  <p>We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. </p> <p>Let <em>Tadd[Q]</em> be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant <em>1</em>, and function symbols for multiplication by
rational constants. Let <em>Tmult[Q]</em> be the analogous theory for the
multiplicative structure, and let <em>T[Q]</em> be the union of the two. We
show that although <em>T[Q]</em> is undecidable, the universal fragment of
<em>T[Q]</em> is decidable. We also show that terms of <em>T[Q]</em>can
fruitfully be put in a normal form. We prove analogous results for theories in
which <em>Q</em> is replaced, more generally, by suitable subfields <em>F</em>
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.</p>
