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.

Lemma socleP (W W' : sG) : reflect (W = W') (W == W')%MS.
Local Notation ssimS := subSocle_semisimple.