W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS: An interactive mathematical proof system. Technical report, The MITRE Corporation, 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....this additional overhead. Consequently, we have preferred the plain type of total HOL functions in our theory, using inverse : rat and : rat rat. There have been several attempts to reformulate the basic ideas of HOL with partiality in mind, e.g. in the Lutins logic underlying IMPS [Farmer et al. 1993]. Additional builtin support for automated totality reasoning is required to turn the basic idea of first class partiality into a practically useful environment. The system of predicate subtypes of PVS [Owre et al. 1996] may get used to model partial functions as well. Definedness reasoning ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS: An interactive mathematical proof system. Journal of Automated Reasoning, 11(2):213--248, Oct 1993.

....mapped in EHDM, in which case it must be mapped to an equivalence relation. In PVS, mappings are provided as a syntactic component of names, and are essentially an extension of theory parameters. Equality is not treated specially, but is handled by mapping a given type to a quotient type. IMPS [FGT90,Far94] also supports theory interpretations. It is similar to EHDM in that it has a special def translation form that takes a source theory, target theory, sort association list, and constant association list, and generates a theory translation. Obligations may be generated that ensure that every ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. In Mark E. Stickel, editor, 10th International Conference on Automated Deduction (CADE), volume 449 of Lecture Notes in Computer Science, pages 653--654, Kaiserslautern, Germany, July 1990. Springer-Verlag.

....express recursive predicates. The Isabelle package might be the first to be based on the Knaster Tarski theorem. 8 Conclusions and future work Higher order logic and set theory are both powerful enough to express inductive definitions. A growing number of theorem provers implement one of these [9, 33]. The easiest sort of inductive definition package to write is one that asserts new axioms, not one that makes definitions and proves theorems about them. But asserting axioms could introduce unsoundness. The fixedpoint approach makes it fairly easy to implement a package for (co)inductive ....

Farmer, W. M., Guttman, J. D., Thayer, F. J., IMPS: An interactive mathematical proof system, J. Auto. Reas. 11, 2 (1993), 213--248

.... Set theory is the standard foundation for mathematics and for formal notations like Z [30] VDM [14] and TLA [15] However, most general purpose mechanised proof assistants support typed higher order logics (type theories) Examples include Alf [17] Coq [7] EHDM [19] HOL [12] IMPS [9], LAMBDA [10] LEGO [16] Nuprl [5] PVS [26] and Veritas [13] For many applications type theory works well, but there are certain classical constructions, like the definition of the natural numbers as the set f; f;g, f; f;gg, f; f;g,f; f;ggg, Delta Delta Delta g, that are essentially ....

W. M. Farmer, J. D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. Journal of Automated Reasoning, 11(2):213--248, 1993.

....and make selections automatically [11, 13, 62, 84, 123] These systems show their strength in and, to a large extent, are limited to Horn theories and theories that have the nature of rewrite rules. Other systems exist that apply knowledge based on hints 64 or other information from the user [36, 53, 61, 84]. There are also existing systems that use analogy: given a proof that is expected to be similar to the proof of the problem at hand, the prover is able to revise or expand the analogous proof [95] So a problem that remains largely open is that of automatically discovering a proof in a large ....

....IPR is not unusual in this way. In order to produce all of its output in natural language, IPR requires format strings to be associated with predicate and term definitions. Any system that translates formulas and terms into natural language must have this information given to it in some form [53]. In terms of the interaction, IPR is similar to interactive theorem proving systems [53] Powerful automatic systems do not offer the non expert user the opportunity to watch a proof in progress and change its direction. With few exceptions [74] automatic systems cannot produce a natural ....

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William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. Journal of Automated Reasoning, 11(2):213--248, August 1993.

....of theorem export : Mark2 allows theorems proved in one theory to be exported to another under suitable conditions. The prerequisites for export of a theory from theory A to theory B is the existence of a view of theory B from theory A (this term is borrowed from the developers of IMPS ([7]) The view is a list of translations of names of axioms and definitions, and possibly of other symbols. It does not need to be exhaustive: Mark2 will match theorems of A with their translations into B and either reject the view and abort the export (if the theorems do not match in form) or extend ....

