Foundational extensible corecursion: a proof assistant perspective

Abstract

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under "friendly" operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic.

Friends with Benefits

“…The last example, about a probabilistic process calculus, takes our tool to its limits: We discuss how to support corecursion through monadic sequencing and mix unbounded recursion with corecursion. All eight formalizations are available online [14], together with our earlier stream examples [17].…”

“…The operations BASE and STEP have already been described in detail and with many examples in our previous paper [17]. Here, we give a brief, self-contained account of them.…”

“…These functions preserve productivity by consuming at most one constructor when producing one. Our previous work [17] presented the category theory underlying friends, based on more expressive blueprints than the one shown above for primitive corecursion. We now introduce a tool, AmiCo, that automates the process of applying and incrementally improving corecursion.…”

“…Another command, friend _ of _ corec, registers existing functions as friendly. Friendliness amounts to the relational parametricity [60,69] of a selected part of the definition [17], which in this paper we call a surface. The tool synthesizes the surface, and the parametricity proof is again either discharged automatically or presented to the user.…”

“…5). It subsumes a crude prototype [17] based on a shell script and template files that automated the corecursor derivation but left the blueprint and surface synthesis problems to the user. Our tool is available as part of the official Isabelle2016-1 release.…”