We describe a procedure to derive equality tests and their correctness proofs from inductive type declarations. Programs and proofs are derived compositionally, reusing code and proofs derived previously. The key steps are two. First, we design appropriate induction principles for data types defined using parametric containers. Second, we develop a technique to work around the modularity limitations imposed by the purely syntactic termination check Coq performs on recursive proofs. The unary parametricity translation of inductive data types turns out to be the key to both steps. Last but not least, we provide an implementation of the procedure for the Coq proof assistant based on the Elpi [3] extension language.