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).
Canonical socle_subType := SubType _ _ _ socle_sort_rect PackSocleK.
Definition socle_eqMixin := Eval hnf in [eqMixin of sG by <:].
Canonical socle_eqType := Eval hnf in EqType sG socle_eqMixin.
Definition socle_choiceMixin := Eval hnf in [choiceMixin of sG by <:].
Canonical socle_choiceType := ChoiceType sG socle_choiceMixin.
Local Notation ssimS := subSocle_semisimple.
End SubSocle.
Definition Socle := (\sum_(W : sG) W)%MS.

Lemma reducible_Socle U :
mxmodule U -> mx_completely_reducible U -> (U <= Socle)%MS.