Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems

Sciences study both individual objects and generic ones. For example, Astronomy studies both the individual planets of the Solar system: Mercury, Venus, etc. determining their radius, mass, composition, etc., but also the motion of generic planets: Kepler’s laws, that do not just apply to the six planets known at the time of Kepler, but also to those that have been discovered after, and those that may be discovered in the future. Computer science studies both algorithms that apply to generic data, but also specific pieces of data. Mathematics mostly studies generic objects, but sometimes also specific ones, such as the number π or the function ζ . Proof theory mostly studies generic proofs. For example, Gentzen’s cut elimination theorem for Predicate logic applies to any proof expressed in Predicate logic, those that were known at the time of Gentzen, those that have been constructed after, and those that will be constructed in the future. Much less effort is dedicated to studying the individual mathematical proofs, with a few exceptions, for example [29]. Considering the proofs that we have, instead of all those that we may build in some logic, sometimes changes the perspective. For example, consider a cut elimination theorem for a logic L . The stronger the logic L , the stronger the theorem. In contrast, consider a specific proof π , say a proof of Fermat’s little theorem, and consider a theorem of the form: the proof π can be expressed in the logic L . In this case, the weaker the logic, the stronger the theorem. So, studying generic proofs leads to focus on stronger and stronger logics, while studying individual proofs, on weaker and weaker ones. In this paper, we present a program of analyzing the formal proofs that have been developed in computerized proof systems such as COQ, MATITA, HOL LIGHT, ISABELLE/HOL , PVS, FOCALIZE, etc. In particular, we want to be able to analyze in which logicsa each of these proofs can be expressed. Such a project is largely inspired by the reverse mathematics project [18, 34], but has some differences. First, we do not propose to classify theorems according to the logics in which they can be proved, but to classify the proofs according to the logics in which they can be expressed. Some theorems, for