R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, Univ. of Edinburgh, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....performance, provide assurance about the objectprograms behavior, or validate meaning preserving transformations. Examples of reasoning meta programs are program analyses such as flow analyses and type checkers. Reasoning meta programs are also used to build theorem proving systems such as LEGO [66], HOL [33] Coq [7] and Isabelle [57] and the study [36] and implementation [24] of logical frameworks such as Elf [60] Twelf [62] LF [61] Pedagogy. A pedagogical use of meta programs is program observation. Computation often proceeds in stages. Inputs arrive in several stages and the ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....HHP92, ACN90] for their subsequent use in machine assisted reasoning about programs. To this purpose it might be helpful to close the gap between the language of category theory and that of type theory. Recent work in program verification in synthetic domain theory using the LEGO proof checker [Pol95] can be found in [Reu95] Relating models of FPC. It is an interesting and rather straightforward observation that two interpretations of the simply typed calculus in a cartesian closed category, for which the base types get isomorphic objects, are essentially the same; in that the ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1995.

....9. CONCLUSION AND FURTHER WORK We have presented a method enabling to separate the syntactic overhead of the AL logic from the actual mathematical content of verification. A first example involving the recursive gcd object from [11] was very promising. While a formalisation [11] in LEGO [23] of the complete proof in AL took three weeks with a resulting proof script of several pages, the generated verification condition was just one line and amounted to the intuitive proof obligation that would arise from a simple imperative program. All the overhead due to objects, methods, etc. has ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....and their properties have already been studied in [10 12] In the next section we will see the principal properties of PTS that we will use in through this paper. An important property to be studied in PTS systems is the Expansion Postponement problem (EP ) which is introduced below. Following [9], let us consider the r and e systems, obtained when the rule ( is substituted, respectively, by ( r ) a:A a:A 0 A A 0 ( e ) a:A A 0 :s a:A 0 A A 0 For arbitrary PTS, the above systems are corrects: e ; r . The Expansion Postponement problem (EP ) presented ....

....: A 0 r b : B 0 0 r x : A:B : s 0 r x : A:b : x : A 0 :B 0 B B 0 ; A A 0 it is easy to obtain 0 r and (by inheritance) S 0 r . But the inclusion 0 r . r needs the S r property and we are faced with a recurrent problem. 1 We can see in [9] that S r fail in the Calculus of Constructions. So, admitting EP the S property is not hereditary. 4 EP via systems with normal types In the following, we will only consider PTS with normalising types. Let us now consider the problem of nding rules for a system that assigns normal ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....reduction. Proposition 10. If Gamma e : A and e fifi oe e 0 then Gamma e 0 : A. Third, the calculus enjoys decidable type checking, provided every legal term is strongly normalising. A key argument in the proof is that convertibility between legal types is decidable, see e.g. [6, 31, 37] for a recent survey of type checking algorithms for dependent type theory. Proposition 11. If every legal term is normalising, then it is decidable whether Gamma e : A is derivable. Finally, the calculus is consistent. This can be established by a standard model construction, e.g. the ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....decidable type checking. For such systems, the question remains whether it is possible to find reasonable, sound and complete, algorithms for type checking. The existence of such algorithms is not obvious and indeed the completeness of the most natural typechecking algorithm, due to R. Pollack [8], remains an open problem. In a nutshell, the problem is caused by the second premise of the abstraction rule, which makes it difficult to prove completeness by induction on the structure of derivations. Nevertheless several authors have proposed type checking algorithms that are sound and ....

....the second premise of the abstraction rule, which makes it difficult to prove completeness by induction on the structure of derivations. Nevertheless several authors have proposed type checking algorithms that are sound and complete for some specific classes of PTSs. In the early 90s, R. Pollack [7, 8] introduced the class of semi full PTSs informally a PTS is semi full if it has enough rules and gave a sound and complete type checking algorithm for PTSs in that class. Unfortunately, many PTSs of interest are not semi full. Later L.S. van Benthem Jutting, J. McKinna and R. Pollack [3, 8] ....

