Magnusson, L. 1995. The implementation of ALF -- a proof editor based on Martin-Lof's monomorphic type theory with explicit substitutions. Ph.D. thesis, Chalmers University of Technology and Goteborg University.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....grammatical annotations to make texts more idiomatic. 1 Introduction Computer algebra systems, such as Mathematica [21] and Maple [14] are widely used by mathematicians and students who do not know the internals of these systems. Proof editors, such as Coq [1] LEGO [2] Isabelle [4] and ALF [15], are less widely used, and require more specialized knowledge than computer algebras. One important reason is, of course, that the structures involved in manipulating algebraic expressions are simpler and better understood than the structures of proofs, and typically much smaller. This difference ....

....proof system Agda. Agda [7] is an implementation of structured type theory (STT) 8] which is based on Martin Lof s type theory [16] The system is implemented completely in Haskell, using the graphical user interface toolkit Fudgets [6] Like its predecessors in the ALF family of proof editors [15], Alfa allows the user to, interactively and incrementally, define theories (axioms and inference rules) formulate theorems and construct proofs of the theorems. All steps in the proof construction are immediately checked by the system and no erroneous proofs can be constructed. Alternatively, ....

L. Magnusson. The Implementation of ALF - a Proof Editor based on MartinL of's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Department of Computing Science, Chalmers University of Technology and University of Goteborg, 1994.

....by the FEMAT Brazilian foundation for research in mathematics, second author supported by the CAPES Brazilian foundation. Email: fayala,flaviog mat.unb.br Email: fairouz cee.hw.ac.uk,fairouz macs.hw.ac.uk c 2002 Published by Elsevier Science B. V. automated deduction and theorem proving [24,25], to proof theory [31] to programming languages [8,20,23,26] and to higher order uni cation HOU [2,13] This paper concentrates on three di erent styles of substitutions: i) The style [1] which introduces two di erent sets of entities: one for terms and one for substitutions. ii) The ....

L. Magnusson. The implementation of ALF - a proof editor based on Martin Lof's Type Theory with explicit substitutions. PhD thesis, Chalmers, 1995.

....partially suported by the FEMAT Brazilian foundation for research in mathematics, second author supported by the CAPES Brazilian foundation. Email: fayala,flaviog mat.unb.br Email: fairouz cee.hw.ac.uk c 2002 Published by Elsevier Science B. V. automated deduction and theorem proving [24,25], to proof theory [31] to programming languages [8,20,23,26] and to higher order uni cation HOU [2,13] This paper concentrates on three di erent styles of substitutions: i) The style [1] which introduces two di erent sets of entities: one for terms and one for substitutions. ii) The ....

L. Magnusson. The implementation of ALF - a proof editor based on Martin Lof's Type Theory with explicit substitutions. PhD thesis, Chalmers, 1995.

....of complete and incomplete proofs allows to identify the proof construction and term synthesis mechanisms. Furthermore, if such a representation supports an e#ective type checking procedure, type inconsistencies can be detected during the whole process of the proof term construction. In [28], Magnusson proposes an extension to the # calculus with place holders and explicit substitutions to represent incomplete proofs. Her ideas were implemented in the theorem prover Alf [2] but a complete meta theoretical study of the system and its properties is missing. A term with place holders ....

....about capture of variables. In the previous example the instantiation of Y with x results in the term #x:A.x, while the substitution of x for Y in #x:A.Y results in #z:A.x. Notice that unless A and B represent the same type, the resulting terms in both cases may be ill typed. As pointed out in [28, 15], open terms in the # calculus reveal new challenges. Assume, for example, that an open term is involved in a # redex. The # rule can create substitutions applied to meta variables that cannot be e#ective while the meta variables are not instantiated. In this case, a notation for suspended ....

[Article contains additional citation context not shown here]

L. Magnusson, The Implementation of ALF---A Proof Editor Based on Martin-Lof's Monomorphic Type Theory with Explicit Substitution, PhD thesis, Chalmers University of Technology and Goteborg University, Jan. 1995.

....strong normalization [30] that is, well typed terms may not terminate in ##. Furthermore, as a rewrite system, ## is not confluent on open terms [7] In constructive logic, explicit substitutions and open terms form a framework to represent incomplete proofs, i.e. proofs under development [29, 32]. In this approach, meta variables are place holders in a proof term, and an explicit substitution notation is necessary to delay the application of substitutions to meta variables waiting to be instantiated. Meta variables have also been used as unification variables in the higher order ....

L. Magnusson, The Implementation of ALF---A Proof Editor Based on Martin-Lof's Monomorphic Type Theory with Explicit Substitution, Ph.D. thesis, Chalmers University of Technology and Goteborg University, Jan. 1995.

....has, nor can their implementations be compared. Thus, it helps to have a choice of explicit substitution systems whose properties have already been established. This is witnessed by the recent theorem prover ALF, which is formally based on Martin Lof s type theory with explicit substitution [Mag95]. Another justification for explicit substitution in theorem proving is that some researchers believe tactics can be replaced by the notion of incomplete proofs, which are believed to need explicit substitutions [MH95, Mag95] The last fifteen years have seen an increasing interest in ....

