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Nominal techniques in Isabelle/HOL
 Proceedings of the 20th International Conference on Automated Deduction (CADE20
, 2005
"... Abstract. In this paper we define an inductive set that is bijective with the ffequated lambdaterms. Unlike deBruijn indices, however, our inductive definition includes names and reasoning about this definition is very similar to informal reasoning on paper. For this we provide a structural induc ..."
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Abstract. In this paper we define an inductive set that is bijective with the ffequated lambdaterms. Unlike deBruijn indices, however, our inductive definition includes names and reasoning about this definition is very similar to informal reasoning on paper. For this we provide a structural induction principle that requires to prove the lambdacase for fresh binders only. The main technical novelty of this work is that it is compatible with the axiomofchoice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for ChurchRosser and strongnormalisation. Keywords. Lambdacalculus, nominal logic, structural induction, theoremassistants.
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.
Formalising in Nominal Isabelle Crary’s Completeness Proof for Equivalence Checking
 LFMTP 2007
, 2007
"... In the book on Advanced Topics in Types and Programming Languages, Crary illustrates the reasoning technique of logical relations in a case study about equivalence checking. He presents a typedriven equivalence checking algorithm and verifies its completeness with respect to a definitional characte ..."
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In the book on Advanced Topics in Types and Programming Languages, Crary illustrates the reasoning technique of logical relations in a case study about equivalence checking. He presents a typedriven equivalence checking algorithm and verifies its completeness with respect to a definitional characterisation of equivalence. We present in this paper a formalisation of Crary’s proof using Isabelle/HOL and the nominal datatype package.
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
Formalising mathematics in UTT: fundamentals and case studies
, 1994
"... We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fullychecked example based on our re ..."
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We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fullychecked example based on our representation of naive set theory. Contents 1 Introduction 1 2 Fundamentals 3 2.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Discrete sets . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 The category of sets . . . . . . . . . . . . . . . . . . . . . 5 2.1.5 Multivariate maps . . . . . . . . . . . . . . . . . . . . . . 6 2.1.6 Predicates and relations . . . . . . . . . . . . . . . . . . . 7 2.1.7 Subsets and powerset . . . . . . . . . . . . . . . . . . . . 7 2.1.8 Quotients . . . . . . . . . . . . . . . ...
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...