We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.
The -calculus forms one of the vertices in Barendregt’s -cube and has been used as the core language for a number of logical frameworks. Following earlier extensions of natural deduction [14], Cousineau and Dowek [11] generalize the denitional equality of this well studied calculus to an arbitrary congruence generated by rewrite rules, which allows for more faithful encodings of foreign logics. This paper motivates the resulting language, the -calculus modulo, as a universal proof language, capable of expressing proofs from many other systems without losing their computational properties. We further show how to very simply and eciently check proofs from this language. We have implemented this scheme in a proof checker called Dedukti. We present a new scheme to translate mathematical developments from HOL Light to Coq, where they can be re-used and re-checked. By relying on a carefully chosen embedding of Higher-Order Logic into Type Theory, we try to avoid some pitfalls of inter-operation between proof systems. In particular, our translation keeps the mathematical statements intelligible. This translation has been implemented and allows the importation of the HOL Light basic library into Coq. 