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R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....performance, provide assurance about the objectprograms behavior, or validate meaning preserving transformations. Examples of reasoning meta programs are program analyses such as ow analyses and type checkers. Reasoning meta programs are also used to build theorem proving systems such as LEGO [66], HOL [33] Coq [7] and Isabelle [57] and the study [36] and implementation [24] of logical frameworks such as Elf [60] Twelf [62] LF [61] Pedagogy. A pedagogical use of meta programs is program observation. Computation often proceeds in stages. Inputs arrive in several stages and the ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....my proposed strategy for its completion. 2 Progress My academic progress in the past year has been almost exclusively concentrated on work relating to the program logic [2] AL) of Abadi and Leino. My initial thesis proposal already described the beginnings of an embedding of the logic in LEGO[9]. Since then, my progress builds on this embedding in two orthogonal directions: I considered a new example, the dining philosophers; and together with Martin Hofmann, we successfully produced a justi cation of the embedding a soundness proof. As expected, over the past year, my ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....with regards to formal methods for verification. This article is a foundational contribution towards the development of formal tools verification for OO languages. We have implemented a program logic for an object calculus, based on the logic from [1, 2] We initially used the proof assistant LEGO[8] for historic reasons, and more recently we have repeated the exercise in PVS, though the techniques can be applied to other theorem provers, for example Isabelle HOL. There are broadly two di#ering approaches to embedding program logics in higher order logic. On one hand, we have the deep style ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....form according to fi ffi conversion are called convertible, and are treated as the same term by the logic. ffi conversion involves the substitution of a constant by its defining term and conversion is automation of inductive definitions. The CIC implemented in Coq differs from that of LEGO [9] by having two sorts of universes, an impredicative universe for sets in which functions are computable, and a predicative universe for types and propositions in which functions (predicates) need not be computable (decidable) Due to the Curry Howard isomorphism, theorem proving corresponds to the ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....object oriented programming in a general setting. In section 3, we give a survey of existing work on verification of programs. In this section, we also present some analysis of these approaches. In section 4, we discuss an experiment in implementing a verification system in the theorem prover LEGO[31]. Finally, in section 5 we propose further work. 2 Object oriented programming Surprisingly given the widespread acceptance of object oriented languages, there is no universally agreed definition of what constitutes an object oriented language. Of course, this state of affairs is not unique: ....

....are other works that I have not studied yet. These works include those of Poetzsch Heffter, Muller, Leavens, Liskov and Wing. It is intended that these will be considered in due course. 4 An experiment: a mechanisation of an object language verification system Inspired by [2] we have used LEGO[31] to implement a verification system. The system is essentially that presented in [2] except we allow transition relations to be higher order logic formulae. In fact, transition relations are coded as predicates in LEGO. We encoded the verification rules as closely as possible and derive from these ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....where no mechanism of renaming has to be considered in order to avoid capture. Further, there is no need for an a priori identification of ff convertible terms for the algorithm to be defined. This latter is, we think, quite a relevant point if one wants to describe an actual implementation. In [Pol94a] Pollack adopts the use of parameters to implement a type checking algorithm for a family of PTS [Bar92] One of the motivations for introducing the notion of parameter and consequently make use of them in the reformulation of the rules of inference of the formal system is to provide a solution ....

....in terms of record inclusion is partially achieved in terms of (the transitive closure of) coercions. Direct successors of this work are the mechanisms implemented by Bailey [Bai97] and Saibi [Sai97] for defining coercions between types or classes of types developed for the proof assistants LEGO [Pol94a] and Coq [Bar97] respectively. They have also formalized corresponding large scale case studies on Galois theory and Category theory. 17 In [Jac95] algebraic structures are formalized in Nuprl s version of type theory [Con86] using sets of unlabeled dependent pairs and subsets. No general ....

R. Pollack. The Theory of LEGO: a proof checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....pseudo terms are denoted by Legal. 8. If Gamma M : s is derivable and s 2 Sr then M is a type. The set of types is denoted by Type. 9. The triple S = T ; C; is the Pure Type System (PTS) induced by S. Note that the side condition A 2 V [ S in the (weakening) rule is due to R. Pollack [24]; omitting it does not change the set of derivable judgments. Normalization will play a central role in the development of this paper. The next denition collects some important aspects of normalization. Denition 3 Let S be a specication. 1. A fi reduction path from M 0 2 T is apossibly ....

.... Gamma R A : B 0 if B fi B 0 (reduction) 2. The relation Gamma r M : A is dened by the rules of Table 1, except for the rule (conversion) which is replaced by Gamma r A : B Gamma r A : B 0 if B fi B 0 (lax reduction) The above denition is taken from R. Pollack s thesis [24], which provides an introductionand an historical account up to 1994to the problem. Lemma 58 (Pollack [24] R r The problem of Expansion Postponement is stated as follows. Open Problem 59 Let fl be r or R . Gamma M : A ) 9A 0 2 T : Gamma fl M : A 0 A fi A 0 ....

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R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....The first task is to select and or develop an appropriate formalism for the machine representation and generalisation of a proof. There are various systems in which inference rules and proofs may be represented and generated, such as Isabelle [37] Prolog[35] ELF[40] Automath[36] LF[25] Lego[41] and various other sequent calculus, natural deduction [10] typed terms [10] tableau based, resolutionbased, matrix[46] hybrid proof systems. Each approach has its own advantages as well as limitations. Analytic proof systems [39] such as resolution, matrix or connection method based systems ....

....programs as formulas paradigms it is desirable and important to present proofs in a way that is at once intuitive, simple to analyse and amenable to automated analysis. It has been demonstrated that proof systems based on natural deduction and sequent calculus (such as NuPRL [5] Isabelle[37] Lego[41]) can be used quite successfully but the degree of automatic support is very weak. Natural deduction [10, 17] provides some advantages as a basis for logic programming [19] comprehensible proofs that are easy understandable for humans; easy provision of control and debugging facilities ( simply ....

Pollack R. The theory of LEGO - a proof checker for the extended calculus of constructions, PhD thesis, University of Edinburgh, 1994.

....where no mechanism of renaming has to be considered in order to avoid capture. Further, there is no need for an a priori identification of ff convertible terms for the algorithm to be defined. This latter is, we think, quite a relevant point if one wants to describe an actual implementation. In [Pol94a] Pollack adopts the use of parameters to implement a type checking algorithm for a family of PTS [Bar92] One of the motivations for introducing the notion of parameter and consequently make use of them in the reformulation of the rules of inference of the formal system is to provide a solution ....

....one presented in [Tas97, BT98] which incorporates the notion of parameters to represent the notion of free names . In that respect, we have situated ourselves closer to the spirit of the calculus presented by Coquand in the work we reference above. The work by McKinna and Pollack, presented in [MP93, Pol94a], concerning the type checking of PTS has also been quite influential in the development of our work. In another direction, Coquand [Coq96] has recently proposed an algorithm for type checking dependent types that, to some extent, conceptually departs from the spirit of the ones above mentioned. ....

R. Pollack. The Theory of LEGO: a proof checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....extended with a declaration for p, being p a fresh parameter. Then, we can see this abstraction rule as having an implicit renaming 10 Alpha Conversion in Simply Typed Lambda Calculus rule. A similar solution for dependent types has been presented and proved correct by Pollack in his thesis [13]. From the logical point of view, it is interesting to study the inhabitation of types. To study the inhabitation of types in the simply typed calculus, it does not make any di erence if we add conversion or not. However in some extensions of the calculus, conversion can increase the set ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....theoretical results, but it does not support their derivation. The calculus of constructions [CH88, PM93] and Martin Lof type theory [ML84, ML84] are type theories different from LF. Based on these type theories, many different proof development systems have been implemented: Coq [C 95] LEGO [LP92, Pol94], Nuprl [C 86] Alf [Mag93, MN94, Mag95] and others. These systems are not designed as programming languages. The aim of this thesis is to define the computational meta logic MLF for a fragment of LF. MLF is based on the intuitionistic sequent calculus with induction. Intuitionistic logic is ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

.... of constructive analysis on real numbers following Bishop s approach to constructive mathematics [Bis67] They represented reals by Cauchy sequences, and proved the completeness theorem using the Nuprl system [Con86] Successively Jones [Jon93] gave some theorems of constructive analysis in Lego [Pol94]. More recently Harrison [Har96] presented a significant part of classical analysis, constructing the real numbers by means of a technique closely related to Cantor s classical method in the context of the system Isabelle HOL [GM93] Later Cederquist, Coquand and Negri [Ced97] approached ....

R.Pollack: "The theory of Lego, a proof checker for the Extended Calculus of Constructions" PhD Thesis, University of Edimburgh, 1994

....which are constructive higher order logics with sophisticated type systems. This area is marked by its proliferation of different although related logics, many of which are supported by their own theorem prover. The most prominent of these provers are Coq [BBC 97] Nuprl [C 86] and Lego [Pol94] Also based on type theory are the Elf and Twelf systems [Pfe91, PS99] Strictly speaking the latter are not fully fledged theorem provers but logical frameworks specifically designed for prototyping but also (automatic) reasoning about deductive systems, in particular operational descriptions ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....with regards to formal methods for verification. This article is a foundational contribution towards the development of formal tools verification for OO languages. We have implemented a program logic for an object calculus, based on the logic from [1, 2] We have used the proof assistant LEGO[6] for historic reasons, though the techniques can be applied to other existing theorem provers, for example PVS and Isabelle HOL. The encoding is notable for using: HOAS for encoding program syntax; and a direct embedding of the object logic into the metalogic. The use of HOAS simplifies ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....to be a complete summary in any sense, but only serves the purpose of illustration. The references, not complete either, may serve as pointers to the work mentioned and to some other related work. 1 What is Lego ffl The Lego proof system Lego is a proof checker and a proof development system [19, 24]. Lego implements expressive type theories ECC [12] and UTT [15, 7] Lego has been used in formalisation of mathematics, program specification verification, and other applications. ffl The development of the system Lego was developed at Edinburgh in projects led by Rod Burstall, and ....

.... UTT (see eg, 15] Typed operational semantics [7, 8] Coercive subtyping [16, 29] cf, implementation of coercions in Lego [2] and in Coq [27] Extensionality and related issues [9] ffl Application examples: Formalisation of Pure Type Systems [3] and verification of proof checker [24, 22, 25] Verification of functional programs [4] imperative programs [11, 28] and concurrent programs [30] Model checking in Lego [31] SN proof of the system F [1] Formalisation of synthetic domain theory in Lego [26] ffl Development of representation schemes methods: Program ....

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R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, Edinburgh University, 1994.

.... Systems Preliminary version Judicael Courant October 96 1 Introduction The notion of Pure Type Systems has been first introduced by Terlouw and Berardi [Bar91] These systems are well suited for expressing specifications and proofs and are the basis of several proof assistants [CCF 95, Pol94, MN94, HHP93] However, there is actually a gap between PTS and the extensions needed for proof assistants. Indeed, PTS are well suited to type theoretic study, but lack some features that a proof assistant needs. A first practical expectation when specifying and proving in a proof assistant is ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....can be found on the Web at http: www.cs.chalmers.se augustss cayenne . 7 Related work There are many logical frameworks (proof checking systems) that are based on dependent types. Some examples, among many, are ALF [MN94,Nor93,ACN90] CoC [CH86,CH88] ELF [Pfe89,Fra91,HHP93] Lego [Pol94] and NuPRL [Con86] All these systems are primarily designed for making (constructive) proofs even if many of them can also execute the resulting proofs or extract a program from them. Our approach is dioeerent in that we want to make a programming language, not a proof system, but of course ....

Robert Pollack. The Theory of Lego A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....last of the above clauses we have adopted a Currystyle definition of the action on binders, in accordance with our decision to treat variable names informally. Such matters may be treated formally in a more elaborate fashion, following the lines of previous work of Pollack and the second author [MP93, Pol94, MP97]. The definition of the action changes, while the candidate closure conditions do not. We may now encapsulate the above mentioned induction loading once and for all, and moreover unify this notion of substitution with the notion of thinning, by introducing the following definition of context ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994. ftp://ftp.dcs.ed.ac.uk/pub/lego/thesis-pollack.ps.Z.

....theoretical results, but it does not support their derivation. The calculus of constructions [CH88, PM93] and Martin Lof type theory [ML84] are type theories different from LF. Based on these type theories, many different proof development systems have been implemented: Coq [C 95] LEGO [LP92, Pol94] Nuprl [C 86] Alf [Mag95] and others. These systems are not designed as programming languages. The aim of this research is it to provide a suitable meta theory supporting theoretical and representation work in LF. This paper will define the computational meta logic for the Horn fragment of ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....not be enough to convince a critical Twelf user of the correctness of a constructed proof. Pollack suggests to use a small trusted proof checker in order to check its correctness. There are many (more or less) trusted proof checkers, as for example HOL [Ge93] LCF ML [Pau87] Coq [C 95] Lego [Pol94] Isabelle [Pau94] or the schema checker for Elf [Roh94] and many others. To employ these proof checkers one has to define an interface converting a Twelf proof into the appropriate syntax, so that the trusted proof checker can easily decide its correctness. Such an interface can be set up in ....

....languages using higher order abstract syntax are no longer inductive. No strong induction principles exist. Hence this technique is only of limited use in the setting of automated proof assistants relying on the existence of strong induction principles as for example Coq, Nuprl [C 86] Lego [Pol94] Alf [AGNvS94, Mag95] PVS [ORS92] and many others. Languages using higher order abstract syntax can still be represented in these systems as defined types, but they cannot be used for inductive reasoning. Attempts to simulate higher order abstract syntax in the setting of strong induction ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....theoretical results, but it does not support their derivation. The calculus of constructions [CH88, PM93] and Martin Lof type theory [ML84] are type theories different from LF. Based on these type theories, many different proof development systems have been implemented: Coq [C 95] LEGO [LP92, Pol94] Nuprl [C 86] Alf [Mag95] and others. These systems are not designed as programming languages. This work was sponsored by the German Academic Exchange Service (DAAD) through the graduate fellowship program HSP II AUFE and by NSF Grant CCR 9303383 This paper defines the computational ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....pure type systems and extends ideas given originally in [1] Then applications are shown using the extended calculus of constructions [11] which contains Sigma types. In [2] and [16] mechanisms for defining coercions between types or classes of types are developed for the proof assistants LEGO [15] and Coq [3] respectively. In [12] a calculus in the spirit of Martin Lof s theory of types is presented, where forms of judgement are introduced, among others, that express the concepts of a kind K being a principal kind of an object k and (proper) kind inclusion. The meaning explanation of the ....

R. Pollack. The Theory of LEGO: a proof checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

.... Subject Reduction, and Strong Normalisation [21] Implemented in the Lego proof development system, UTT has been applied to verification of functional programs [10, 11] imperative programs [42] and concurrent programs [44] specification and data refinement [29] and formalisation of mathematics [40]. 2.3 Definitional equality and computational equality We use LF seriously as a meta level language (see Section 9.1.2 of [30] for a discussion) Along the same line, we make a distinction between the notion of definitional equality (abbreviational equality, reflected as fij equality for ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, Edinburgh University, 1994.

....in e is syntactic sugar for let x ( fix f:vfx : fg in e. We have chosen the syntax with fix to simplify the translation. The intended operational semantics is call by value, which we formalize by defining 4 Actually, Pure Type Systems with Universe Inclusion have been studied and formalized in [41] and it would make sense to use them as a basis. E : j E e j v E j if0 E e e j succ E j pred E (x:e) v 7 efx : vg (fix f:e) v 7 eff : fix f:eg v let x = v in e 7 efx : vg if0 0 e 1 e 2 7 e 1 if0 (i 1) e 1 e 2 7 e 2 succ i 7 (i 1) pred (i 1) 7 i pred 0 7 0 l 7 r ) ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....pure type systems and extends ideas given originally in [1] Then applications are shown using the extended calculus of constructions [12] which contains Sigma types. In [3] and [17] mechanisms for defining coercions between types or classes of types are developed for the proof assistants LEGO [16] and Coq [9] respectively. In [13] a calculus in the spirit of Martin Lof s theory of types is presented, where forms of judgement are introduced, among others, that express the concepts of a kind K being a principal kind of an object k and (proper) kind inclusion. The meaning explanation of the ....

Robert Pollack. The Theory of LEGO: a proof checker for the Extended Calculus of Constructions. Doctor of Philosophy thesis, University of Edinburgh, 1994.

....systems of interest have decidable typechecking [5] For such systems, the question remains whether it is possible to find reasonable, sound and complete, algorithms for type checking. Over the last years, several authors have proposed such algorithms for some specific classes of Pure Type Systems [3, 6, 13, 16, 17, 18]. However, the situation is in our view unsatisfactory because these algorithms either impose strong restrictions on the Pure Type Systems or make use of a complex derivability relation, see Section 6. In fact, the completeness of the most natural type checking algorithm, due to R. Pollack, ....

