proposition: Lemma real_leif_AGM2 x y : x \is real -> y \is real -> x * y <= mid x y ^+ 2 ?= iff (x == y).
proof: Proof. move=> Rx Ry; rewrite -(mono_leif (ler_pM2r (ltr_nat F 0 4))). rewrite mulr_natr (natrX F 2 2) -exprMn divfK ?pnatr_eq0 //. exact: real_leif_AGM2_scaled. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq choice) mathcomp(ssrAC div fintype path bigop order finset fingroup) mathcomp(ssralg poly)