Property-Based Testing via Proof Reconstruction Work-in-progress

International audienceProperty-based testing is a technique for validating code against an executable specification by automatically generating test-data. From its original use in programming languages, this technique has now spread to most major proof assistants to complement theorem proving with a preliminary phase of conjecture testing. We present a proof theoretical reconstruction of this style of testing for relational specifications (such as those used in the semantics of programming languages) and employ the Foundational Proof Certificate framework to aid in describing test generators. We do this by presenting certain kinds of " proof outlines " that can be used to describe the shape and size of the generators for the conditional part of a proposed property. Then the testing phase is reduced to standard logic programming search. After illustrating our techniques on simple, first-order (algebraic) data structures, we lift it to data structures containing bindings using λ-tree syntax. The λProlog programming language is capable of performing both the generation and checking of tests. We validate this approach by tackling benchmarks in the metatheory of programming languages coming from related tools such as PLT-Redex

Implementing HOL in an Higher Order Logic Programming Language

International audienceWe present a proof-of-concept prototype of a (constructive variant of an) HOL interactive theorem prover written in a Higher Order Logic Programming (HOLP) language, namely an extension of λProlog. The prototype is meant to support the claim, that we reinforce , that HOLP is the class of languages that provides the right abstraction level and programming primitives to obtain concise implementations of theorem provers. We identify and advocate for a programming technique, that we call semi-shallow embedding, while at the same time identifying the reasons why pure λProlog is not sufficient to support that technique, and it needs to be extended

Translating Between Implicit and Explicit Versions of Proof

International audienceThe Foundational Proof Certificate (FPC) framework can be used to define the semantics of a wide range of proof evidence. For example , such definitions exist for a number of textbook proof systems as well as for the proof evidence output from some existing theorem proving systems. An important decision in designing a proof certificate format is the choice of how many details are to be placed within certificates. Formats with fewer details are smaller and easier for theorem provers to output but they require more sophistication from checkers since checking will involve some proof reconstruction. Conversely, certificate formats containing many details are larger but are checkable by less sophisticated checkers. Since the FPC framework is based on well-established proof theory principles, proof certificates can be manipulated in meaningful ways. In this paper, we illustrate how it is possible to automate moving from implicit to explicit (elaboration) and from explicit to implicit (distillation) proof evidence via the proof checking of a pair of proof certificates. Performing elaboration makes it possible to transform a proof certificate with details missing into a certificate packed with enough details so that a simple kernel (without support for proof reconstruction) can check the elaborated certificate. We illustrate how trust in only a single, simple checker of explicitly described proofs can be used to provide trust in a range of theorem provers employing a range of proof structures