....in Mark2; banning the converse rewriting annotation operators ( and ) would restrict the equational logic of Mark2 to rewriting logic. The theorem export system of 74 Mark2 implements the same insights as the little theories approach to theory modularity of the the developers of IMPS ([7]) though the implementation in IMPS is far more elegant. Though we had developed our system of theorem export already when we encountered the IMPS work, the IMPS developers writings made it much clearer to us what we had done, and also made it clear what further developments would be necessary. ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer, "IMPS: an interactive mathematical proof system", Journal of Automated Reasoning , vol. 11 (1993), pp. 213-48.

....mapped in EHDM, in which case it must be mapped to an equivalence relation. In PVS, mappings are provided as a syntactic component of names, and are essentially an extension of theory parameters. Equality is not treated specially, but is handled by mapping a given type to a quotient type. IMPS [FGT90,Far94] also supports theory interpretations. It is similar to EHDM in that it has a special def translation form that takes a source theory, target theory, sort association list, and constant association list, and generates a theory translation. Obligations may be generated that ensure that every ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. In Mark E. Stickel, editor, 10th International Conference on Automated Deduction (CADE), volume 449 of Lecture Notes in Computer Science, pages 653--654, Kaiserslautern, Germany, July 1990. Springer-Verlag.

....This complexity can enormously enlarge the search space one confronts when one tries to proves these theorems. On the other hand, tactic based theorem provers, beginning with LCF [27] and including systems such HOL [28, 29] Nuprl [21] the Calculus of Constructions [22] Isabelle [44] and IMPS [23], have paid considerable attention to user interaction and to the problem of formulating and supporting expressive languages for the formalization of mathematics. Techniques developed for first order theorem proving, however, have been essentially ignored with the exception of unification, which ....

Farmer, W. M., Guttman, J. D., and Thayer, J.: IMPS: An interactive mathematical proof system, J. Automated Reasoning 11 (1993), 213--248.

....This paper illustrates by examples from abstract algebra how this combination works and argues that it enables modular reasoning. 1 Motivation Modules for theorem provers are a means for organizing theories of applications. Generic interactive theorem provers like PVS [OSRSC98] IMPS [FGT93] and HOL [GM93] define their applications as object logics. Modules are used to maintain and structure these object logics. Being a classical software engineering concept for re usability and structuring, modules are the obvious method for organizing formalizations of theorem provers. Apart ....

....to the meta logic. Thereby reuse of the locale of groups is possible. 4 Conclusion 4.1 Related Work The proof of the theorem of Lagrange has been performed with the Boyer Moore Prover [Yu90] E. Gunter formalized group theory in HOL [Gun89] In the higher order logic theorem prover IMPS [FGT93] some portion of abstract algebra including Lagrange is proved. Mizar s [Try93] library of formalized mathematics contains probably more abstract algebra theorems than any other system. However, to our knowledge we were the first to mechanically prove Sylow s first theorem. Since it uses ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993.

....notions like schemas or abstract machines. In classical approaches modules for theorem provers are outside the logic: they do not have a logical representation, instead serve an efficient organization of theories. Nevertheless, most of the theorem provers that have powerful module systems (e.g.[OSR93,FGT93,GH93]) suggest to use their modules as representations for (algebraic) structures. Although the encapsulation and abstraction achieved by packaging structures into modules is sensible, it does not constitute an adequate representation. This becomes obvious once one leaves the scope of toy examples ....

....languages, finite automata, and the like, also as algebraic structures. Certainly, logical theories can as well be seen as algebraic structures, but it is not our aim to express logics like that. In some respects our view of algebraic structures corresponds to the notion of Little Theories [FGT93] in IMPS, but does not try to capture the notion of a logical theory. In this section we characterize our notion of simple algebraic structure and higher order structure. We use an informal notion of signature instead of modules because that is what the latter basically are. We do not use a ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993.

....which is rather unnatural. Then we would have to parameterize over all variables of the right hand side. In our example we would get something like M(G; p; ff) which is almost as bad as the original formula. 1. 1 Related Work There are several theorem provers that support modules, e.g. IMPS [FGT93] PVS [OSRSC98,ORR 96] and Larch [GH93] The authors of these systems suggest to use their modules for the representation of mathematical structures, for example abstract algebraic structures like groups. This representation by modules is often not adequate because the modules have no ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993.

