Logic for exact real arithmetic

Continuing earlier work of the first author with U.Berger, K.Miyamoto and H.Tsuiki, it is shown how a division algorithm for real numbers given as either a stream of signed digits or via Gray code can be extracted from an appropriate formal proof. The property of being a real number represented in either of these forms is formulated by means of coinductively defined predicates, and formal proofs involve coinduction. The proof assistant Minlog is used to generate the formal proofs and extract their computational content, both as Scheme and Haskell programs.