End Aux.
Section Ratio.
Variable R : realFieldType.
Variable n : nat.
Implicit Types (v : (@Vector R n)).
Definition ratio v1 v2 :=
if [pick k : 'I_(n.+1) | v2 0 k != 0] is Some k
then v1 0 k / v2 0 k else 0.

Lemma ratio_neq0 v1 v2:
v1 != 0 -> v2 != 0 -> v1 == ratio v1 v2 *: v2 -> ratio v1 v2 != 0.