any two representations of reals which are acceptable are actually computably isomorphic,please see here for referencewhere may proof of  this theorem   be found, and what is the the computational complexity of the computably isomorphic maps?
Where can I find the proof of the theorem and what is the computational complexity of the computably isomorphic map?
cc.complexity theory;reference request;computability;computable analysis
The first theorem of the form you are asking about was proved by Y. Moschovakis in Notation systems and recursive ordered fields, Compositio Mathematica 17:4071 (1965). Then in the context of Type Two Effectivity a similar theorem was proved by P. Hertling, see A real number structure that is effectively categorical, Mathematical Logic Quarterly, 45(2):147182 (1999). A generalization of these results (and several others) was proved by A. Bauer and J. Blanck in Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions, Journal of Universal Computer Science, vol 16:24962522 (2010).The paper I wrote with Jens is very general and maybe less accessible. Another good source to read about this is Peter Hertling's The Real Number Structure is Effectively Categorical.I have nothing intelligent to say about computational complexity.