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 literature contains many results about exchanging proofs between proof assistant using deep or shallow embeddings @cite_7 @cite_6 . Boessplug, Cerbonneaux and Hermant propose to use the @math -calculus as a universal proof language which can express proof without losing their computational properties @cite_5 . To our knowledge, these works do not focus on the readability of proofs.
@cite_5 @cite_6 @cite_7
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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. 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. Modern proof assistants such as Coq and Isabelle provide high degrees of expressiveness and assurance because they support formal reasoning in higher-order logic and supply explicit machine-checkable proof objects. Unfortunately, large scale proof development in these proof assistants is still an extremely difficult and time-consuming task. One major weakness of these proof assistants is the lack of a single language where users can develop complex tactics and decision procedures using a rich programming model and in a typeful manner. This limits the scalability of the proof development process, as users avoid developing domain-specific tactics and decision procedures. In this paper, we present VeriML - a novel language design that couples a type-safe effectful computational language with first-class support for manipulating logical terms such as propositions and proofs. The main idea behind our design is to integrate a rich logical framework - similar to the one supported by Coq - inside a computational language inspired by ML. The language design is such that the added features are orthogonal to the rest of the computational language, and also do not require significant additions to the logic language, so soundness is guaranteed. We have built a prototype implementation of VeriML including both its type-checker and an interpreter. We demonstrate the effectiveness of our design by showing a number of type-safe tactics and decision procedures written in VeriML. We continue the investigation of interactive proofs with bounded communication, as initiated by Goldreich & Hastad (1998). Let L be a language that has an interactive proof in which the prover sends few (say b) bits to the verifier. We prove that the complement L has a constant-round interactive proof of complexity that depends only exponentially on b. This provides the first evidence that for NP- complete languages, we cannot expect interactive provers to be much more "laconic" than the standard NP proof. When the proof system is further restricted (e.g., when b = 1, or when we have perfect completeness), we get significantly better upper bounds on the complexity of L.