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A Framework for Defining Logics
 JOURNAL OF THE ASSOCIATION FOR COMPUTING MACHINERY
, 1993
"... The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. T ..."
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Cited by 696 (39 self)
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The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higherorder judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logicindependent tools such as proof editors and proof checkers can be constructed.
Using Typed Lambda Calculus to Implement Formal Systems on a Machine
 Journal of Automated Reasoning
, 1992
"... this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its a ..."
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Cited by 83 (14 self)
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this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.
A HOL theory of Euclidean space
 In Hurd and Melham [7
, 2005
"... Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in ..."
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Cited by 14 (0 self)
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Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in a simple and useful way, even though HOL does not permit dependent types. In the resulting theory the HOL type system, far from getting in the way, naturally imposes the correct dimensional constraints, e.g. checking compatibility in matrix multiplication. Among the interesting later developments of the theory are a partial decision procedure for the theory of vector spaces (based on a more general algorithm due to Solovay) and a formal proof of various classic theorems of topology and analysis for arbitrary Ndimensional Euclidean space, e.g. Brouwer’s fixpoint theorem and the differentiability of inverse functions. 1 1 The problem with R N
A certified, corecursive implementation of exact real numbers
 Theoretical Computer Science
, 2006
"... We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecu ..."
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Cited by 13 (0 self)
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We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecursive proofs. Thus we obtain reliable, corecursive algorithms for computing on real numbers.
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
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The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.
Notes at Conferences on Intelligent Computer Mathematics 2010
, 2010
"... If you read nothing else, read footnote 1 on page 10 to learn that, in Excel, ‘paste ’ is function application. The “impact police ” might be amused by note 3 (page 57). Updated 16.7.2010 to include some factual corrections by DPC to the OpenMath discussion (at which he was unable to be present).Con ..."
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If you read nothing else, read footnote 1 on page 10 to learn that, in Excel, ‘paste ’ is function application. The “impact police ” might be amused by note 3 (page 57). Updated 16.7.2010 to include some factual corrections by DPC to the OpenMath discussion (at which he was unable to be present).Contents
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"... Abstract. The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed Acalculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof’s system of a ..."
Abstract
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Abstract. The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed Acalculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof’s system of arities. The treatment of rules and proofs focuses on his notion of a judgment. Logics are represented in LF via a new principle, the judgrrzents as ~pes principle, whereby each judgment is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higherorder judgments and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logicindependent tools,
The Formal System λδ
, 2008
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
Abstract
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The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.
The Formal System λδ
, 2008
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
Abstract
 Add to MetaCart
The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.