Section PseriesDefs.
Variables (pis : seq nat_pred) (gT : finGroupType) (A : {set gT}).
Definition pcore_mod pi B := coset B @*^-1 'O_pi(A / B).
Canonical pcore_mod_group pi B : {group gT} :=
Eval hnf in [group of pcore_mod pi B].
Definition pseries := foldr pcore_mod 1 (rev pis).

Lemma sdprod_pcore_HallP H G :
pi^'.-Hall(G) H -> reflect ('O_pi(G) ><| H = G) (pi.-Hall(G) 'O_pi(G)).