....introduction to the systems PVS, IMPS, and Larch. The system descriptions in Section 2 are not complete. They omit lots of basic details and are more detailed in aspects interesting for the global subject. For more comprehensive introductions to PVS and IMPS the reader is referred to [OSR93] and [FGT93a] and for Larch to [GH93] Section 3 is then concerned with theories in those systems. There the notion of theories and the mechanisms available to handle those theories are introduced and in the following Section 4 some major aspects are compared. Finally, Section 5 tries to reflect some issues ....

....realistic case study Abstract Algebra offers all kinds of difficulties which are characteristic for the former application. 1 For example, this version does neither incorporate polymorphic types nor axiomatic type classes 2 2 Systems The Interactive Mathematical Proof System IMPS [FGT92a, FGT93a] developed at MITRE is mainly designed to the interactive machine supported proof of mathematical reasoning. It tries to emphasize the linking of axiomatic theories as the main method of mathematical reasoning [FGT92b] The heart of IMPS is its higher order logic LUTINS, a Logic of Undefined ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-- 248, 1993.

....to express recursive predicates. The Isabelle package might be the first to be based on the Knaster Tarski theorem. 8 Conclusions and future work Higher order logic and set theory are both powerful enough to express inductive definitions. A growing number of theorem provers implement one of these [9, 33]. The easiest sort of inductive definition package to write is one that asserts new axioms, not one that makes definitions and proves theorems about them. But asserting axioms could introduce unsoundness. The fixedpoint approach makes it fairly easy to implement a package for (co)inductive ....

Farmer, W. M., Guttman, J. D., Thayer, F. J., IMPS: An interactive mathematical proof system, J. Auto. Reas. 11, 2 (1993), 213--248

....formalized. Most automated reasoning systems are first order at best, while mathematics makes heavy use of higher order notations. We have conducted our work in Isabelle [20] which provides for higher order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS [6], HOL [8] and Coq [5] We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC) Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abrial and Laffitte [1] Grabczewski has mechanized the first two chapters of Rubin and Rubin s famous ....

....A class relation is a binary predicate and has the Isabelle type i # (i # o) A class function is traditionally represented by its graph, a single valued class predicate [12, page 25] it is more easily formalized in Isabelle as a meta level function, an object of type i # i. See Paulson [18, 6] for an example involving the Replacement Axiom. Because Isabelle ZF is built upon first order logic, quantification over variables of types i # o, i # i, etc. is forbidden. And it should be; allowing such quantification in uses of the Replacement Axiom would be illegitimate. However, ....

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William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. Journal of Automated Reasoning, 11(2):213--248, 1993.

....are presented as sequences of structures called verification lines (VLs) representing a graph structure. Each VL has an identifier, a formula, a set of dependencies, and a justification. A VL may be used as a premiss in several rule applications, providing a sharing of sub deductions. In the IMPS [25] system, derivations are represented as deduction graphs labelled graphs with two sorts of nodes, sequent and rule nodes. A sequent may be linked as premiss to any number of rule nodes (possibly none) giving sharing of sub deductions. A sequent may be the conclusion of zero or more rule nodes, ....

....finite set formation, conjunction and implication. A rule is correct for a system if it is valid in the system. Notions of a rule following from a set of rules and a rule being derivable from a set of rules are also defined. Reasoning structures generalize the deduction graphs used in IMPS [56, 25] in several ways: a richer domain of sequents; using constraints for provisional reasoning; and nesting. The work presented in this paper is an attempt at an axiomatic presentation of a wide class of deductive systems in the spirit of the work on general logics [53] Other meta logical frameworks ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993.

....development process. These objects are arranged in a taxonomy which is structured by a subsumption relation (see [SLW96] and [Lut95] for details) In the case of parameterized specifications, the subsumption relation is a covariant refinement relation. In a style partly inspired by the IMPS system [FGT93], theorems can be inherited from more general to more specific theories along theory morphisms. Acknowledgments The design of the Typelab language and system has to a great extent been influenced by Holger Pfeifer, Harald Rue and Detlef Schwier. Matthias Wagner has contributed a lot to the ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. J. of Automated Reasoning, 11:213-- 248, 1993.

