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Some lambda calculus and type theory formalized
 Journal of Automated Reasoning
, 1999
"... Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention ..."
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Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.
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.
Formalising formulasastypesasobjects
 Types for Proofs and Programs
, 2000
"... Abstract. We describe a formalisation of the CurryHowardLawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current ..."
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Abstract. We describe a formalisation of the CurryHowardLawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current assumption list Γ and the conclusion A and organise numerous useful lemmas about proof trees categorically. We prove categorical properties about proof trees up to (syntactic) identity as well as up to βηconvertibility. We prove that our notion of proof trees is equivalent in an appropriate sense to more traditional representations of lambda terms. The formalisation is carried out in the proof assistant ALF for MartinLöf type theory. 1
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...