English
A deep constructive characterization: linear independence is equivalent to a disjoint Finset condition with two finite index sets s and t, and a forcing that coefficients vanish on both sides.
Русский
Линейная независимость эквивалентна раздельному условию на конечных множествах s и t и затем лосящие нули коэффициентов на обоих сторонах.
LaTeX
$$$\text{linearIndependent}_R v \iff \forall (s t : Finset ι) (f : ι \to R), Disjoint s t \rightarrow \sum i\in s, f_i v_i = \sum i\in t, f_i v_i \rightarrow \big( \forall i\in s, f_i = 0 \big) \wedge \forall i\in t, f_i = 0$$$
Lean4
theorem not_linearIndependent_iffₒₛ :
¬LinearIndependent R v ↔
∃ (s t : Finset ι) (f : ι → R), Disjoint s t ∧ ∑ i ∈ s, f i • v i = ∑ i ∈ t, f i • v i ∧ ∃ i ∈ s, 0 < f i :=
by
simp only [linearIndependent_iffₒₛ, pos_iff_ne_zero]
set_option push_neg.use_distrib true in push_neg
refine ⟨fun ⟨s, t, f, hst, heq, h⟩ => ?_, fun ⟨s, t, f, hst, heq, hi⟩ => ⟨s, t, f, hst, heq, .inl hi⟩⟩
rcases h with ⟨i, hi, hfi⟩ | ⟨i, hi, hgi⟩
· exact ⟨s, t, f, hst, heq, i, hi, hfi⟩
· exact ⟨t, s, f, hst.symm, heq.symm, i, hi, hgi⟩
