Section recursor.
Variables (F : nat -> Prop) (G : nat -> nat -> nat -> Prop).
Fixpoint μ_rec n :=
match n with
| 0   => F
| S n => fun x => exists y, μ_rec n y /\ G n y x
end.
Fact μ_rec_inv n x : μ_rec n x -> exists s, F (s 0)
/\ s n = x
/\ forall i, i < n -> G i (s i) (s (S i)).

Theorem μ_rec_eq n x : μ_rec n x <-> exists s, F (s 0)
/\ s n = x
/\ forall i, i < n -> G i (s i) (s (S i)).