....with rules for interactive proving. PVS has a rich type system and provides the ability to postpone type checking, by making presumptions, analogous to verification conditions. The Ontic rule compiler [42] compiles sets of rules of suitable form into efficient decision procedures. The IMPS system [16] uses theory interpretation maps to import results from one theory into another theory, and macetes (tricks) to express theory specific rules. Hyperproof [3]provides two representations of information, diagrams and first order sentences, to reason about simple blocks worlds problems. Inference ....

....finite set formation, conjunction and implication. A rule is correct for a system if it is valid in the system. Notions of a rule following from a set of rules and a rule being derivable from a set of rules are also defined. Reasoning structures generalize the deduction graphs used in IMPS [46, 16] in several ways: a richer domain of sequents; using constraints for provisional reasoning; and nesting. The work presented in this paper is an attempt at an axiomatic presentation of a wide class of deductive systems in the spirit of the work on general logics [43] Other meta logical frameworks ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS: an Interactive Mathematical Proof System. Jorunal of Automated Reasoning, 11:213--248, 1993.

....specific problem encountered when we step up from univariate to multivariate Algebra but, as pointed out in [Har96] of multivariate calculus and analysis in multidimensional Euclidean spaces as well. As suggested by Harrison this problem can be handled and indeed is handled accordingly in IMPS [FGT90] by using arbitrary (that is polymorphic) types together with a constraining axiomatization on these types. We believe that neither this approach nor the approach of defining all theories (and their signatures) explicitly are adequate at least not for the Omega mega system. The first ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. Imps: An interactive mathematical proof system. System report, MITRE Corporation, Bedford, MA01730 USA, 1990.

....axiomatizations) so that these systems show a nonmonotonic characteristic in the sense that they are no longer capable of proving some theorems if further (irrelevant) axioms are added. In interactive systems, structured theories are more commonly used. Isabelle [15, 16] HOL [8] and IMPS [6, 4] for instance use a layered approach to knowledge bases (cf. little theories in IMPS [5] Theories, however, are in a sense atomic, so that for a proof one must either explicitly enter a theory and use its methods solely, or gets the sum of all theories logically underneath. In the sequel it is ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer, `IMPS: An Interactive Mathematical Proof System', Journal of Automated Reasoning, 11(2), 213--248, (October 1993).

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Technical Report M90-19, The mitre Corporation, 1991.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

....than one axiomatic base for a string theory, and that these bases can be proven to be equivalent. Given the steep learning curve associated with theorem proving systems, we also wished to assess the benefit of learning how to use one. With the aid of the Interactive Mathematical Proof System, IMPS [FGT1993a] [FGT1993b] FGT1996] we were able to develop the beginnings of a theory, compare it to the beginnings of a theory used by Manna and Waldinger in The Logical Basis for Computer Programming [MW1985] and verify that the two theories are equivalent. By showing that the two theories are equivalent ....

Farmer, W. M., Guttman, J. D., and Thayer, F. J., " IMPS: an Interactive Mathematical Proof System," Journal of Automated Reasoning, volume 11, pages 213-248, 1993. 135

....of Peano arithmetic, a famous axiomatic theory that represents the standard model of the natural numbers. The computer theorem proving framework is mechanized by a wide range of di erent kinds of computer theorem provers. Examples include Automath [44] Coq [2] eves [15] hol [35] imps [28], Isabelle [46] Mizar [49] Nqthm [5] Nuprl [14] Otter [42] and pvs [45] Most theorem provers are primarily used to prove conjectures in the context of an axiomatic theory. Other aspects of the mathematics process are usually not well supported. However, some can be used to manage the ....

....soundness. And networks of biform theories are developed by creating biform theories, linking them with interpretations, and installing theorems, theoremoids, and de nitions in them. Many of the ideas and mechanisms used in ffmm are inspired by the imps Interactive Mathematical Proof System [25, 28, 29] and the Axiom [39] computer algebra system. The mechanization of ffmm is not discussed in this paper. We believe that ffmm can be mechanized using ideas embodied in computer theorem proving systems like imps and computer algebra systems like Axiom and Maple. This paper presents an overview of ....

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Farmer, W. M., J. D. Guttman, and F. J. Thayer: 1993a, `imps: An Interactive Mathematical Proof System'. Journal of Automated Reasoning 11, 213-248.

