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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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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...
MachineAssisted MetaTheory of SequentStyle Calculi in Coq
"... . A formalisation of the implicational fragments of two sequent calculi and a sequentstyle presentation of natural deduction in Coq is presented. The systems presented are all typed lambda calculi. Based on this formalisation, some general comments on the feasibility of performing metatheoretic pro ..."
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. A formalisation of the implicational fragments of two sequent calculi and a sequentstyle presentation of natural deduction in Coq is presented. The systems presented are all typed lambda calculi. Based on this formalisation, some general comments on the feasibility of performing metatheoretic proofs about typed lambda calculi in a proof assistant based on higherorder type theory are made. Conclusions about the suitability of Coq for this work, as well as some indications as to the future use of completely formal developments in metatheoretic reasoning, are drawn. 1 Introduction Research into the properties of logics (metatheory) is a rich and diverse field. Despite the tools available for implementing object logics in Logical Frameworks [HP91, HP93] that have been developed over the last twenty years, little has been done in the area of machine assistance for metatheory. Investigations of the properties of calculi [Coq93, Hue94, Alt93] and typechecking Pure Type Systems [vBJM...