proposition: Lemma leif_AGM (I : finType) (A : {pred I}) (E : I -> F) : let n := #|A| in let mu := (\sum_(i in A) E i) / n%:R in {in A, forall i, 0 <= E i} -> \prod_(i in A) E i <= mu ^+ n ?= iff [forall i in A, forall j in A, E i == E j].
proof: Proof. move=> n mu Ege0; have [n0 | n_gt0] := posnP n. by rewrite n0 -big_andE !(big_pred0 _ _ _ _ (card0_eq n0)); apply/leifP. pose E' i := E i / n%:R. have defE' i: E' i *+ n = E i by rewrite -mulr_natr divfK ?pnatr_eq0 -?lt0n. have /leif_AGM_scaled (i): i \in A -> 0 <= E' i *+ n by rewrite defE' => /Ege0. rewrite -/n -mulr_suml (eq_bigr _ (in1W defE')); congr (_ <= _ ?= iff _). by do 2![apply: eq_forallb_in => ? _]; rewrite -(eqr_pMn2r n_gt0) !defE'. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq choice) mathcomp(ssrAC div fintype path bigop order finset fingroup) mathcomp(ssralg poly)