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A thread of HOL development
 Computer Journal
"... The HOL system is a mechanized proof assistant for higher order logic that has been under continuous development since the mid1980s, by an everchanging group of developers and external contributors. We give a brief overview of various implementations of the HOL logic before focusing on the evoluti ..."
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Cited by 11 (7 self)
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The HOL system is a mechanized proof assistant for higher order logic that has been under continuous development since the mid1980s, by an everchanging 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.
On Extensibility of Proof Checkers
 in Dybjer, Nordstrom and Smith (eds), Types for Proofs and Programs: International Workshop TYPES'94, Bastad
, 1995
"... This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. Howeve ..."
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Cited by 6 (2 self)
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This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. However, we are quite rigid about this: only a derivation in our given formal system will do; nothing else counts as evidence! Thus it is not a collection of judgements (provability), or a consequence relation [Avr91] (derivability) we are interested in, but the derivations themselves; the formal system used to present a logic is important. This viewpoint seems forced on us by our intention to actually do formal mathematics. There is still a question, however, revolving around whether we insist on objects that are immediately recognisable as proofs (direct proofs), or will accept some metanotations that only compute to proofs (indirect proofs). For example, we informally refer to previously proved results, lemmas and theorems, without actually inserting the texts of their proofs in our argument. Such an argument could be made into a direct proof by replacing all references to previous results by their direct proofs, so it might be accepted as a kind of indirect proof. In fact, even for very simple formal systems, such an indirect proof may compute to a very much bigger direct proof, and if we will only accept a fully expanded direct proof (in a mechanical proof checker for example), we will not be able to do much mathematics. It is well known that this notion of referring to previous results can be internalized in a logic as a cut rule, or Modus Ponens. In a logic containing a cut rule, proofs containing cuts are considered direct proofs, and can be directly accepted by a proof ch...
TANSTAAFL (with partial functions)
, 1996
"... . Partial operators and functions are ubiquitous in program specifications, designs and arguments about both. There are several ways to ensure that logics can safely handle partiality. This paper identifies the costs associated with each of a variety of approaches to reasoning about partial function ..."
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Cited by 1 (0 self)
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. Partial operators and functions are ubiquitous in program specifications, designs and arguments about both. There are several ways to ensure that logics can safely handle partiality. This paper identifies the costs associated with each of a variety of approaches to reasoning about partial functions. 1 The problem Much of classical mathematics focuses on total functions where given some function f , one can write an application of that function to a value in its domain and know that f (x ) denotes a value of a type fixed by the signature of the function. This property is easier to achieve in texts which are concerned with restricted families of functions; for example, a book on number theory would involve natural numbers and a small collection of operators. There are inconveniences such as `division by zero' but, if these are sufficiently limited, they can often be handled by textual comments. When writing specifications of computer systems, one is often faced with many different ty...
Desiderata for Interactive Verification Systems
, 1994
"... 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 suppor ..."
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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...