## Little Theories (1992)

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Venue: | Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science |

Citations: | 49 - 16 self |

### BibTeX

@INPROCEEDINGS{Farmer92littletheories,

author = {William M. Farmer and Joshua D. Guttman and F. Javier Thayer},

title = {Little Theories},

booktitle = {Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science},

year = {1992},

pages = {567--581},

publisher = {Springer-Verlag}

}

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### Abstract

In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.

### Citations

601 |
A Mathematical Introduction to Logic
- Enderton
- 1972
(Show Context)
Citation Context ...a fundamental concept of mathematics and logic. As we mentioned in the introduction, this notion is formalized using certain syntactic translations that are known in logic as \theory interpretations&q=-=uot; [6, 25-=-]. Intuitively, a theory interpretation from T to T 0 species one of the (possibly many) ways of embedding T in T 0 , while preserving theorems. Logicians have used theory interpretations to prove met... |

478 | Basic Algebra - Jacobson - 1974 |

269 |
Foundations of Modern Analysis
- Dieudonné
- 1960
(Show Context)
Citation Context ...ghtforward proof in B.2 Another theorem which can be proved with a technique similar to the one we have outlined here is the Picard-Lindelof existence theorem for ordinary dierential equations (see [5=-=]-=-). Its proof involves an application of the Contraction Principle to a space of continuous functions on an interval. Much of this proof (which involves the construction of a large network of theories)... |

217 |
An introduction to the theory of lists
- Bird
- 1987
(Show Context)
Citation Context ...an then be applied to a large class of iterated operators, including the normal numerical sum and product operators. Many interesting facts about algorithms can be developed in this general framework =-=[1-=-]. Another very useful technique is to interpret a theory into itself or one of its subtheories. For example, many similar theorems about algebraic groups are just dierent instantiations of a particul... |

161 |
The semantics of Clear, a specification language
- Burstall, Goguen
- 1980
(Show Context)
Citation Context ... proving. The little theories idea is, however, a familiar ingredient in work on specification languages, probably first introduced by Burstall and Goguen [2]. It was a central tenet of work on Clear =-=[3, 4]-=-, and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical frameworks context, however, it appears in an unusual g... |

144 |
Putting Theories Together to Make Specifications
- Burstall, Goguen
- 1977
(Show Context)
Citation Context ...n path of previous work in mechanized theorem proving. The little theories idea is, however, a familiar ingredient in work on specification languages, probably first introduced by Burstall and Goguen =-=[2]-=-. It was a central tenet of work on Clear [3, 4], and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical framewo... |

129 | A guide to LP, the Larch prover
- Garland, Guttag
- 1991
(Show Context)
Citation Context ... to prove the images of the source theory axioms. Moreover, because of obj's equational logic, its usefulness as a theorem prover is, in our opinion, highly restricted. Curiously, the Larch Prover lp =-=[12]-=- does not give strong support for the little theories approach: there is only one theory available in the prover at a time, and thus theory interpretations cannot be used in proofs. E. Gunter [15] has... |

121 |
Parameterized Programming
- Goguen
- 1983
(Show Context)
Citation Context ...histicated network of interrelated theories used in the proof. 5.1 The Network of Theories The theories are constructed step-by-step using theory extension and theory instantiation (as recommended in =-=-=-[13]), beginning from a few general mathematical theories: A theory of an abstract ring. A theory of an abstract module over a ring. A theory of an abstract metric space (denoted M). A theory of r... |

106 |
Introducing OBJ3
- Goguen, Winkler
- 1988
(Show Context)
Citation Context ...in an extended version of lcf [24]. However, according to the the published description, theory interpretation and parameterized theories had not yet been implemented [24, pp. 384, 389]. Although obj =-=[14] inco-=-rporates a translation (\view") mechanism, the user is responsible for deciding whether the translation is in fact a theory interpretation: obj itself makes no attempt to prove the images of the ... |

79 | IMPS: An interactive mathematical proof system
- Farmer
- 1993
(Show Context)
Citation Context ...ed to expressing and proving sophisticated mathematical statements. One example, a simple inverse function theorem for Banach spaces, is presented in Section 5. For a general description of imps, see =-=[9, 10]-=-. Examples of imps proofs are found in [11, 10]. All of the examples given below (except where explicitly noted) represent material we have developed using imps. All concept formulation, calculation, ... |

