Limits of real numbers in the binary signed digit representation

We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams binary signed digits. The main objective paper is the formalisation proof that numbers are closed with respect to limits. All proofs theorem and first application implemented in Minlog system extracted terms further translated into Haskell. compare two approaches. approach direct proof. In second make by Cauchy-sequence rationals. Utilizing translations between represenation using completeness Cauchy-reals, very short. both cases Minlog's program extraction mechanism automatically formally transforms converging sequence reals, i.e.~a digits together modulus convergence, digit its limit. correctness follows directly soundness extraction. As Heron's method construct an algorithm computes square roots representation. convergence show under multiplication.