Dealing with logical complexity in mathematics has a long tradition of building models for complex logics from models for simpler logics, using construction such as the sheaf construction. In computer science, there is a dual practice which consists in giving meaning to a complex language using a compilation phase into a simpler one.

In my presentation, I will show how the compilation phase approach also makes sens in logic, via the Curry-Howard isomorphism, and is more precise than the model extension approach because it takes computation into account. To illustrate this, I will show examples of program transformations in type theory, with applications to the Coq proof assistant.