....M : s ) sort( GammajM ) s Then we use elmt( j: and sort( j: to eliminate the problematic clause in the (abstraction) rule of Pure Type Systems and obtain a sound and complete algorithm for type checking. Besides, we show that the same idea also applies to the problem of Expansion Postponement [4, 14, 16]. Contents The remaining of the paper is organized as follows: in Section 2, we provide a brief overview of Pure Type Systems. In Section 3, we present two motivating open problems, namely the completeness of Pollack s type checking algorithm and Expansion Postponement. In Section 4, we present a ....

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R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....These inference rules are the introductory clauses of five relations which are simultaneously inductively defined. For example, the typing judgement, Gamma a : A, is a ternary relation between raw contexts, raw terms, and raw types. This approach to syntax has been used extensively by Pollack [16] for mechanizing metamathematical proofs about type theory inside his type theory implementation LEGO. If we focus on the last question, various categorical notions of model come to mind: categories with attributes a la Cartmell, locally cartesian closed categories a la Seely, etc. We could try to ....

....one is that it is standard Another is that it is used in implementations. An argument for our intrinsic syntax is that it is much more compact, since it unifies the generation of raw syntax and of derivations of judgement. Standard syntax fully formalized is actually very complex, see Pollack [16] Another argument, just as important, is that it is closer to a notion of model. We shall make this precise by relating it to various categorical notions of model of dependent types. Moreover, the intrinsic approach comes out of taking the Curry Howard identification seriously on the syntactic ....

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R. Pollack. The Theory of Lego A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....language and deductive system for proving a collection of theorems, it is possible to support this by declaring the language and deductive system, and automatically scoping each (meta)theorem proved for them. This could be similar to the kind of support provided by theorem provers like LEGO [13] where a user begins (in the metalogic of the Edinburgh LF) by declaring a language and deductive system, and then proves a sequence of theorems in the scope of this context; afterwards, the system closes the scope by discharging the context, universally quantifying over all the term and proof ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....Subject Reduction property for a language of expressions similar to ours (see [Hol83] In some of his proofs (as for example in the proof of the Substitution Lemma) he shows that the conclusion of a theorem holds by informally manipulating the derivations in the type system. In his PhD thesis [Pol94], Pollack studies and formalises the Subject Reduction property for Pure Type Systems (PTS) in the proof checker LEGO. In his thesis, he distinguishes between bound variables that he calls variables and free variables that he calls parameters. Parameters and variables are disjoint sets. These two ....

R. Pollack. The Theory of Lego A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....Define an algorithm ty Gamma (M) which returns, when it exists, a type for M in context Gamma and returns otherwise. The algorithm is given in Figure 8; it makes use of an auxiliary function leg( Gamma) which checks whether a context is legal. More efficient algorithms can be derived, see [77]. 2 4.3.2 Injective CPTSs The central result of this section is a classification lemma for injective specifications. As usual with this kind of result, we partition the set of variables V as S s2S V s in such a way that each V s is countably infinite and V s V s 0 = for s 6j s ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994. Available by anonymous ftp from ftp.cs.chalmers.se in directory pub/users/pollack.

.... ALF [ACN90] Coq [D 91] NuPRL [C 86] and Lego [LP92] The experience of proof development shows that, to apply our technology of proof development such as systems based on type theory to large real world application problems (e.g. in formalisation of a large body of mathematics (cf. Pol94] and in verification of safety critical software components) it is very useful and indeed necessary to develop methods and supporting tools for reuse of proofs, programs, and the formalisation or development processes. Many proofs and many proof development processes have striking similarities. ....

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, Edinburgh University, 1994.

....developed a substantial body of formal knowledge, including Church Rosser and standardization theorems for beta reduction, and the basic theory of Pure Type Systems ( PTS ) leading to the strengthening theorem and type checking algorithms for PTS . Some of this work is reported in [MP93, vBJMP94, Pol94b, Pol95] In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, and simplified, proof of standardization for beta reduction, not ....

....development. In [MP93] we used formal notation, verbatim text manually extracted from LEGO source files; no translation errors occur, but there is no reason to believe the verbatim text in the paper actually appears in the files. Indeed, the the document and the files drifted apart over time. In [Pol94b] we again used formal notation, mechanically extracting marked sections of the source files, following the idea of Knuth s WEB. We could rerun the extraction to update the document to the formal source, but many readers complained the document was as unreadable as the formal source. Presenting a ....

