Lemma irr_degreeE i : 'n_i = \rank (socle_base i). Proof. by []. Qed.
Lemma irr_degree_gt0 i : 'n_i > 0.

(* Specifications: *)
Proof. by rewrite lt0n mxrank_eq0; case: (socle_simple i). Qed.
by rewrite linearZ /= gring_projE // eq_sym (negPf ne_yx) s