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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 ..."
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Cited by 422 (47 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 ...
Set Theory for Verification: II  Induction and Recursion
 Journal of Automated Reasoning
, 2000
"... A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. ..."
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Cited by 43 (21 self)
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A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Cited by 23 (7 self)
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
Kripke Resource Models of a DependentlyTyped, Bunched lambdaCalculus (Extended Abstract)
, 1999
"... The lLcalculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Second ..."
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Cited by 8 (6 self)
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The lLcalculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proofobjects of BI, the logic of bunched implications. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The lLcalculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we describe the categorical semantics of the lLcalculus. This is given by Kripke resource models, which are monoidindexed sets of functorial Kripke models, the monoid giving an account of resource consumption. We describe a class of concrete, settheoretic models. The models are given by the category of families of sets, parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.
Validity concepts in prooftheoretic semantics
 ProofTheoretic Semantics. Special issue of Synthese
"... Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in det ..."
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Cited by 5 (4 self)
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Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in detail various notions of prooftheoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart. 1. Introduction: Prooftheoretic
2004).On the notion of assumption in logical systems
 In R
"... When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either ass ..."
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Cited by 2 (2 self)
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When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either assumptions or axioms, formulas of the sort (ii) are conclusions of (proper) inference rules. Axioms may be viewed as conclusions of (improper) inference rules, viz. inference rules without premisses. In what follows I refer to conclusions of proper or improper inference rules as assertions. 1 In natural deduction systems, inference rules deal both with assumptions and assertions, as the assumptions on which the conclusion of an inference rule depends, are not necessarily given by the collection of all assumptions on which the premisses depend, in case the rule permits the discharging of assumptions. For example, the rule of implication introduction
c ○ Peter SchroederHeisterPreface
, 1987
"... and Neil Tennant for helpful comments and discussions on topics related to this work. ..."
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and Neil Tennant for helpful comments and discussions on topics related to this work.