There are several ways to actually implement an explanation-based constraints system. The main issues are addressed on a page dedicated to design issues for implementing e-constraints systems.
Here are some implementations:
- DECorum: Deduction-based constraint relaxation
management
DECorum is a explanation-based constraint solver developped in C++ from 1995 to 1998. Is was (it is no longer maintained nor available) available as a C++ library. Some more information on this page along with a demo application.go to the DECorum page 
PaLM: Propagation and Learning with Move
PaLM is under developement since 2000. It is an explanation-based constraint system build on top of choco [Laburthe, 2000]. It is available both as a claire [Caseau and Laburthe, 1996] module and as an integrated module in the Java version of choco. The PaLM page gives more information, some download instructions and an online manual.go to the PaLM page
Bibliography
- [Caseau and Laburthe, 1996] Yves Caseau and François Laburthe. CLAIRE: Combining objects and rules for problem solving. Proceedings of JICSLP, workshop on multi-paradigm logic programming, 1996.
- [Laburthe, 2000] François Laburthe. Choco: implementing a cp kernel. In CP00 Post Conference Workshop on Techniques for Implementing Constraint programming Systems (TRICS), Singapore, September 2000.