....of Peano arithmetic, a famous axiomatic theory that represents the standard model of the natural numbers. The computer theorem proving framework is mechanized by a wide range of di erent kinds of computer theorem provers. Examples include Automath [41] Coq [2] eves [14] hol [31] imps [25], Isabelle [43] Mizar [46] Nqthm [5] Nuprl [13] Otter [38] and pvs [42] Most theorem provers are primarily used to prove conjectures in the context of an axiomatic theory. Other aspects of the mathematics process are usually not well supported. However, some can be used to manage the ....

....soundness. And networks of biform theories are developed by creating biform theories, linking them with interpretations, and installing theorems, theoremoids, and de nitions in them. Many of the ideas and mechanisms used in ffmm are inspired by the imps Interactive Mathematical Proof System [22, 25, 26]. The mechanization of ffmm is not discussed in this paper. We believe that ffmm can be mechanized using ideas embodied in computer theorem proving systems like imps and computer algebra systems like Maple. There is a large body of work related to our proposal concerning (1) logical frameworks ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

....functions. Our scheme is presented within a formal higher order logic called LUTINS [2, 3, 4, 8] that admits undefined terms and partial functions and that contains a definite description operator. The scheme has been implemented and tested in the IMPS Interactive Mathematical Proof System [7, 8] which has LUTINS as its logic. IMPS is equipped with an automatic mechanism for syntactically checking whether a functional is monotone. Many common functions can be defined in IMPS by functionals on which the monotonicity check succeeds. As a result, defining functions in IMPS by recursion is ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993.

....argued (e.g. see Andrews remarks in [2] that cstt, with its strong 1 support for reasoning with functions, is a more practical reasoning system than traditional Zermelo Fraenkel set theory. cstt is the basis of the logics used in several computer theorem proving systems including hol [8] imps [6, 7], pvs [10] and tps [3] This paper presents an extended version of cstt called Basic Extended Simple Type Theory (bestt) It adds the following facilities to cstt: 1) Type variables for forming polymorphic types and expressions as in the hol logic [8] 2) New type constructors, expression ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

....machine and vice versa, in a certain sense of correspond. To state it, we will use # to mean is de ned, and we will say that s t (read s is quasi equivalent to t ) if (s # t #) s = t. Quasi equivalence says that the terms have the same denotation or lack thereof, and in our logic [3, 2] it is the condition that justi es substitution of s for t, wherever s is free for t. If C i is a computation of the implementation, then an abstract computation C corresponds to C i if there is a non decreasing function f : N N onto the domain of C such that abstr(C i (j) C(f(j) for all ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11(2):213-248, October 1993.

....1 Introduction In this paper we develop a new approach to process algebra and CSP. One goal of the paper is to provide a foundation for subsequent work in process speci cation, and to this end, the development has been done entirely using the imps system. For an overview of the imps system, see [6, 7]. Moreover, we have departed from the usual formulations in which external behaviors are modeled as sequences of events called traces. In the following presentation, we have adopted a mathematically more satisfying and more general approach which replaces the set of traces by the set of elements ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, October 1993.

....Second, it illustrates how virtual memory systems can be speci ed and, at least at the rst re nement levels, veri ed. Lower re nement levels would, however, raise additional issues of concurrency and of hardware dependencies. 1 Introduction imps, an Interactive Mathematical Proof System [6], aims to provide mechanical support for traditional methods and activities of mathematics, and for traditional styles of classical mathematical proof. The bulk of imps work has focused on mathematics [7, 5] However, the same broadly understandable techniques are also valuable for formal methods. ....

....preserves this invariant, and returns a value within the subsort vstate. But this work must generally still be done using other approaches to formalization; in imps one then bene ts from the system s ability to use sorting information e ectively in reasoning about the domain and range of functions [6]. The sort de nition that follows, generated automatically by imps, introduces a subsort of pre vstate named vstate. The theory being extended by this de nition is vm spec. The members of the new sort are those satisfying the predicate that follows, namely those :pre vstate such that the ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-248, 1993.

....functions what we will call the traditional approach to partial functions. Even though the traditional approach is well established in mathematics practice, very few contemporary mechanized mathematics systems support it. One major exception is imps, an Interactive Mathematical Proof System [6] developed by William M. Farmer, Joshua D. Guttman, and F. Javier Thayer F abrega. This paper describes the traditional approach to partial functions; illustrates how predicate logic can be modified to support it; and discusses mechanisms for implementing formalisms that support the traditional ....

