Certified mathematical hierarchies: the FoCal system

The focal language (formerly Foc) allows a programmer to incrementally build mathematical structures and to formally prove their correctness. focal encourages a development process by refinement, deriving step-by-step implementations from specifications. This refinement process is realized using an inheritance mechanism on structures which can mix primitive operations, axioms, algorithms and proofs. Inheritance from existing structures allows to reuse their components under some conditions, which are statically checked by the compiler. In this paper, we first present the main constructions of the language. Then we show a shallow embedding of these constructions in the Coq proof assistant, which is used to check the proofs made in Focal. Such a proof can be either an hand-written Coq script, made in an environment set up by the Focal compiler, or a Coq term given the zenon theorem prover, which is partly developed within Focal. Last, we present a formalization of focal structures and show that the Coq embedding is conform