Section Fip.
Variables (X: eqType) (sigma: list X -> X -> bool) (R: list X).
Inductive fip : X -> Prop :=
| fip_intro A x : (forall x, x el A -> fip x) -> sigma A x -> x el R -> fip x.
Definition fip_monotone := forall A B x, A <<= B -> sigma A x -> sigma B x.
Definition fip_closed A := forall x, x el R -> sigma A x -> x el A.

Lemma fip_least A x :
fip_monotone -> fip_closed A -> fip x -> x el A.