## Modular Reasoning in Isabelle (1999)

Citations: | 13 - 2 self |

### BibTeX

@TECHREPORT{Kammüller99modularreasoning,

author = {Florian Kammüller},

title = {Modular Reasoning in Isabelle},

institution = {},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

The concept of locales for Isabelle enables local definition and assumption for interactive mechanical proofs. Furthermore, dependent types are constructed in Isabelle/HOL for first class representation of structure. These two concepts are introduced briefly. Although each of them has proved useful in itself, their real power lies in combination. This paper illustrates by examples from abstract algebra how this combination works and argues that it enables modular reasoning.

### Citations

529 |
A lattice-theorethic Fixpoint Theorem and its applications
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- 1955
(Show Context)
Citation Context ...he first class representation of structures allows the reduction of the proposition and hence improves the proof process. 6.4. OPERATIONS ON MODULES 107 6.3.5 Tarski The fixed point theorem of Tarski =-=[Tar55]-=- is well known in computer science. Yet the form of the theorem which is usually proved is an older version from 1928. This theorem says that the least upper bound of all fixed points of a monotonic f... |

441 |
The formulae-as-types notion of construction
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(Show Context)
Citation Context ...ils the proposition we just proved, i.e. that the quotient is a group. If we interpret the sets that are our structures as types, then we see how the Curry-Howard isomorphism of propositions-as-types =-=[How80]-=- is embodied in a statement like above. In contrast to type theory, we do not need to state this isomorphism as a paradigm --- it is inherent because we use sets: from the above we can derive the logi... |

389 | Modular elliptic curves and Fermat’s last theorem
- Wiles
- 1995
(Show Context)
Citation Context ...pletely different domains of mathematics; a connection that was independently conjectured some 30 years before by Taniyama and Shimura but only solved by Wiles in order to prove Fermat's last theorem =-=[Wil95]-=-. We believe that formalization of mathematics plays a similar role in theorem proving. Attempts to construct good tools for the formalization of mathematics can produce techniques of general interest... |

198 |
Melham, editors. Introduction to HOL: A Theorem Proving Environment for Higher Order Logic
- Gordon, F
- 1993
(Show Context)
Citation Context ... 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 ar... |

164 |
Isabelle: A generic theorem prover, volume 828
- Paulson
- 1994
(Show Context)
Citation Context ...rate the use of locales and dependent types and validate that the combination of these concepts enables modular reasoning. 2 Prerequisites and Concepts Isabelle is a higher order logic theorem prover =-=[Pau94]-=-. It is generic, that is, it can be instantiated to form theorem provers for a wide range of logics. These can be made known to the prover by defining theories that contain sort and type declarations,... |

147 |
LEGO Proof Development System: User's Manual
- Luo, Pollack
- 1992
(Show Context)
Citation Context ...e are not adequate; we can only reason about a restricted set of aspects of the (mathematical) world. In rich type theories there is the concept of dependent types. Systems like Coq [D + 93] and LEGO =-=[LP92]-=- implement such type theories. If the hierarchies of the type theory are rich enough then dependent types are first class citizens. Usually, type theories do not have advanced module concepts as they ... |

125 | Using dependent types to express modular structure
- MacQueen
- 1986
(Show Context)
Citation Context ...heories do not have advanced module concepts as they are known in interactive theorem provers, like PVS and IMPS. However, it is well known that dependent types may be used to represent modules (e.g. =-=[Mac86]-=-). We verified by case studies (e.g. [KP99]) that a module system where the modules are first class citizens is actually necessary for an adequate representation of (mathematical) structures in the lo... |

99 |
Principia Mathematica
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- 1957
(Show Context)
Citation Context ... all kinds of practical tasks in software and hardware engineering. Higher order logic is usually traced back to Church's paper [Chu40], but its origin is to be seen in Russell's and Whitehead's work =-=[WR62]-=-, which has been motivated by a crisis in the foundations of mathematics shortly characterized by Russell's paradox about the set of all sets. Besides a contribution to the improvement of module syste... |

78 | IMPS: An interactive mathematical proof system
- Farmer, Guttman, et al.
- 1990
(Show Context)
Citation Context ...nd 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 structu... |

55 |
de Bruijn. A survey of the project AUTOMATH
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- 1980
(Show Context)
Citation Context ... results about finite sets to reduce it to simpler theorems about cosets. Hence, compared to [Yu90] our proof of Lagrange is simpler. Locales implement a sectioning device similar to that in AUTOMATH =-=[dB80]-=- or Coq [Dow90]. In contrast to this kind of sections, locales are defined statically. Also, optional pretty printing syntax is part of the concept. The HOL system [GM93] has a concept of abstract the... |

35 | Locales: A sectioning concept for Isabelle - Kammüller, Wenzel, et al. - 1999 |

30 |
The machine-checked literate formalisation of algebra in type theory
- Bailey
- 1998
(Show Context)
Citation Context ...ect logics --- not just Isabelle/HOL --- and have been used already in many applications apart from the ones presented here. 2.2 Dependent Types as First Class Modules In rich type theories, e.g. UTT =-=[Bai98]-=-, groups can be represented as \Sigma G : set: \Sigma e : G: \Sigma ffi : map 2 G G G: \Sigma \Gamma1 : map G G: group axioms where group axioms abbreviates the usual rules for groups, corresponding t... |

