Definition in_Crat_span s x :=
exists a : rat ^ size s, x = \sum_i QtoC (a i) * s`_i.
Fact Crat_span_subproof s x : decidable (in_Crat_span s x).
Definition Crat_span s : pred algC := Crat_span_subproof s.

Lemma Crat_spanP s x : reflect (in_Crat_span s x) (x \in Crat_span s).
Fact Crat_span_zmod_closed s : zmod_closed (Crat_span s).