Translating between implicit and explicit versions of proof

The 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 (distil-lation) 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.