Some Lambda Calculus and Type Theory Formalized

Abstract

"This paper is about our hobby." That is the first sentence of [MP93], the first report on our formal development of lambda calculus and type theory, written in autumn 1992. We have continued to pursue this hobby on and off ever since, and have developed a substantial body of formal knowledge, including Church-Rosser and standardization<br />theorems for beta reduction, and the basic theory of<br />Pure Type Systems (PTS) leading to the strengthening theorem and type checking algorithms for PTS. Some of this work is reported in [MP93, vBJMP94, Pol94b, Pol95]. In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, and simplified, proof of standardization for beta reduction, not previously published, that does<br />not mention redex positions or residuals. On the general issues, we emphasize the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts. The LEGO Proof Development System [LP92] was used to check the work in an implementation of the Extended Calculus of Constructions<br />(ECC) with inductive types [Luo94]. LEGO is a refinement style<br />proof checker, publicly available by ftp and WWW, with a User's Manual [LP92] and a large collection of examples. Section 1.3 contains information on accessing the formal development described in this paper. Other interesting examples formalized in LEGO include program specification and data refinement [Luo91], strong normalization of System F [Alt93], synthetic domain theory [Reu95, Reu96], and operational semantics for imperative programs [Sch97].

Viewing λ-terms through maps

“…If x = y and x P , then M {x\N }{y\P } = M {y\P }{x\N {y\P }}. 2 Other representations have this same property [12,1,15]. Proof.…”

“…11, this rule mentions a free parameter in the premise. This rule is also reminiscent of the representations discussed in [12,1,20]. It is not only rule ξ that poses this problem for the map representation.…”

“…Conversely, could we modify a local representation such as [1,12,15] to be name-free by allowing some other constants (analogous to unbound ) in wellformed terms? The answer seems to be no, as all the cited local representations depend on having no free local names.…”

“…Now we can draw a comparison between the mask abstraction of our representation, and the abstraction of various local representations [1,12,15] that may illuminate the rôle of unbound boxes. For example consider the locally nameless representation [1].…”

“…We could, if desired, remove unbound boxes from the notion of well-formed mask terms, using a function such as lam of Definition 4 that replaces a free name by boxes, similarly to the abstraction of [1,12,15], but we would lose the name-free aspect of mask terms, and they would behave very much like the local representations [1,12,15].…”