A direct proof of the completeness of SLDNF-resolution

We give a direct proof of the following theorem: if a goal Gσ is a logical consequence of the partial completion of an arbitrary normal logic program P, then each fair, non-floundering SLDNF-tree T for G yields an answer substitution θ which is more general than σ. If the negation G is a logical consequence of the partial completion of P, then T is finitely failed. A tree is fair, if each negative main branch ends in failure or each literal in the branch is selected at a certain point. A tree is floundering if it contains a positive node that consists of negative, non-ground literals only.