Treating Partiality in a Logic of Total Functions

Abstract

The need to use partial functions arises frequently in formal descriptions of computer systems. However, most proof assistants are based on logics of total functions. One way to address this mismatch is to invent and mechanize a new logic. Another is to develop practical workarounds in existing settings. In this paper we take the latter course: we survey and compare methods used to support partiality in a mechanization of a higher order logic featuring only total functions. The techniques we discuss are generally applicable and are illustrated by relatively large examples.

A case-study in algebraic manipulation using mechanized reasoning tools

“…Now termination is not trivial, and thus the user has to provide an explicit proof of the function termination. These termination proofs can be even omitted and supplied later (see for details [18]). In some cases, it is enough to 'suggest' the system a proper measure function, and the system will automatically try to check the termination of the recursive function using that measure (in the example below, the size, or the length, of the list of terms, is enough to prove termination; clearly, the size of the list of terms decreases in each of the defining equations).…”

A Mechanized Proof of the Basic Perturbation Lemma

“…Under these circumstances, we should modify either the representation of homomorphisms or the equality in our setting. The problem of representing partial domains in a total setting is well-known, and different solutions can be considered (for a thorough study, see, for instance [25]). One of them would consist in defining a new equality between functional objects.…”

A reflective functional language for hardware design and theorem proving

“…In the HOL logic, we would usually express this kind of partially-defined object as an 'under-specified' total function (Müller & Slind, 1997). Formally, one uses a selection operator (Leisenring, 1969) to construct an expression 'ε x. P [x]' with the meaning 'an x such that P [x], or a fixed but unknown value if no such x exists'.…”