Abstract not found

The HOL system is a mechanized proof assistant for higher order logic that has been under continuous development since the mid-1980s, by an ever-changing group of developers and external contributors. We give a brief overview of various implementations of the HOL logic before focusing on the evolution of certain important features available in a recent implementation. We also illustrate how the module system of Standard ML provided security and modularity in the construction of the HOL kernel, as well as serving in a separate capacity as a useful representation medium for persistent, hierarchical logical theories.

this paper, we present a formalization of the proof in the HOL theorem prover [5], which is based on simple type theory. The formalization is considerably simpler than the ALF formalization, both theoretically and practically. Just like ALF, HOL does not directly support equational reasoning in diagram chasing. However, HOL comes with the functional programming language ML. This made it possible with a very limited programming effort to support diagram chasing at such a level that proofs in HOL were more or less as abstract as informal proofs on paper. The user does not have to worry about applying transitivity, congruence and associativity rules, only about specifying the main steps, just like in a paper-and-pencil proof. The ideas for the tool support are inspired by Paulson's higher order conversions for rewriting [11] and might be useful for other purposes than category theory. They only require the presence of a congruence (i.e. equality-like) relation as in, for instance, bisimularity proofs in program verification. Below, we first give a brief introduction to category theory in Section 2 and to the HOL system in Section 3. In Section 4 the formalization of a monoid of binary words is presented, including the normalization theorem. The formalization of a monoidal category of binary words is presented in Section 5 and in Section 6 the proof of a coherence theorem for this category is presented. Section 7 presents the tool support for diagram chasing. 2 Category Theory

What facilities should an interactive verification system provide? We take the pragmatic view that the particular logic underlying a proof system is not as important as the support that is provided. Although a plethora of logics have been implemented, we think that there is a common kernel of support that a proof system ought to provide. Towards this end, we give detailed suggestions for verification support in three major areas: formalization, proof, and interface. Although our perspective comes from experience with highly expressive logics such as set theory, higher order logic, and type theory, we think our analyses apply more generally. Introduction Currently, theorem provers are used in the verification of both hardware and software [GM93, ORS92, BM90, HRS90, FFMH92], the formalization of informal mathematical proofs [FGT90, CH85, Pau90b], the teaching of logic[AMC84], and as tools of mathematical and metamathematical research [WWM + 90, CAB + 86]. 1 In this paper we describ...