ML Pattern-Matching, Recursion, and Rewriting: From FoCaLiZe to Dedukti

Abstract

Abstract. The programming environment FoCaLiZe allows the user to specify, implement, and prove programs with the help of the theorem prover Zenon. In the actual version, those proofs are verified by Coq. In this paper we propose to extend the FoCaLiZe compiler by a backend to the Dedukti language in order to benefit from Zenon Modulo, an extension of Zenon for Deduction modulo. By doing so, FoCaLiZe can benefit from a technique for finding and verifying proofs more quickly. The paper focuses mainly on the process that overcomes the lack of local pattern-matching and recursive definitions in Dedukti.

First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice

“…Combining the computational power of β-reduction (function application) and rewriting results with the possibility to define shallow embeddings of many systems, including Coq [27], Matita [3], HOL Light [4,3,5], FoCaLiZe [46,47], etc. Dedukti is mature enough to be able to check large libraries coming from those systems.…”

FoCaLiZe and Dedukti to the Rescue for Proof Interoperability

“…In particular, Holide [3], Coqine [2], and Focalide [10] are translators from respectively the OpenTheory format for ITPs in the HOL family, the Coq proof assistant and the FoCaLiZe framework. Some ATPs also produce Dedukti files, e.g.…”

“…The historic one produces Coq code and requires the use of the ATP Zenon which can output proofs as Coq terms. A more recent backend, called Focalide, produces Dedukti code [10] and requires to use Zenon Modulo [8], an extension of Zenon which produces Dedukti proofs [11]. In this work, we only use the Focalide backend.…”