R is strict

when

any interpretation of the derivation that is a counterexample lurking in a gap to which the rule R is applied lurks in some child of the gap

R is sound

when

any interpretation of the derivation that it is a counterexample lurking in a gap to which the rule R is applied and in all ancestors of this gap lurks in some child of the gap

R is safe

when

any interpretation of the derivation that is a counterexample lurking in a child of a gap to which the rule R has been applied lurks in the parent gap