Local Notation sG := socle_sort.
Local Notation e0 := (socle_base_enum sG0).
Definition socle_base W := let: PackSocle W _ := W in e0`_(index W socle_enum).
Coercion socle_val W : 'M[F]_n := component_mx (socle_base W).
Definition socle_mult (W : sG) := (\rank W %/ \rank (socle_base W))%N.

(* Specifications: *)
Lemma socle_simple W : mxsimple (socle_base W).
Definition socle_module (W : sG) := mxsimple_module (socle_simple W).