A Computational Approach to Pocklington Certificates in Type Theory

Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some well-defined syntactical rules. In most formalisms, a formal proof is viewed as some tree-structure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧B: