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39
Proof pearl: de bruijn terms really do work
 In TPHOLs, volume 4732 of LNCS
, 2007
"... Abstract. Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λcalculus (à laHuet,Nipkowand Shankar) is isomorphic to an αquotiented λcalculus. In order to establish the link, we introduce an “indexcarrying ” abstraction mechanism over de Bruijn t ..."
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Abstract. Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λcalculus (à laHuet,Nipkowand Shankar) is isomorphic to an αquotiented λcalculus. In order to establish the link, we introduce an “indexcarrying ” abstraction mechanism over de Bruijn terms, and consider it alongside a simplified substitution mechanism. Relating the new notions to those of the αquotiented and the proper de Bruijn formalisms draws on techniques from the theory of nominal sets. 1
A HigherOrder Specification of the πCalculus
, 2000
"... We present a formalization of a typed picalculus in the Calculus of Inductive Constructions. We give the rules for typechecking and for evaluation and formalize a proof of type preservation in the Coq system. The encoding of the picalculus in Coq uses Coq fonctions to represent bindings of variab ..."
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We present a formalization of a typed picalculus in the Calculus of Inductive Constructions. We give the rules for typechecking and for evaluation and formalize a proof of type preservation in the Coq system. The encoding of the picalculus in Coq uses Coq fonctions to represent bindings of variables. This kind of encoding is called a higherorder specication. It provides a concise description of the calculus, leading to simple proofs. The specification we propose for the picalculus formalizes communication by means of function application.
Recursion principles for syntax with bindings and substitution
 In ICFP
, 2011
"... We characterize the data type of terms with bindings, freshness and substitution, as an initial model in a suitable Horn theory. This characterization yields a convenient recursive definition principle, which we have formalized in Isabelle/HOL and employed in a series of case studies taken from the ..."
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Cited by 5 (4 self)
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We characterize the data type of terms with bindings, freshness and substitution, as an initial model in a suitable Horn theory. This characterization yields a convenient recursive definition principle, which we have formalized in Isabelle/HOL and employed in a series of case studies taken from the λcalculus literature.
Contributions to the Theory of Syntax with Bindings and to Process Algebra
, 2010
"... We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abst ..."
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We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abstract Syntax) and tries to take advantage of the best of both worlds. The connection between FOAS and HOAS follows some general patterns and is presented as a (formally certified) statement of adequacy. We also develop a general technique for proving bisimilarity in process algebra Our technique, presented as a formal proof system, is applicable to a wide range of process algebras. The proof system is incremental, in that it allows building incrementally an a priori unknown bisimulation, and patternbased, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. All the work presented here has been formalized in the Isabelle theorem prover. The formalization is performed in a general setting: arbitrary manysorted syntax with bindings and arbitrary SOSspecified process algebra in de Simone format. The usefulness of our techniques is illustrated by several formalized case studies: a development of callbyname and callbyvalue λcalculus with constants, including ChurchRosser theorems, connection with de Bruijn representation, connection with other Isabelle formalizations, HOAS representation, and contituationpassingstyle (CPS) transformation; a proof in HOAS of strong normalization for the polymorphic secondorder λcalculus (a.k.a. System F). We also indicate the outline and some details of the formal development. ii to Leili R. Marleene iii
Basic category theory for models of syntax, Course notes for
 Summer School on Generic Programming, SSGP'02
, 2002
"... Abstract. These notes form the basis of four lectures given at the Summer ..."
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Abstract. These notes form the basis of four lectures given at the Summer
External and internal syntax of the λcalculus
 In: Buchberger, Ida, Kutsia (Eds.), Proc. of the AustrianJapanese Workshop on Symbolic Computation in Software Science, SCSS 2008. No. 08–08 in RISCLinz Report Series
"... There is growing interest in the study of the syntactic structure of expressions equipped with a variable binding mechanism. The importance of this study can be justified for various reasons, e.g. educational, scientific and engineering reasons. This study is educationally important since in logic a ..."
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There is growing interest in the study of the syntactic structure of expressions equipped with a variable binding mechanism. The importance of this study can be justified for various reasons, e.g. educational, scientific and engineering reasons. This study is educationally important since in logic and computer science, we cannot avoid teaching the
Strong Induction Principles in the Locally Nameless Representation of Binders (Preliminary Notes)
"... Abstract. When using the locally nameless representation for binders, proofs by rule induction over an inductively defined relation traditionally involve a weak and strong version of this relation, and a proof that both versions derive the same judgements. In these notes we demonstrate with examples ..."
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Abstract. When using the locally nameless representation for binders, proofs by rule induction over an inductively defined relation traditionally involve a weak and strong version of this relation, and a proof that both versions derive the same judgements. In these notes we demonstrate with examples that it is often sufficient to define just the weak version, using the infrastructure provided by the nominal Isabelle package to automatically derive (in a uniform way) a strong induction principle for this weak version. The derived strong induction principle offers a similar convenience in induction proofs as the traditional approach using weak and strong versions of the definition. From our experience, we conjecture that our technique can be used in many rule and structural induction proofs. 1
A Formalization of a Concurrent Object Calculus Up to AlphaConversion
, 1999
"... We experiment a method for representing a concurrent object calculus in the Calculus of Inductive Constructions. Terms are first defined in de Bruijn style, then names are reintroduced in binders. The terms of the calculus are formalized in the mechanized logic by suitable subsets of the de Bruijn ..."
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We experiment a method for representing a concurrent object calculus in the Calculus of Inductive Constructions. Terms are first defined in de Bruijn style, then names are reintroduced in binders. The terms of the calculus are formalized in the mechanized logic by suitable subsets of the de Bruijn terms; namely those whose de Bruijn indices are relayed beyond the scene. The ffequivalence relation is the Leibnitz equality and the substitution functions can de defined as sets of partial rewriting rules on these terms. We prove induction schemes for both the terms and some properties of the calculus which internalize the renaming of bound variables . We show that, despite that the terms which formalize the calculus are not generated by a last fixed point relation, we can prove the desire inversion lemmas. We formalize the computational part of the semantic and a simple type system of the calculus. At least, we prove a subject reduction theorem and see that the specications and proofs have the nice feature of not mixing de Bruijn technical manipulations with real proofs.
Coding binding and substitution explicitly in isabelle
 University of Cambridge Computer Laboratory
, 1995
"... Logical frameworks provide powerful methods of encoding objectlogical binding and substitution using metalogical λabstraction and application. However, there are some cases in which these methods are not general enough: in such cases objectlogical binding and substitution must be explicitly code ..."
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Logical frameworks provide powerful methods of encoding objectlogical binding and substitution using metalogical λabstraction and application. However, there are some cases in which these methods are not general enough: in such cases objectlogical binding and substitution must be explicitly coded. McKinna and Pollack [MP93] give a novel formalization of binding, where they use it principally to prove metatheorems of Type Theory. We analyse the practical use of McKinnaPollack binding in Isabelle objectlogics, and illustrate its use with a simple example logic. 1