FoCaLiZe and Dedukti to the Rescue for Proof Interoperability

Abstract

Abstract. Numerous contributions have been made for some years to allow users to exchange formal proofs between different provers. The main propositions consist in ad hoc pointwise translations, e.g. between HOL Light and Isabelle in the Flyspeck project or uses of more or less complete certificates. We propose in this paper a methodology to combine proofs coming from different theorem provers. This methodology relies on the Dedukti logical framework as a common formalism in which proofs can be translated and combined. To relate the independently developed mathematical libraries used in proof assistants, we rely on the structuring features offered by FoCaLiZe, in particular parameterized modules and inheritance to build a formal library of transfer theorems called MathTransfer. We finally illustrate this methodology on the Sieve of Eratosthenes, which we prove correct using HOL and Coq in combination.

HolPy: Interactive Theorem Proving in Python

“…While this design is certainly natural, it has the disadvantage that it is difficult for other tools to interpret the files -any such tool must re-implement a large part of the core of the proof assistant. Various proof exchange formats, such as OpenTheory [14] and Dedukti [5], have the potential to solve this problem. However, as they are developed independently, there is always difficulty translating theories between them and the major proof assistants.…”

“…OpenTheory [14] and Dedukti [5] both provide a universal language for representing theories in a very general logic, at least in part with the aim to allow communication between different proof assistants. While we also provide a common format for theories, our aim is different.…”

Sharing a Library between Proof Assistants: Reaching out to the HOL Family

“…Suppose that the proof of a theorem thm A expressed in a logic L A needs a proof of a theorem thm B already proven in the logic L B . A first solution explored, for instance, by Cauderlier and Dubois [3] is to have the combined proof inside the logical framework by encoding the proofs of thm A and thm B in it. In this solution, proofs are not exported outside of the logical framework.…”

“…Cauderlier and Dubois already used Dedukti for interoperability in [3]. Their goal was to prove the sieve of Eratosthenes using HOL and Coq in combination.…”

Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems

“…These details should be ignored by the arithmetic library, that should be exported to any system that contains a notion of natural number, an induction principle and a way to define functions by induction, regardless the way this induction principle is proved and this induction operator is defined there [36]. Using such an abstract definition of the natural numbers, R. Cauderlier and C. Dubois [10] have built a proof of the correctness of Eratosthenes' sieve in the expression of FOCALIZE in DEDUKTI, using definitions coming from COQ and lemmas coming from HOL LIGHT.…”