75 |
A partial functions version of church’s simple theory of types
- Farmer
- 1990
(Show Context)
Citation Context ... about partial (and total) functions, and is equipped with a system of types and subtypes. Types and subtypes are collectively called sorts. The treatment of partial functions in lutins is studied in =-=[7]-=-, while the treatment of sorts and interpretations is the subject of [8]. For a detailed presentation of the syntax and semantics of lutins, see [17]. A language is built in imps from a signature|a li... |

59 |
Larch in five easy pieces
- Guttag, Horning, et al.
- 1985
(Show Context)
Citation Context ...amiliar ingredient in work on specification languages, probably first introduced by Burstall and Goguen [2]. It was a central tenet of work on Clear [3, 4], and it was also a motivating idea in Larch =-=[16]-=-. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical frameworks context, however, it appears in an unusual guise in which not just theories but also logics m... |

38 |
An informal introduction to specifications using Clear
- Burstall, Goguen
- 1981
(Show Context)
Citation Context ... proving. The little theories idea is, however, a familiar ingredient in work on specification languages, probably first introduced by Burstall and Goguen [2]. It was a central tenet of work on Clear =-=[3, 4]-=-, and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical frameworks context, however, it appears in an unusual g... |

30 |
A Simple Type Theory with Partial Functions and Subtypes
- Farmer
- 1993
(Show Context)
Citation Context ... theories approach, including its use of theory interpretations. On the contrary, the logic of theory interpretations is well understood, and a version for the particular logic we use is available in =-=[8-=-]. Interpretations have been eectively used in the logical literature since at least the 1950's [27], and in mathematics for much longer. Indeed, this makes interpretations especially attractive to us... |

30 | Structured theories in LCF
- Sannella, Burstall
- 1983
(Show Context)
Citation Context ...blems. There has also been some work on supporting little theories in mechanized theorem proving. Sannella and Burstall undertook to implement some of the ideas of Clear in an extended version of lcf =-=[24]-=-. However, according to the the published description, theory interpretation and parameterized theories had not yet been implemented [24, pp. 384, 389]. Although obj [14] incorporates a translation (\... |

16 |
The semantics of Clear, a speci language
- Burstall, Goguen
- 1980
(Show Context)
Citation Context ...rem proving. The little theories idea is, however, a familiar ingredient in work on speci cation languages, probablysrst introduced by Burstall and Goguen [2]. It was a central tenet of work on Clear =-=[3, 4]-=-, and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical frameworks context, however, it appears in an unusual g... |

15 |
Foundations of Geometry, Open
- Hilbert
- 1971
(Show Context)
Citation Context ...ng consistency and independence, the little theories approach has become a deeply entrenched way of organizing mathematical knowledge. For instance, in the introduction to The Foundations of Geometry =-=[1-=-9], Hilbert wrote that one of his aims was: to bring out as clearly as possible the signicance of the dierent groups of axioms and the scope of the conclusions to be derived from the individual axioms... |

15 |
A Proposed Interface Logic for Verification Environments
- Guttman
- 1991
(Show Context)
Citation Context ...atment of partial functions in lutins is studied in [7], while the treatment of sorts and interpretations is the subject of [8]. For a detailed presentation of the syntax and semantics of lutins, see =-=[17]-=-. A language is built in imps from a signature—a list of sort and constant declarations. A theory consists of a language L plus a set of axioms (i.e., sentences of L). Theories are the basic units in ... |

14 | IMPS: System description
- Farmer, Guttman
- 1992
(Show Context)
Citation Context ...eory. The goal of this paper is to demonstrate the usefulness of the little theories approach in mechanized reasoning. In addition, we will indicate how imps, an Interactive Mathematical Proof System =-=[9]-=-, supports the approach. We will not focus on logical issues related to the little theories approach, including its use of theory interpretations. On the contrary, the logic of theory interpretations ... |

10 |
Introduction to Axiomatic Set Theory
- Krivine
- 1971
(Show Context)
Citation Context ...ism for synthesizing a suitable theory. 13 ical practice. For instance, the consistency of the Generalized Continuum Hypothesis with ZF is sometimes proved in essentially this way (see, for instance, =-=[21-=-]). 8 In this case, both T and T 0 happen to be the same theory, namely ZF. 5 Example In this section we discuss an imps proof which exemplies a number of advantages of the little theories approach. O... |

