It is well known in the Constraint Programming community that any non-binary constraint satisfaction problem (with finite domains) can be transformed into an equivalent binary one. One of the most wellknown translations is the Hidden Variable Encoding. In this paper we formalize this encoding in the proof assistant Coq and prove that any solution of the binary constraint satisfaction problem makes it possible to build a solution of the original problem and vice-versa. This formal development is used to complete the formally verified constraint solver developed in Coq by Carlier, Dubois and Gotlieb in 2012, making it a tool able to solve any n-ary constraint satisfaction problem, The key of success of the connection between the translator and the Coq binary solver is the genericity of the latter.