  Real number calculations on elementary functions are remarkably difficult to
handle in mechanical proofs. In this paper, we show how these calculations can
be performed within a theorem prover or proof assistant in a convenient and
highly automated as well as interactive way. First, we formally establish upper
and lower bounds for elementary functions. Then, based on these bounds, we
develop a rational interval arithmetic where real number calculations take
place in an algebraic setting. In order to reduce the dependency effect of
interval arithmetic, we integrate two techniques: interval splitting and taylor
series expansions. This pragmatic approach has been developed, and formally
verified, in a theorem prover. The formal development also includes a set of
customizable strategies to automate proofs involving explicit calculations over
real numbers. Our ultimate goal is to provide guaranteed proofs of numerical
properties with minimal human theorem-prover interaction.
