Definition test_coinc (ss : SS.t) (st : STt) (cp : cartesianPower positive (S (S (S n)))) : bool :=
  let newss := identify_varieties ss st  in
    SS.exists_ (fun s => memCP cp s) newss.

(* Specifications: *)
Lemma pick_variety_aux_memCPAux1 : forall s1 s2 m (cp : cartesianPower positive (S (S m))), pick_variety_auxCP (S.inter s1 s2) cp = true -> memCPAux m cp s1 = true.
Lemma pick_variety_aux_memCPAux2 : forall s1 s2 m (cp : cartesianPower positive (S (S m))), pick_variety_auxCP (S.inter s1 s2) cp = true -> memCPAux m cp s2 = true.