.... is formally based on Martin Lof s type theory with explicit substitution [Mag95] Another justification for explicit substitution in theorem proving is that some researchers believe tactics can be replaced by the notion of incomplete proofs, which are believed to need explicit substitutions [MH95, Mag95]. The last fifteen years have seen an increasing interest in formalising substitution explicitly; various calculi including new operators to denote substitution have been proposed. Amongst these calculi we mention COE [dB78] the calculi of categorical combinators [Cur86] oe [ACCL91] oe ....

Magnusson. The implementation of ALF - a proof editor based on Martin Lof's Type Theory with explicit substitutions. PhD thesis, Chalmers, '95.

....has, nor can their implementations be compared. Thus, it helps to have a choice of explicit substitution systems whose properties have already been established. This is witnessed by the recent theorem prover ALF, which is formally based on Martin Lof s type theory with explicit substitution [Mag95]. Another justification for explicit substitution in theorem proving is that some researchers believe tactics can be replaced by the notion of incomplete proofs, which are believed to need explicit substitutions [Hur96b, Mag95] The last 15 years have seen an increasing interest in formalizing ....

.... is formally based on Martin Lof s type theory with explicit substitution [Mag95] Another justification for explicit substitution in theorem proving is that some researchers believe tactics can be replaced by the notion of incomplete proofs, which are believed to need explicit substitutions [Hur96b, Mag95]. The last 15 years have seen an increasing interest in formalizing substitution explicitly; various calculi, including new operators to denote substitution, have been proposed. Among these calculi we mention COE [dB78] the calculi of categorical combinators [Cur86] oe [ACCL91] oe [CHL96] ....

L. Magnusson. The implementation of ALF---a proof editor based on Martin-Lof's monomorphic type theory with explicit substitution. PhD thesis, Chalmers University of Technology and Goteborg University, January 1995.

....Martin L of s type theory, of which there exist di erent formulations (for example, Mar75] Mar84] NPS90] and [Tas97] The exposition here should suit all of them. The proof of the fan theorem presented here has been written down in full detail with the assistance of the proof editor ALF [Mag94] which is an implementation of the formulation of type theory given in [Tas97] The rest of this article is organized as follows. Section 2 introduces some notations and de nitions to be used in the whole article, and gives an informal presentation of bar induction and the fan theorem. Section 3 ....

L. Magnusson. The Implementation of ALF - a Proof Editor Based on MartinL of's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Department of Computing Science, Chalmers University of Technology and University of Goteborg, 1994. 14

....of de nitions and elimination of dummy de nitions. n x t n : x n : x let x=t in n let x=t in x 6 Chapter 2 Open Term Systems for Theorem Proving In this chapter we will discuss several systems that are speci cally designed to facilitate interactive theorem proving. 2. 1 ALF ALF [18] is an interactive proof editor based on Martin L of s monomorphic type theory. It employs placeholders and explicit substitutions in order to represent open terms. The theory of open terms of ALF was developed in order to support the main feature of the tool which is allowing the user to directly ....

....are contexts containing incomplete types. Each placeholder has an unique expected type (possibly incomplete) and an unique local context (also possibly incomplete) This means that the placeholder 1 Nevertheless, there are some other problems with conversion, discussed in section 4. 4 of [18] 8 may be re ned by a term whose free variables are from the local context and in this context this term should have a type identical to the expected type of the placeholder. The typechecking in this system poses speci c problems, some of which are not encountered in other systems because of ....

[Article contains additional citation context not shown here]

Lena Magnusson. The Implementation of ALF - a Proof Editor based on Martin-Lof Monomorphic Type Theory with Explicit Substitutions. PhD thesis, Chalmers University of Technology / Goteborg University, 1995.

....with de Bruijn indices presented in Section 2.3, then it terminates in the s calculus) If the s calculus is extended with open terms (variables that range over terms) then the reduction rules need also to be extended to guarantee con uence. This extension is essential for implementations, see [34, 36, 37]. Adding the 6 items of Lemma 43 as oriented rewriting rules results in the calculus s e which is con uent on open terms [28] Like of [1] this calculus does not satisfy PSN [18] De nition 50 (The s e calculus) Terms of the s e calculus are given by: s op : VjIN j( s op s op )j( s ....

E. Magnusson. The implementation of ALF - a proof editor based on Martin Lof 's Type Theory with explicit substitutions. PhD thesis, Chalmers, 1995.

.... The importance of the notion of substitution led to an explosion of work on making substitutions explicit in recent years [1, 7, 24, 26, 19, 9, 21] Moreover, a number of works have been devoted to establishing the usefulness of explicit substitution to automated deduction and theorem proving [32, 34], to proof theory [43] to programming languages [29, 6, 8] and to HOU [16] The latter paper [16] shows that in the HOU framework, if substitution was made 1 Work carried out during a year study leave visit of this author at the ULTRA Group, CEE, Heriot Watt University, Edinburgh, Scotland. This ....

