nil — Mathlib

English

Another variant of the append criterion with a quotient-modified tail.

Русский

Ещё одна версия условия добавления через фактор-модуль хвоста.

LaTeX

$$$IsWeaklyRegular(M, rs1 ++ rs2) \\iff IsWeaklyRegular(M, rs1) \\land IsWeaklyRegular(Quot(M, rs1), rs2)$$$

Lean4

theorem nil [Nontrivial M] : IsRegular M ([] : List R)
    where
  toIsWeaklyRegular := IsWeaklyRegular.nil R M
  top_ne_smul
    h := by
    rw [Ideal.ofList_nil, bot_smul, eq_comm, subsingleton_iff_bot_eq_top] at h
    exact not_subsingleton M ((Submodule.subsingleton_iff _).mp h)

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