Truly Modular (Co)datatypes for Isabelle/HOL

Abstract

Abstract. We extended Isabelle/HOL with a pair of definitional commands for datatypes and codatatypes. They support mutual and nested (co)recursion through well-behaved type constructors, including mixed recursion-corecursion, and are complemented by syntaxes for introducing primitive (co)recursive functions and by a general proof method for reasoning coinductively. As a case study, we ported Isabelle's Coinductive library to use the new commands, eliminating the need for tedious ad hoc constructions.

Friends with Benefits

“…The coinduction proof method [16] instantiates the bisimulation witness of the given coinduction rule before applying it backwards. Without corec, the rule +.coinduct of coinduction up to congruence had to be stated and proved manually, including the manual inductive definition of the congruence closure under +.…”

“…When defining functions on these objects, we previously had to break them into a recursive and a corecursive part, using Isabelle's primcorec command for the latter [16]. Since our tool supports mixed recursion-corecursion, we can now express functions on stream processors more directly.…”

“…Starting from the primitive corecursor provided by Isabelle [16], our tool derives corecursors up to larger and larger sets of friends. The corecursion state includes the set of friends F and the corecursor corec F .…”

“…Isabelle/HOL is the first of these systems to also offer codatatypes. It took two years, and about 24 000 lines of Standard ML, to move from an understanding of the mathematics [18,67] to an implementation that automates the process of checking high-level user specifications and producing the necessary corecursion and coinduction theorems [16].…”

“…In Isabelle, this check is replaced by a deeper analysis. The primcorec command [16] transforms a user specification into a blueprint object: the coalgebra b = λn. (n, n + 1).…”