Section SocleDef.
Variable sG0 : socleType.
Definition socle_enum := map component_mx (socle_base_enum sG0).
Inductive socle_sort : predArgType := PackSocle W of W \in socle_enum.
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.
Definition socle_module (W : sG) := mxsimple_module (socle_simple W).
Definition socle_repr W := submod_repr (socle_module W).

Lemma socle_mem (W : sG) : (W : 'M_n) \in socle_enum.