English
A general induction principle for finitely generated submodules: if P holds for spans of single elements and is stable under sums, then P holds for all FG submodules.
Русский
Усиленная индукция FG: если P верно для порождений单 элементов и сохраняется при сложении, то верно для всех FG-подмодулей.
LaTeX
$$Submodule.fg_induction$$
Lean4
theorem fg_induction (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M] (P : Submodule R M → Prop)
(h₁ : ∀ x, P (Submodule.span R { x })) (h₂ : ∀ M₁ M₂, P M₁ → P M₂ → P (M₁ ⊔ M₂)) (N : Submodule R M) (hN : N.FG) :
P N := by
classical
obtain ⟨s, rfl⟩ := hN
induction s using Finset.induction with
| empty =>
rw [Finset.coe_empty, Submodule.span_empty, ← Submodule.span_zero_singleton]
exact h₁ _
| insert _ _ _ ih =>
rw [Finset.coe_insert, Submodule.span_insert]
exact h₂ _ _ (h₁ _) ih
