Definition ss_ok (ss : SS.t) (interp: positive -> COINCpoint) :=
  forall s, SS.mem s ss = true ->
  forall cp, memCP cp s = true ->
    app coinc (interp_CP cp interp).

Lemma mca_pick_variety_aux_pca : forall m cp s1 s2 ss x interp, ss_ok ss interp -> SS.In s1 ss -> SS.In s2 ss -> memCPAux (S m) cp (S.union s1 s2) = true -> pick_variety_aux (S.inter s1 s2) x = true -> pred_conj_aux coinc (S (S m)) (interp_CP cp interp) (interp_CP x interp).