L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In Proceedings of the First Workshop on Logical Frameworks, pages 39--42, Antibes, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is a full reflective subcategory of the topos. Analogous embeddings for weaker axiomatisations are under investigation by the author, Plotkin and Power. Type theory. An important direction of research is the formalisation of semantic developments in logical frameworks [dB80, Con86, CH88, 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 ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Informal Proceedings of the First Workshop on Logical Frameworks, Antibes, 1990.

....of the theorem prover Gandalf and several experimental results are described. 1 Introduction The subject of this paper is the problem of automated theorem proving in Martin Lof s monomorphic type theory [19, 8] which is the underlying logic of the interactive proof development system ALF [2, 14]. In the scope of our paper the task of automated theorem proving in type theory is understood as demonstrating that a certain type is inhabited by constructing a term of that type. The problem of inhabitedness of a type A is understood in the following way: given a set of judgements Gamma ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990.

.... higher order logic and constructive proofs are combined in constructive type theories, Martin Lof, 1970 ] There are now a number of theorem provers based on constructive type theories, e.g. Coq [ Dowek et al. 1991 ] NUPRL [ Constable et al. 1986 ] LEGO [ Luo and Pollack, 1992 ] ALF, Augustsson et al. 1990 ] Due to the di#culty of automating interesting proofs in some of these logics, most of these theorem provers are interactive (see 8) A wide variety of di#erent logics have been developed for formal methods. These include many di#erent: constructive type theories, temporal logics for ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990. 12

....well enough to experiment with both nameless representation and Coquand s named representation. By that time, however, inductive types had become a hot topic in implementation as well as in theory. I had seen Martin L of s encoding of inductive types implemented in the first version of ALF [ACN90] and, after discussions with Coquand and Luo, had a crude implementation of inductive types in LEGO by spring 1991. I should have also discussed it with Paulin Mohring, because I didn t understand about annotating the dependency of an inductive type on its constructors, which her implementation ....

....LEGO supports inductively defined types and relations [Alt93a,Dyb94,Luo94,CPM90,PM93] I especially recommend [DFH 93] for tutorial treatment of inductive types. The inductive types in LEGO are coded in the style of Martin L of, and the implementation owes much to work of Coquand [ACN90] For an example, consider the inductive type of natural numbers, corresponding to the ML datatype datatype NN = Z of NN S of NN NN; An inductive type is specified by giving the names and types of each of its constructors. We will see in a moment that more information is required to ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of another logical framework. In Informal Proceedings of First Workshop on Logical Frameworks, Antibes, May 1990. Available by ftp.

....need to consider various concrete data types as types in the predicative universes (c.f. similar research for Martin2 Lof s type theory, for example, ML84, Acz86, Dyb88, Dyb91] Pollack s Lego proof development system allows users to introduce new inductive data types in a style similar to ALF [ACN90], which conforms with Martin Lof s philosophical openness as explained above. However, it is necessary in applications to have a general mechanism in the theory which provides a large class of inductive data types. 1 Such studies of general mechanisms for inductive data types in the context of ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and G. Plotkin, editors, Preliminary Proc of Logical Frameworks, 1990.

....bugs. The output of most model checkers including the model checker of PVS for a correct system is only a TRUE. People can only choose to believe that TRUE as a pure act of faith, or not at all. On the other hand, the proofs of type theory based theorem provers, such as LEGO [LP92] ALF [ACN90, Mag92] Coq [D 91] and Nuprl [C 86] are proof terms ( terms) which in principle can be justified by different proof checkers so that people can have more confidence on formal proofs. Moreover, proof terms provide a common interface for different tools so that we can easily integrate ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and G. Plotkin, editors, Preliminary Proc. of Logical Frameworks, 1990.

.... higher order logic and constructive proofs are combined in constructive type theories, Martin Lof, 1970 ] There are now a number of theorem provers based on constructive type theories, e.g. Coq [ Dowek et al. 1991 ] NUPRL [ Constable et al. 1986 ] LEGO [ Luo and Pollack, 1992 ] ALF, Augustsson et al. 1990 ] Due to the difficulty of automating interesting proofs in some of these logics, most of these theorem provers are interactive (see x8) A wide variety of different logics have been developed for formal methods. These include many different: constructive type theories, temporal logics for ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990. 12

....the second, a smallest reflexive relation. Key words: coherence, unique identity proofs, type theory, categories 1 Introduction 1.1 Type theory. Martin Lof s type theory [13, 14] may be seen as a framework for expressing constructive mathematical reasoning. The interactive proof assistant ALF [2, 1, 12] can be used to implement a formalization of type theory and to develop formal type theory proofs. The fundamental ingredients of type theory are those of set, element of a set, family of sets and dependently typed 1 function. 1 The type of the result of a function application may depend on the ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and G. Plotkin, editors, Proceedings of the First Workshop on Logical Frameworks, Antibes, May 1992.

