Definition pures_same phi1 phi2 := forall loc k pp, phi1 @ loc = PURE k pp <-> phi2 @ loc = PURE k pp.

Lemma pures_same_eq_r phi1 phi2 phi2' :
level phi2 = level phi2' ->
pures_same phi2 phi2' ->
pures_eq phi1 phi2 ->
pures_eq phi1 phi2'.