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A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF c ..."
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Cited by 217 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Proving Correctness of the Translation from MiniML to the CAM with the Coq Proof Development System
 with the Coq Proof Development System. Research report RR2536, INRIA, Rocquencourt
, 1995
"... In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (MiniML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to ..."
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In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (MiniML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to axiomatise the semantics of our languages. The axiomatisations of inferences systems and of the languages is nicely performed by the mechanism of inductive definitions in the Coq system. Unfortunately both the source and the target semantics involve nested structures that cannot be formalised inductively. We have overcome this problem by making some slight modifications of both the source and target semantics and show how the changes in the source and target semantics are related. For the remaining tranlation we explain how we can use the Coq system to formalize nonterminating programs and incorrect programs, objects that are impossible to explain with only the formalism of natural semantic...
A Formalisation Of Weak Normalisation (With Respect To Permutations) Of Sequent Calculus Proofs
, 1999
"... rule). This is also the case for NJ and LJ as defined in this formalisation. This is due to the particular nature of the logics in question, and does not necessarily generalise to other logics. In particular, a formalisation of linear logic would not work in this fashion, and a more complex variable ..."
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Cited by 3 (0 self)
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rule). This is also the case for NJ and LJ as defined in this formalisation. This is due to the particular nature of the logics in question, and does not necessarily generalise to other logics. In particular, a formalisation of linear logic would not work in this fashion, and a more complex variablereferencing mechanism would be required. See Section 6 for a further discussion of this problem. Other operations, such as substitutions (sub in Table 2) and weakening, require lift and drop operations as defined in [27] to ensure the correctness of the de Bruijn indexing.
Approaches to Formal MetaTheory
, 1997
"... . We present an overview of three approaches to formal metatheory: the formal study of properties of deductive systems. The approaches studied are: nameless dummy variables (also called de Bruijn indices) [dB72], first order abstract syntax for terms with higher order abstract syntax for judgements ..."
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. We present an overview of three approaches to formal metatheory: the formal study of properties of deductive systems. The approaches studied are: nameless dummy variables (also called de Bruijn indices) [dB72], first order abstract syntax for terms with higher order abstract syntax for judgements [MP93, MP97], and higher order abstract syntax [Pfe91]. 1 Introduction Formal metatheory, the machine assisted proof of theorems about logical systems, is a relatively new field. While some approaches ([dB72]) have been known about for some time, large developments have been rare until recently. Starting with [Alt93, Coq93] we have some formalisations of strong normalisation for natural deduction calculi using de Bruijn indices. The body of work in Elf [Pfe91] includes some formal metatheory using the higher order abstract syntax method which is integral to the LF approach. The work of McKinna, Pollack and others in [vBJMR94, MP93, MP97] demonstrates a slightly different approach using a ...
MetaTheory of SequentStyle Calculi in Coq
, 1997
"... We describe a formalisation of proof theory about sequentstyle calculi, based on informal work in [DP96]. The formalisation uses de Bruijn nameless dummy variables (also called de Bruijn indices) [dB72], and is performed within the proof assistant Coq [BB + 96]. We also present a description of ..."
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We describe a formalisation of proof theory about sequentstyle calculi, based on informal work in [DP96]. The formalisation uses de Bruijn nameless dummy variables (also called de Bruijn indices) [dB72], and is performed within the proof assistant Coq [BB + 96]. We also present a description of some of the other possible approaches to formal metatheory, particularly an abstract named syntax and higher order abstract syntax. 1 Introduction Formal proof has developed into a significant area of mathematics and logic. Until recently, however, such proofs have concentrated on proofs within logical systems, and metatheoretic work has continued to be done informally. Recent developments in proof assistants and automated theorem provers have opened up the possibilities for machinesupported metatheory. This paper presents a formalisation of a large theory comprising of over 200 definitions and more than 500 individual theorems about three different deductive system. 1 The central dif...