....model checker itself is rarely posed. In this paper, we take the position that this is an important question, whenever the proof environment we use should be highly reliable. This question gains even more importance in the context of provers based on intuitionistic type theory such as Coq [3] Alf [1] and Lego [12] where explicit proof objects (i.e. terms) are constructed during the proof. These proof objects are then verified by an inference engine implementing the basic proof rules. Since there are only a few rules and the correctness of any proof depends only on the correct implementation ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and P. G., editors, Preliminary Proceedings of Logical Frameworks, 1990.

....contain bugs. The output of most model checkers including the model checker of PVS for a correct system is only a TRUE. People can only choose to believe that TRUE as a pure act of faith, or not at all. On the other hand, the proofs of type theory based theorem provers, such as LEGO [18] ALF [1, 19], Coq [9] and Nuprl [6] are proof terms( terms) which in principle can be justified by different proof checkers so that people can have more confidence on formal proofs. Moreover, proof terms provide a common interface for different tools so that we can easily integrate various tools to complete ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and G. Plotkin, editors, Preliminary Proc. of Logical Frameworks, 1990.

....contain bugs. The output of most model checkers including the model checker of PVS for a correct system is only a TRUE. People can only choose to believe that TRUE as a pure act of faith, or not at all. On the other hand, the proofs of type theory based theorem provers, such as LEGO [16] ALF [1, 17], Coq [8] and Nuprl [6] are proof terms( terms) which in principle can be justified by different proof checkers so that people can have more confidence on formal proofs. Moreover, proof terms provide a common interface for different tools so that we can easily integrate various tools to complete ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and G. Plotkin, editors, Preliminary Proc. of Logical Frameworks, 1990.

....Dybjer and Ver onica Gaspes y Chalmers University of Technology and University of Gothenburg September 16, 1994 1 Introduction Peter Aczel [1] and Gerard Huet [8] have implemented the category of sets in LEGO and Coq respectively. Here we show an implementation of the category of sets in ALF [2], a proof assistant based on Martin Lof s logical framework (or theory of logical types) 10] We used WINDOW ALF. This system allows one to manipulate the proof term in order to refine it until it is complete. Some facilities are provided which show the term in a readable way (special symbols ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In Proceedings of the First Workshop on Logical Framewrks, Antibes, pages 39--42, 1990.

....Logic Draft: please do not distribute. Tanel Tammet and Jan M. Smith May 12, 1995 1 Introduction The subject of this paper is the problem of automated theorem proving in Martin Lof s monomorphic type theory [17, 8] which is the underlying logic of the interactive proof development system ALF [2, 14]. The task of automated theorem proving in type theory is to show that a certain type is inhabited by constructing a term of that type. We consider the question of axiomatizing the problem of inhabitedness of a type as a formula both in the Horn fragment of the standard first order logic (where ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990.

....for computer assisted formal reasoning. The intensive international effort to develop type theory and the associated technology for proof development has resulted in a number of impressive systems based on type theory for program verification and formalisation of mathematics, including 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 ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of another logical framework. In G. Huet and G. Plotkin, editors, Preliminary Proc. of Logical Frameworks, 1990.

....a full reflective subcategory of the topos. Analogous embeddings for weaker axiomatisations are under investigation by the author, Plotkin and Power. Type theory. An important direction of research is the formalisation of semantic developments in logical frameworks [dB80, CAB 86, CH88, 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 ....

L. Augustsson, Th. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Informal Proceedings of the First Workshop on Logical Frameworks, Antibes, 1990.

....could be implemented once in the logical framework avoiding duplication of efforts. Many logical frameworks have been developed, and there are already available computer running versions of them. Some examples are Edinburgh s ELF [HHP87] and LEGO [ZLT90] INRIA s COQ [GHPM89] and Goteborg s ALF [ACN90]. ELF is Pi typed calculus a weak constructive type theoryand was implemented by Harper, Honsell and Plotkin. LEGO and COQ are both based on an extension of Coquand s and Huet s Calculus of Construction. The former was developed by Luo, Pollack and Taylor and the latter by Huet, Coquand, Dowek ....

Lennart Augustsson, Thierry Coquand, and Bengt Nordstrom. A Short Description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, pages 39--42, Antibes, 1990.

....is devoted to a listing of the programs that compute the proof. I would not be surprised if proofs have been developed also using other systems, but I am unaware of that. This paper presents another machine assisted proof of the very same theorem. The proof was developed using the ALF system [1]. This is an implementation of a logical framework where it is possible to represent theories like Martin Lof s set theory, which is the setting of the proofs presented here. In addition, ALF provides a nice window interface, which makes the user s task much easier and more fun. It is not by ....