.... Theta ff n ff n 1 . oe(t) ff means, if t is defined, the value of t is a member of D ff . That is, if a term is defined, its assigned sort gives some immediate information about its value which is very useful to both the human user and the computer. Sorts are discussed in more detail in [2, 4, 5, 6]; 2] and [5] present sort systems for a partial simple type theory and a partial set theory, respectively. 5 Definedness Checking In a logic like pfol that does not assume that all functions are total and all terms are defined, many questions about the definedness of terms must be answered in ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993. This article was processed using the L A T E X macro package with LLNCS style

....approach has been used in both mathematics and computer science (see [10] for references) In [10] we argue that the little theories method o ers important advantages for mechanized mathematics. Many of these advantages have been demonstrated by the imps Interactive Mathematical Proof System [9, 11] which supports the little theories method. A mechanized mathematics system based on the little theories method requires a di erent infrastructure than one based on the big theory method. In the big theory method all reasoning is performed within a single theory, while in the little theories ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-248, 1993.

....in contemporary mechanized mathematics systems. Computer algebra systems, such as Axiom [16] Macsyma [15] Maple [4] and Mathematica [24] o er a rich collection of techniques for performing symbolic computations. Theorem proving systems, such as Automath [19] Coq [1] eves [6] hol [14] imps [10, 11], Isabelle [20] Mizar [22] nqthm [3] Nuprl [5] Otter [17] and pvs [21] have much of technology that an iml needs, but they are more narrow in scope than an iml and are very dicult to use without a fairly deep understanding of formal mathematics. These systems are a signi cant step toward an ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-248, 1993.

....The MITRE Corporation 202 Burlington Road Bedford, MA 01730 1420, USA ffarmer,guttman,men,jtg mitre.org 5 July 1994 Abstract. This paper introduces the imps proof script mechanism and some practical methods for exploiting it. 1 Introduction imps, an Interactive Mathematical Proof System [4, 2], is intended to serve three ultimate purposes: To provide mathematics education with a mathematics laboratory for students to develop axiomatic theories, proofs, and rigorous methods of symbolic computation. To provide mathematical research with mechanized support covering a range of ....

....fundamental role in the imps proof system. Sections 3 5 describe the imps proof script mechanism and ways it can be put to use. Section 6 brie y compares imps proof scripts and macetes with traditional tactics. And Section 7 contains a conclusion. 2 Macetes Macetes supplement the imps simpli er [4, 5] in order to provide more exibility to the user. The simpli er applies universally quanti ed equalities as rewrite rules in a manner which is usually beyond the user s control. In particular, it is not possible for the user to direct the simpli er to apply only those theorems that belong to a ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-248, 1993.

....System Description William M. Farmer, Joshua D. Guttman, F. Javier Thayer The MITRE Corporation 3 November 1992 imps, an Interactive Mathematical Proof System [5], aims at computational support for traditional techniques of mathematics. It is based on three observations about rigorous mathematics: Mathematics emphasizes the axiomatic method. Characteristics of mathematical structures are summarized in axioms. Theorems are derived for all structures ....

....can be used to transport a theorem from the theory it is proved in to any number of other theories. Theory interpretations are also used in imps to show relative consistency of theories, to formalize symmetry and duality arguments, and to prove universal facts about polymorphic operators [5, 4, 1]. The great majority of the theory interpretations needed by the imps user are built by software without user assistance. For example, when a theorem is applied outside of its home theory via a transportable macete, imps automatically builds the required theory interpretation if needed. 3 ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

....value, and includes lambda notation with term constructors for function application and function abstraction. In short, it includes both the set theoretic machinery of nbg set theory and the function theoretic machinery of lutins [1 3] the logic of the imps Interactive Mathematical Proof System [6, 7]. For the purposes of this paper, the reader does not need an intimate understanding of stmm. A language of stmm contains two kinds of expressions: terms which may be unde ned and formulas which denote true or false and are always de ned. An axiomatic theory of stmm is a pair (L; where L is a ....