L. Magnusson. The implementation of ALF - a proof editor based on Martin Lof's Type Theory with explicit substitutions. PhD thesis, Chalmers, 1995. 7. FUTURE WORK AND CONCLUSION 555

....has been done in the proof checker Half, developed by 1 The complete proofs are obtainable from the URL: ftp: ftp.cs.chalmers.se pub users ceder heineb hb.tar. Thierry Coquand, using a type checker and an emacs interface implemented by Dan Synek. The Half system is a successor to ALF [M]. It is a logical framework based on Martin Lof s polymorphic type theory with one universe [ML] extended by a theory mechanism (similar to the theory mechanism in PVS [OSR] and letexpressions (cf. C, Br, Ba] The system has three levels; Set, Type and Kind. Set is an element and a subset of ....

L. Magnusson. "The Implementation of ALF - a Proof Editor based on MartinL of's Monomorphic Type Theory with Explicit Substitution", Chalmers University of Technology and University of Goteborg, PhD Thesis, 1995.

....possible to prove properties about an already written program, as well as to construct a program satisfying a given specification. During the eighties, the research effort on type theories gave rise to different prototypes of interactive environments for the development of certified programs. ALF [52], Coq [20] Isabelle [71] Lego [51] and Nuprl [13] are among the most well known proof environments based on type theory. These prototypes have allowed researchers to jump from simple pencil and paper examples to the construction of verification proofs for non trivial cases of study see for ....

....in the sense of an abbreviation for an object that has been already constructed. On the contrary, it stands for an object under construction, which is only partially determined. It can be considered as a sort of existential variable or placeholder , like those used in certain proof editors [20, 52]. The description of a positive method of construction consists in providing a partial approximation of the object x , assuming that the same method may be used to provide a further approximation of it. This description leads us to introduce the types of the approximations and further ....

[Article contains additional citation context not shown here]

L. Magnusson. The implementation of ALF-- a Proof Editor based on Martin-Lof 's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Chalmers University of Goteborg, 1994.

....explicit substitutions have been studied in [25, 3] A polymorphic calculus with explicit substitutions has been proposed in [6] In contrast to our work, in those approaches explicit substitutions are not firstclass objects and atomic implicit substitutions are required by the type systems. In [18], Magnusson uses meta variables and explicit substitutions to represent incomplete proof terms in the Martin Lof Type Theory, but a complete metatheoretical study of the system is missing. More recently, Strecker has developed in [26] a complete meta theory for a variant of the Extended Calculus ....

L. Magnusson. The Implementation of ALF---A Proof Editor Based on MartinL of's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Chalmers and Goteborg Univ., 1995.

No context found.

Magnusson, L. 1995. The implementation of ALF -- a proof editor based on Martin-Lof's monomorphic type theory with explicit substitutions. Ph.D. thesis, Chalmers University of Technology and Goteborg University.

No context found.

Lena Magnusson. The Implementation of ALF---A Proof Editor Based on MartinL of's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Chalmers University of Technology and Goteborg University, January 1995.

No context found.

Magnusson, L. (1995). The Implementation of ALF---A Proof Editor Based on Martin-Lof's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Chalmers University of Technology and Goteborg University.

No context found.

L Magnusson. The implementation of ALF - a proof editor based on Martin-Lof's monomorhic type theory with explicit substitution. PhD thesis, Chalmers University, 1994.

No context found.

Lena Magnusson. The Implementation of ALF---a Proof Editor Based on Martin-Lof's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Chalmers University of Technology and Goteborg University, 1995.

No context found.

L. Magnusson. The Implementation of ALF - a Proof Editor based on Martin-Lof's Monomorphic Type Theory with Explicit Substitution. PhD thesis, Department of Computing Science, Chalmers University of Technology and University of Goteborg, 1994.

No context found.

Lena Magnusson. The Implementation of ALF - a Proof Editor based on MartinL of Monomorphic Type Theory with Explicit Substitutions. PhD thesis, Chalmers University of Technology / Goteborg University, 1995.

No context found.

L. Magnusson. The implementation of ALF - a proof editor based on Martin Lof's Type Theory with explicit substitutions. PhD thesis, Chalmers, 1995.

No context found.

Lena Magnusson. The Implementation of ALF - a Proof Editor based on MartinL of Monomorphic Type Theory with Explicit Substitutions. PhD thesis, Chalmers University of Technology / Goteborg University, 1995. 538, 541

No context found.

Lena Magnusson. The Implementation of ALF - a Proof Editor based on MartinL of's Monomorphic Type Theory with Explicit Substitutions. PhD thesis, Chalmers University of Technology / Goteborg University, 1995.

No context found.

Magnusson. The implementation of ALF - a proof editor based on Martin Lof's Type Theory with explicit substitutions. PhD thesis, Chalmers, '95.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC