Let neg_unity_root n : (n > 1)%N -> exists2 w : C, w ^+ n = 1 & 'Re w < 0.

(* More properties of n.-root will be established in cyclotomic.v. *)
(* The proper form of the Arithmetic - Geometric Mean inequality. *)
Lemma lerif_rootC_AGM (I : finType) (A : pred I) (n := #|A|) E :
{in A, forall i, 0 <= E i} ->
n.-root (\prod_(i in A) E i) <= (\sum_(i in A) E i) / n%:R
?= iff [forall i in A, forall j in A, E i == E j].