proposition: Lemma lt_le_rank_trans u v w : lt_rank u v -> le_rank v w -> lt_rank u w.
proof: Proof. rewrite /lt_rank => /andP [uv H] vw. rewrite (le_rank_trans uv vw) /=; apply: contra H => /eqP ?. subst w; by rewrite (@le_rank_asym u v) // uv. Qed.
imports: mathcomp(all_ssreflect fingroup perm)