма shortRoot и longRoot с коэффициентами 1 и 1: shortAddLongRoot = shortRoot + longRoot.

LaTeX

$$$shortAddLongRoot = shortRoot + longRoot$$$

Lean4

theorem allRoots_nodup : (allRoots P).Nodup :=
  by
  have hli : Injective (Fintype.linearCombination ℤ ![shortRoot P, longRoot P]) :=
    by
    rw [← linearIndependent_iff_injective_fintypeLinearCombination]
    exact (linearIndependent_short_long P).restrict_scalars' ℤ
  rw [allRoots_eq_map_allCoeffs, nodup_map_iff hli]
  decide

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