9 |
Structure and representation in LF
- Harper, Sannella, et al.
- 1989
(Show Context)
Citation Context ...d by Burstall and Goguen [2]. It was a central tenet of work on Clear [3, 4], and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks =-=[18]-=-. In the logical frameworks context, however, it appears in an unusual guise in which not just theories but also logics may be combined. It is not clear whether this additional generality will prove t... |

9 |
Inference rules using local contexts
- Monk
- 1988
(Show Context)
Citation Context ...ges of the monoid axioms under I may not be theorems of B. Although I is not properly a theory interpretation, it is what we call a context theory interpretation from M to B relative to the \context&q=-=uot; [22, 10]-=- containing the assumptions f' 1 ; : : : ; ' n g. Context theory interpretations 12 are used in imps in much the same as way as ordinary theory interpretations, so long as our position in a proof lice... |

8 | Two computersupported proofs in metric space topology
- Farmer, Thayer
- 1991
(Show Context)
Citation Context ...hematical statements. One example, a simple inverse function theorem for Banach spaces, is presented in Section 5. For a general description of imps, see [9, 10]. Examples of imps proofs are found in =-=[11, 10]-=-. All of the examples given below (except where explicitly noted) represent material we have developed using imps. All concept formulation, calculation, and inference in imps is performed with respect... |

7 |
Putting theories together to make speci cations
- Burstall, Goguen
- 1977
(Show Context)
Citation Context ...main path of previous work in mechanized theorem proving. The little theories idea is, however, a familiar ingredient in work on speci cation languages, probablysrst introduced by Burstall and Goguen =-=[2]-=-. It was a central tenet of work on Clear [3, 4], and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical framewo... |

6 |
The Implementation and Use of Abstract Theories in HOL
- Gunter
- 1990
(Show Context)
Citation Context ... lp [12] does not give strong support for the little theories approach: there is only one theory available in the prover at a time, and thus theory interpretations cannot be used in proofs. E. Gunter =-=[15]-=- has made a start on implementing little theories within hol. So far as we know, imps, an Interactive Mathematical Proof System, is thesrst interactive theorem prover to have been designed from the st... |

5 |
A lattice of interpretability types of theories
- Mycielski
- 1977
(Show Context)
Citation Context ...e classic work of Tarski, Mostowski, and Robinson [27], for example, illustrates how theories can be proved undecidable by means of theory interpretation. References on theory interpretations include =-=[23, 26, 28, 29]-=-. 3 Pronounced as the word in French. 6 2.1 Theory Interpretations in IMPS The notion of a theory interpretation in lutins is similar to the standard notion insrst-order logic (see [6, 25]). However, ... |

4 |
Larch in easy pieces
- Guttag, Horning, et al.
- 1985
(Show Context)
Citation Context ...a familiar ingredient in work on speci cation languages, probablysrst introduced by Burstall and Goguen [2]. It was a central tenet of work on Clear [3, 4], and it was also a motivating idea in Larch =-=[16]-=-. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical frameworks context, however, it appears in an unusual guise in which not just theories but also logics m... |

4 |
A proposed interface logic for veri environments
- Guttman
- 1991
(Show Context)
Citation Context ...atment of partial functions in lutins is studied in [7], while the treatment of sorts and interpretations is the subject of [8]. For a detailed presentation of the syntax and semantics of lutins, see =-=[17]-=-. A language is built in imps from a signature|a list of sort and constant declarations. A theory consists of a language L plus a set of axioms (i.e., sentences of L). Theories are the basic units in ... |

3 |
An informal introduction to speci using Clear
- Burstall, Goguen
- 1981
(Show Context)
Citation Context ...rem proving. The little theories idea is, however, a familiar ingredient in work on speci cation languages, probablysrst introduced by Burstall and Goguen [2]. It was a central tenet of work on Clear =-=[3, 4]-=-, and it was also a motivating idea in Larch [16]. The idea is also an ingredient in more recent work on logical frameworks [18]. In the logical frameworks context, however, it appears in an unusual g... |