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Abstract:
We propose inference systems for binary relations with
composition laws of the form $S\circ T\subseteq U$
in the context of resolution-type theorem proving.
Particulary interesting examples include transitivity,
partial orderings, equality
and the combination of equality with other transitive relations.
Our inference mechanisms are based on standard techniques from term rewriting
and represent a refinement of chaining methods.
We establish their refutational completeness and also prove
their compatibility with the usual
simplification techniques used in rewrite-based theorem provers.
A key to the practicality of chaining techniques is
the extent to which so-called variable chainings can be restricted.
We demonstrate that rewrite techniques considerably restrict
variable chaining, though we also show that they cannot be
completely avoided in general.
If a binary relation under consideration satisfies additional properties,
such as symmetry, further restrictions are possible.
In particular,
we discuss orderings and partial congruence relations