19 |
Abstract theories in HOL
- Windley
(Show Context)
Citation Context ...presented by separately compiled ML structures. In contrast to our main objective of adequacy, HOL theories address the efficient organization of proof developments. Earlier work on abstract theories =-=[Win93]-=- is more along our lines (see Section 5.4). Another possible example for a comparison was Eves [Saa89, KPS + 92, SC91]. However, after having studied the relevant publications we came to the conclusio... |

13 |
Doing algebra in simple type theory
- Gunter
- 1989
(Show Context)
Citation Context ...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... |

13 |
Théorèmes sur les groupes de substitutions
- Sylow
(Show Context)
Citation Context ...for the order of G. The proof displayed here and used as the basis for the formal proof is due to Wielandt [Wie59]. It generalizes the original form found by the Norwegian mathematician Sylow in 1872 =-=[Syl72]-=-. We give the proof following [Her64] but go much more into detail to prepare the description of the formalization. Proof The proof is presented in three major parts. In the second part the existence ... |

11 | Translating dependent type theory into higher order logic
- Jacobs, Melham
(Show Context)
Citation Context ...proving. Although the dependent types are only modelled as typed sets of Isabelle/HOL we get the "expressive advantage". In contrast to earlier mechanizations of dependent types in higher or=-=der logic [JM93]-=- our embedding is relatively lightweight as it is based on a simple set-theoretic embedding. At the same time the \Pi and \Sigma -types are strong enough to express higher-level modular notions, like ... |

11 |
Using Axiomatic Type Classes
- Wenzel
(Show Context)
Citation Context ... is declared as a member of the type class, then all axioms are inherited. The mechanism of defining type classes together with axioms is an explicit device of the Isabelle theory definition language =-=[Wen95]-=-. A type class in Isabelle is a device for grouping types. For example, semigroup ! term defines a type class as a subclass of the built-in class term of HOL. Type classes can be defined directly toge... |

9 |
A formal proof of Sylow’s first theorem – an experiment in abstract algebra with Isabelle HOL
- Kammüller, Paulson
- 1999
(Show Context)
Citation Context ... as they are known in interactive theorem provers, like PVS and IMPS. However, it is well known that dependent types may be used to represent modules (e.g. [Mac86]). We verified by case studies (e.g. =-=[KP99]-=-) that a module system where the modules are first class citizens is actually necessary for an adequate representation of (mathematical) structures in the logic of a theorem prover. Yet, it turns out ... |

9 |
Computer proofs in group theory
- Yu
- 1990
(Show Context)
Citation Context ...s are reflected 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 form... |

6 |
Naming and scoping in a mathematical vernacular
- Dowek
- 1990
(Show Context)
Citation Context ... finite sets to reduce it to simpler theorems about cosets. Hence, compared to [Yu90] our proof of Lagrange is simpler. Locales implement a sectioning device similar to that in AUTOMATH [dB80] or Coq =-=[Dow90]-=-. In contrast to this kind of sections, locales are defined statically. Also, optional pretty printing syntax is part of the concept. The HOL system [GM93] has a concept of abstract theories based on ... |

6 | The Implementation and Use of Abstract Theories in HOL - Gunter - 1990 |

6 |
Ein beweis für die existenz der sylowgruppen. Archiv der Mathematik, 10:401–402
- Wielandt
- 1959
(Show Context)
Citation Context ...contains a subgroup of order p ff . In the following we write a j b for a divides b and o(G) for the order of G. The proof displayed here and used as the basis for the formal proof is due to Wielandt =-=[Wie59]-=-. It generalizes the original form found by the Norwegian mathematician Sylow in 1872 [Syl72]. We give the proof following [Her64] but go much more into detail to prepare the description of the formal... |

5 | Modular Structures as Dependent Types in Isabelle
- Kammuller
- 1998
(Show Context)
Citation Context ... concepts whose scope is limited or temporary. Locality and adequacy are separate concerns that do not coincide generally. We propose to use separate devices, i.e. locales [KWP99] and dependent types =-=[Kam99b]-=-. We have designed and implemented them for Isabelle. In this paper, we show that in combination they realize modular reasoning. In Section 2.1 we shortly introduce the concept of locales for Isabelle... |

4 | Some Features of the Mizar Language
- Trybulec
- 1993
(Show Context)
Citation Context ...yer 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. Sinc... |

2 | Formal Verification of an Avioniocs Microprocessor - Srivas, Miller - 1995 |

1 |
Working Group Meeting
- Types
- 1998
(Show Context)
Citation Context ...ing automorphisms in Section 6.3.4. 5.4.2 Records In an earlier version of this work we used products as base type for simple structures. Due to a suggestion of P. Martin-Lof at the TYPES 98 workshop =-=[Typ98]-=- we employ here extensible records, which were first presented at the workshop. Although Martin-Lof suggested to use records as an alternative to HOL sets, we use the concept of records just in additi... |