Section FCI.
Variables (X: eqType) (sigma: list X -> X -> bool) (R: list X).
Definition F (A : list X) : list X.
destruct (pick A) as [[x _]|_].
- exact (x::A).
- exact A.
Defined.
Definition C := it F (card R) nil.

Theorem fip_dec x :  (* Proof using FCI *)
fip_monotone sigma -> dec (fip sigma R x).