L. Augustsson, T. Coquand, B. Nordstrom. A short description of Another Logical Framework. Proceedings of the First workshop on Logical Frameworks, pp.39-42.

....HOL s type theory takes on more the role of a metalogic. Recently, M. Gordon has also been investigating techniques for merging HOL with set theory [Gor94] 13 Other theorem provers that use a constructive type theory and encode logic using the propositions as types correspondence include Alf [ACN90] LEGO [Pol90] and Coq [DFH 91] Alf uses a more recent type theory of Martin Lof s. Both LEGO and Coq use the Calculus of Constructions (CoC) CH85] extending it with a kind of inductive definitions [Luo89, PPM89] Bailey developed a concrete theory of polynomials in one variable in the ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In Proc. of First Annual Workshop on Logical Framworks, pages 38--41, Sophia-Antipolis, France, 1990.

....As a consequence of this, editors and automatic tools for using such formalized languages have been developed. This work is an experiment with proposals arising from both these fields: Martin Lof s set theory [ML84] as a foundational framework for mathematics, and ALF (Another Logical Framework) [ACN90] as editor and checker of a basic formalized language in which the above can be expressed. The example we chose is the proof that Ackermann s function is not primitive recursive. The proof is mathematically simple in the sense that it only uses induction and some basic properties of the addition ....

....Proof in Martin Lof s Set Theory In this section, we will present a complete formalization of the proof presented above for t primitive recursive functions. The formalization will be done in Martin Lof s set theory. The proof was written in an implementation of the (monomorphic) set theory in ALF [ACN90] which was also used to edit and check the proof. A complete presentation of Martin Lof s set theory can be found in [NPS90] We will start by recalling some notational conventions. Then we will show how we formalized the proof in Martin Lof s set theory. We will first present the definitions of ....

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L. Augustsson, T. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In G. Huet and G. Plotkin, editors, Informal Proceedings of the First Workshop on Logical Frameworks, pages 39--42. Esprit Basic Research Action 3245, May 1990.

....interpretation) Finally, appendix A shows the equivalence between IRT and the formulation of the Intuitionistic Ramsey Theorem given in [13] It justifies the unprovability of IRT in Type Theory. 2 The proofs All the proofs presented in subsections 2.1 and 2. 2 have been formalized in ALF (see [1, 6]) which is an implementation of Martin Lof s Type Theory (see [10] The ALF system itself is the appropriate tool to read the completely formal proofs. In the presentation of the proofs we write t 0 ; t n : T for t 0 ; t n are objects of type T . Set is a type: the type of sets or ....

Lennart Augustsson, Thierry Coquand and Bengt Nordstrom. A short description of Another Logical Framework. Proceedings of the workshop on types for proofs and programs, Antibes, 1990.

....of the theorem prover Gandalf and several experimental results are described. 1 Introduction The subject of this paper is the problem of automated theorem proving in Martin Lof s monomorphic type theory [16, 8] which is the underlying logic of the interactive proof development system ALF [2, 11]. In the scope of our paper the task of automated theorem proving in type theory is understood as demonstrating that a certain type is inhabited by constructing a term of that type. The problem of inhabitedness of a type A is understood in the following way: given a list of judgements Gamma ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990.

....without any deep semantic properties. The function type for hidden arguments should not be viewed as a new type. It is completely compatible with the normal function type. It only serves as a marker to aid the insertion of the hidden arguments. This view of hidden arguments was presented in [ACN90] and later used in other systems like Lego, LP92] where the concept was formalized. Similar mechanism exist e.g. in Quest, Car94] and Russell, BDD89] The current implementation of hidden arguments is quite weak and cannot always nd the hidden arguments even when it seems reasonable that ....

....program imakes sensej. A snapshot of the current implementation 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 ....

L. Augustsson, T. Coquand, and B. Nordstr#m. A short description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 3942, 1990.

....type theory have been implemented which have the important property that they can serve as logical frameworks: rules and axiom of various theories can easily be expressed in them. The systems Coq [16] and LEGO [31] are both based on the Calculus of Constructions; in this paper we will describe ALF [2, 32], which is based on Martin Lof s type theory. These computer systems have now been in use for about three years and among the examples carried out in them we just mention the following. The correctness of a data link protocol developed and used by Philips, modeled using I O automata theory, has ....

L. Augustsson, T. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990.

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L. Augustsson, T. Coquand, and B. Nordstrom. A short description of Another Logical Framework. In Proceedings of the First Workshop on Logical Frameworks, Antibes, pages 39--42, 1990.

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L. Augustsson, T. Coquand, and B. Nordstrom. A short description of another logical framework. In Proceedings of the First Workshop on Logical Frameworks, pages 39--42, Antibes, 1990.

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