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The Foundation of a Generic Theorem Prover
 Journal of Automated Reasoning
, 1989
"... Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabelle is ..."
Abstract

Cited by 420 (46 self)
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Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabelle is now based on higherorder logic  a precise and wellunderstood foundation. Examples illustrate use of this metalogic to formalize logics and proofs. Axioms for firstorder logic are shown sound and complete. Backwards proof is formalized by metareasoning about objectlevel entailment. Higherorder logic has several practical advantages over other metalogics. Many proof techniques are known, such as Huet's higherorder unification procedure. Key words: higherorder logic, higherorder unification, Isabelle, LCF, logical frameworks, metareasoning, natural deduction Contents 1 History and overview 2 2 The metalogic M 4 2.1 Syntax of the metalogic ......................... 4 2.2 ...
The Semantics of Entailment Omega
, 2002
"... This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the twea ..."
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Cited by 4 (2 self)
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This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B intersect T of B+ , which saves implication and conjunction but drops disjunction. The filter models of the lambdacalculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models of these calculi. (The logician's Theory is the algebraist's Filter.) The coincidence extends to a dual interpretation of key particles  the subtype translates to provable >, type intersection to conjunction, function space > to implication and whole domain omega to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper and to be expected that type union U should correspond to the logical disjunction \/ of B+ . But the simulation of functional application by a fusion (or modus ponens product) operation o on theories leaves the key Bubbling lemma of work on ITD unprovable for the \/prime theories now appropriate for the modelling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of lambda and CL.
M. H. Newman’s Typability Algorithm for LambdaCalculus
, 2006
"... This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambdacalculus term is typable without actually computing its principal type. Newman’s algorithm seems to have been completely neglected by ..."
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This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambdacalculus term is typable without actually computing its principal type. Newman’s algorithm seems to have been completely neglected by the typetheorists who invented their own rather different typability algorithms over 15 years later. 1