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Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

....range over a type, SS, with decidable (boolean valued) equality. In the SML examples I used type string, which is built in to SML with boolean valued equality. In LEGO I am working parametrically in any equality type . I will not discuss the machinary of equality types in LEGO (see [MP93, Pol94] The type of contexts is lists (LL) of prop. As in the second, more expressive LCF variant, the theorems are inductively defined, but here they are an inductive relation instead of just an inductive type. Inductive [thm:context prop Type] Constructors ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994. Available by anonymous ftp from ftp.cs.chalmers.se in directory pub/users/pollack.

....since, and have developed a substantial body of formal knowledge, including Church Rosser and standardization theorems for beta reduction, and the basic theory of Pure Type Systems (PTS) leading to the strengthening theorem and type checking algorithms for PTS. Some of this work is reported in [30, 49, 36, 37]. In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, simplified, proof of standardization for beta reduction, not previously published, ....

....formal development. In [30] we used formal notation, verbatim text manually extracted from LEGO source files; no translation errors occur, but there is no reason to believe the verbatim text in the paper actually appears in the files. Indeed, the document and the files drifted apart over time. In [36] we again used formal notation, mechanically extracting marked sections of the source files, following the idea of Knuth s WEB. We could rerun the extraction to update the document to the formal source, but many readers complained the document was as unreadable as the formal source. Presenting a ....

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Pollack, R.: 1994b, `The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions'. Ph.D. thesis, University of Edinburgh.

....formal systems. The user gives a definition of the object logic to be checked (details vary) and either the implementation of the framework itself becomes a checker, or we program a checker in the internal language of the framework which is formally proved to be correct w.r.t. the definition (Pollack 1994, Barras 1996) The latter of these variations can be seen as an application of LCF style to type systems more expressive than SML (Pollack 1995) and supports LCF style tactics. For technical reasons, classical LCF tactics must be expanded to official proofs (which can be very costly) even when ....

....that shares some common substructures by construction. Martin Lof (1971) gives an algorithm for type synthesis that transforms official derivations to avoid duplication of context validity checking. This idea is used by Huet (1989) in the Constructive Engine, and abstractly explained and proved by Pollack (1994). The other computational expense in checking a formal system is the side conditions. For example in CC the rule of type conversion has convertibility of two well typed terms as a side condition. This is decidable, but certainly not feasible in general, so neither is proof checking. For the ....

Pollack, R. (1994). The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions, PhD thesis, University of Edinburgh.

....type synthesis in his impredicative system (now called ) that transforms official derivations to avoid duplicate work. This idea is used in Huet s Constructive Engine [Hue89] an abstract explanation and machine checked proof of correctness of this transformation on type systems is given in [Pol94] section 4.4.10. Another common technique for improving efficiency of checking a formal system is annotation of judgements so that a full derivation of a judgement can be mechanically constructed from the judgement itself. The use of decidable type checking as a tool for proof checking uses this ....

.... these needs [Gor93, GM96] However useful this approach turns out to be, it represents expressions only up to alpha conversion (as with de Bruijn indexes, HOAS, and another alternative [Sat83] A more intensional approach, using parameters and variables, is suggested in [Coq91] and formalized in [MP93, Pol94]. As just suggested, these representations are not all isomorphic : a presentation of type theory using de Bruijn indexes has different theorems than one using parameters and variables 6 , but by formalizing the relationship between them, theorems can be stated and proved in their most natural ....

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994. ftp://ftp.dcs.ed.ac.uk/pub/lego/thesis-pollack.ps.Z.

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R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, Univ. of Edinburgh, 1994.

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R. Pollack. The theory of LEGO A proof checker for the extended calculus of construction. Technical Report ECS-LFCS-95-323, LFCS, Computer Science Dept., University of Edinburgh, April 1995.

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Pollack, R. (1994). The Theory of LEGO: a proof checker for the Extended Calculus of Constructions. Ph.D. thesis, University of Edinburgh. 2

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Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

No context found.

Robert Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

No context found.

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

No context found.

R. Pollack. The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

No context found.

R. Pollack. The Theory of Lego A Proof Checker for the Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1994.

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