### Pure Type Systems in Rewriting Logic: Specifying Typed Higher-Order Languages in a First-Order Logical Framework [chapter]

Mark-Oliver Stehr, José Meseguer
<span title="">2004</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The logical and operational aspects of rewriting logic as a logical framework are tested and illustrated in detail by representing pure type systems as object logics. More precisely, we apply membership equational logic, the equational sublogic of rewriting logic, to specify pure type systems as they can be found in the literature and also a new variant of pure type systems with explicit names that solves the problems with closure under α-conversion in a very satisfactory way. Furthermore, we
more &raquo; ... e rewriting logic itself to give a formal operational description of type checking, that directly serves as an efficient type checking algorithm. The work reported here is part of a more ambitious project concerned with the development of the open calculus of constructions, an equational extension of the calculus of constructions that incorporates rewriting logic as a computational sublanguage. This paper is a detailed study on the ease and naturalness with which a family of higher-order formal systems, namely pure type systems (PTSs) [6, 50] , can be represented in the first-order logical framework of rewriting logic [36] . PTSs generalize the λ-cube [1], which already contains important calculi like λ→ [12], the systems F [23,43] and Fω [23], a system λP close to the logical framework LF [24] , and their combination, the calculus of constructions CC [16] . PTSs are considered to be of key importance, since their generality and simplicity makes them an ideal basis for representing higher-order logics, either via the propositions-as-types interpretation [21] , or via their use as a higher-order logical framework in the spirit of LF [24, 20] or Isabelle [39] .
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