Module SSWP := WPropertiesOn SetOfSetsOfPositiveOrderedType SS.
Module SSWEqP := WEqPropertiesOn SetOfSetsOfPositiveOrderedType SS.
Module Import PosSort := Sort PosOrder.
Section Coinc_refl.
Context {AR : Arity}.
Definition pick_variety_auxCP {m : nat} (s : SS.elt) (cp : cartesianPower positive (S (S m))) : bool.

Lemma pick_variety_auxCP_existsF {m : nat} :
forall s (cp : cartesianPower positive (S (S m))),
pick_variety_auxCP s cp = false <-> (exists p, InCP p cp /\ S.mem p s = false).