....in some cases possibly unsound) transformers using constructors that always produce sound transformers. Third, a transformer can be manually de ned and then manually proven to be sound. These ways of de ning transformers are illustrated with examples inspired by the macete mechanism of imps (see [7, 9]. 7 6.1 Generating Transformers from Theorems There are several ways that sound transformers can be automatically generated from theorems. We will give two representative examples. Example 3 (Implication Transformers) Suppose T j= 8x 1 1 ; x n n : A 0 A 00 : Let 1 ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-248, 1993.

....the abstractly de ned solution is usually itself more dicult than the theory of the exact solution. The basic idea of our method is to write a numerical program in the PreScheme programming language [9] and then translate it into a representation in the imps Interactive Mathematical Proof System [2] so that conjectures 1 concerning the correctness of the program can be investigated with the help of imps. The representation of the Pre Scheme program in imps is based on the standard for numerical datatypes proposed by M. Payne, C. Scha ert, and B. Wichmann [8] We have produced a software ....

.... (simplify antecedent with(r:rr,not(0 =r) simplify antecedent with(m:zz,r:rr,not(r m) simplify antecedent with(n:zz,r:rr,not(r =n) block (script comment direct inference at (1) instantiate universal antecedent with(p:prop,forall(k:zz,p and p implies (p implies p) [ 2] m ) simplify antecedent with(r:rr,not(0 =r) simplify antecedent with(m:zz,r:rr,not(r m) simplify antecedent with(n:zz,r:rr,not(r =n) def compound macete apply machine axioms (repeat machine arithmetic extension axiom 0 machine arithmetic extension axiom 1 ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213{ 248, 1993.

....functions what we will call the traditional approach to partial functions. Even though the traditional approach is well established in mathematics practice, very few contemporary mechanized mathematics systems support it. One major exception is imps, an Interactive Mathematical Proof System [6] developed by William M. Farmer, Joshua D. Guttman, and F. Javier Thayer F abrega. This paper describes the traditional approach to partial functions; illustrates how predicate logic can be modi ed to support it; and discusses mechanisms for implementing formalisms that support the traditional ....

.... (f) 1 n n 1 . t) means, if t is de ned, the value of t is a member of D . That is, if a term is de ned, its assigned sort gives some immediate information about its value which is very useful to both the human user and the computer. Sorts are discussed in more detail in [2, 4, 5, 6]; 2] and [5] present sort systems for a partial simple type theory and a partial set theory, respectively. 5 De nedness Checking In a logic like pfol that does not assume that all functions are total and all terms are de ned, many questions about the de nedness of terms must be answered in the ....

[Article contains additional citation context not shown here]

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213-248, 1993. This article was processed using the L A T E X macro package with LLNCS style

.... However, nbg is finitely axiomatizable, while zf is not (see [11] or [13] for a proof) stmm admits undefined terms and has the same kind of machinery for reasoning with functions and for classifying terms as lutins [4, 5, 6] the logic of the imps Interactive Mathematical Proof System [10]. lutins closely corresponds to mathematics practice and has proven to be an effective logic for formalizing traditional mathematics (e.g. see [9] In particular, stmm is equipped with operators for forming definite descriptions, function applications, and function abstractions and a sort ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213--248, 1993.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS: An interactive mathematical proof system. Technical report, The MITRE Corporation, 1990.

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W.M.Farmer, J.D.Guttman and F.J.Thayer, IMPS: AN Interactive Mathematical Proof System, Technique Report, The MITRE Corporation, 1990.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS : an interactive mathematical proof system. Journal of Automated Reasoning, 9(11):213--248, 1993.

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W. Farmer, J. Guttman, and J. Thayer. IMPS: an interactive mathematical proof system. LNCS 449, 1990.

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William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. Journal of Automated Reasoning, 11(2):213--248, October 1993.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. IMPS : an interactive mathematical proof system. Journal of Automated Reasoning, 9(11):213-248, 1993.

No context found.

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. IMPS: An Interactive Mathematical Proof System. Journal of Automated Reasoning, 11(2):213-248, October 1993.

No context found.

W. M. Farmer, J. D. Guttman and F. J. Thayer. "IMPS: an Interactive Mathematical Proof System." Technical Report M90-19, MITRE Corporation, 1991.

No context found.

W. M. Farmer, J. D. Guttman and F. J. Thayer. "IMPS: an Interactive Mathematical Proof System." Technical Report M90-19, MITRE Corporation, 1991.

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