## Universal Algebra in Type Theory (1999)

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Venue: | Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs '99, volume 1690 of LNCS |

Citations: | 8 - 6 self |

### BibTeX

@INPROCEEDINGS{Capretta99universalalgebra,

author = {Venanzio Capretta},

title = {Universal Algebra in Type Theory},

booktitle = {Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs '99, volume 1690 of LNCS},

year = {1999},

pages = {131--148},

publisher = {Springer-Verlag}

}

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### Abstract

We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ...

### Citations

552 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...o or Alf. Although Coq is based on the Extended Calculus of Constructions (see [15]) everything could be formalized in a weaker system. Any Pure Type System that is at least as expressive as P! (see [2]) endowed with inductive types (see [22]), or Martin-Lof's Type Theory with at least two universes (see [16], [17] or [19]) is enough. We assume that we have two universes of types s for sets and ... |

280 |
Constructive mathematics and computer programming
- Martin-Löf
- 1982
(Show Context)
Citation Context ...malized in a weaker system. Any Pure Type System that is at least as expressive as λP ω (see [2]) endowed with inductive types (see [22]), or Martin-Löf’s Type Theory with at least two universes (see =-=[16]-=-, [17] or [19]) is enough. We assume that we have two universes of types ∗ s for sets and ∗ p for propositions (Set and Prop in the syntax of Coq), and that they both belong to the higher universe 2 (... |

149 |
Intuitionistic type theory," Bibliopolis
- Martin-Lof
- 1984
(Show Context)
Citation Context ...d in a weaker system. Any Pure Type System that is at least as expressive as λP ω (see [2]) endowed with inductive types (see [22]), or Martin-Löf’s Type Theory with at least two universes (see [16], =-=[17]-=- or [19]) is enough. We assume that we have two universes of types ∗ s for sets and ∗ p for propositions (Set and Prop in the syntax of Coq), and that they both belong to the higher universe 2 (Type i... |

116 |
Inductively defined types
- Coquand, Paulin-Mohring
- 1989
(Show Context)
Citation Context ...orresponding subtree; this is done by giving a function h :(Cb) → (W BC). c1 c2 c3 c4 (C b) (h c1) (h c2) (h c3) (h c4) b Formally we can define (W BC) in the Calculus of Inductive Constructions (see =-=[7]-=- and [9]) as the inductive type (W B C) with one constructor sup : (b : B)((C b) → (W B C)) → (W B C). As for any inductive definition, we automatically get principles of recursion and induction assoc... |

114 |
Computation and reasoning. A type theory for computer science., volume 11
- Luo
- 1994
(Show Context)
Citation Context ...alized inside Coq, but it could have equally easily been formalized in other proof systems based on Type Theory, like Lego or Alf. Although Coq is based on the Extended Calculus of Constructions (see =-=[15]-=-) everything could be formalized in a weaker system. Any Pure Type System that is at least as expressive as λP ω (see [2]) endowed with inductive types (see [22]), or Martin-Löf’s Type Theory with at ... |

82 |
Constable et al., Implementing Mathematics with the NuPRL
- L
- 1986
(Show Context)
Citation Context ...tient algebras we are led to consider a more suitable solution. In some version of (extensional) type theory notions of subtype and quotient type are implemented (for example in the Nuprl system, see =-=[6]-=-), but the version of (intensional) type theory implemented in Coq does not. Nevertheless a model of extensional type theory inside intensional type theory has been constructed by Martin Hofmann (see ... |

51 | Using Reflection to Build Efficient and Certified Decision Procedures
- Boutin
- 1997
(Show Context)
Citation Context ...n be used to program tactics inside the system (see [12]). An application of this reflection mechanism to algebra was developed by Samuel Boutin in Coq for the simplification of ring expressions (see =-=[5]-=-). In the present work the need to parameterize the construction of the syntactic level on the type of signatures posed an additional problem. A very general type construction similar to Martin-Löf’s ... |

42 | Typing algorithm in type theory with inheritance
- Saı̈bi
- 1997
(Show Context)
Citation Context ...g that s eq is an equivalence relation over the set s el. We often identify a setoid S with its carrier set (s el S). In Coq this identification is realized through the use of implicit coercions (see =-=[21]-=-). Similar implicit coercions are also used to identify an algebraic structure with its carrier. If a, b : S (i.e. as we said x, y :(s el S)), we use the simple notation x = y in place ofsUniversal Al... |

37 |
Computational Metatheory in Nuprl
- Howe
- 1988
(Show Context)
Citation Context ...oning. This method was already used by Douglas Howe to construct a partial syntactic model of the Type Theory of Nuprl inside Nuprl itself, which can be used to program tactics inside the system (see =-=[12-=-]). An application of this re ection mechanism to algebra was developed by Samuel Boutin in Coq for the simplication of ring expressions (see [5]). In the present work the need to parameterize the con... |

28 | Constructive category theory
- Huet, Säıbi
- 2000
(Show Context)
Citation Context ...res. Previous work on Algebra in Type Theory was done by Paul Jackson using the proof system Nuprl (see [14]), by Peter Aczel on Galois Theory (see [1]) and by Huet and Sabi on Category Theory (see [1=-=-=-3]). A large class of algebraic structures has been developed in Coq by Loc Pottier. Another aim is the use of a two level approach to the derivation of propositions about algebraic objects (see [4]).... |

27 |
A groupoid model refutes uniqueness of identity proofs
- Hofmann, Streicher
- 1994
(Show Context)
Citation Context ... that the form of the element is an application of one of the constructors corresponding to that type. This was proved for thesrst time by Hofmann and Streicher in the case of the equality types (see =-=[11]-=-). Therefore we just took the transitive closure of the above relation. Functions. We have constructed an interpretation of the sorts of a signature in setoids of terms as syntax trees. We still have ... |

22 |
Exploring abstract algebra in constructive type theory
- Jackson
- 1994
(Show Context)
Citation Context ...enterprise. We decided to develop Universal Algebra as a general tool to dene algebraic structures. Previous work on Algebra in Type Theory was done by Paul Jackson using the proof system Nuprl (see [14]), by Peter Aczel on Galois Theory (see [1]) and by Huet and Sabi on Category Theory (see [13]). A large class of algebraic structures has been developed in Coq by Loc Pottier. Another aim is the ... |

22 |
A set constructor for inductive sets in Martin-Löf’s type theory
- Petersson, Synek
- 1989
(Show Context)
Citation Context ...ypes. To deal with multi-sorted signatures we need to generalize the construction. The General Trees type constructor that we use is very similar to that introduced by Kent Petersson and Dan Synek in =-=[20]-=- (see also [19], chapter 16). In the multi-sorted case we have to define not just one type of terms, but n types, if n is the number of sorts. These types are mutually inductive. So we define a family... |

19 | A two-level approach towards lean proof-checking
- Barthe, Ruys, et al.
- 1996
(Show Context)
Citation Context ... [13]). A large class of algebraic structures has been developed in Coq by Loc Pottier. Another aim is the use of a two level approach to the derivation of propositions about algebraic objects (see [4=-=-=-]). In this approach, statements about objects are lifted to a syntactic level where they can be manipulated by operators. An example is the simplication of expressions and automatic equational reason... |

17 | A tutorial on recursive types in coq
- Gimenez
- 1998
(Show Context)
Citation Context ...g subtree; this is done by giving a function h : (C b) ! (W B C). (h c1) (h c2) (h c3) (h c4) c1 c2 c3 c4 (C b) b Formally we can dene (W B C) in the Calculus of Inductive Constructions (see [7] and [=-=9-=-]) as the inductive type (W B C) with one constructor sup : (b : B)((C b) ! (W B C)) ! (W B C). As for any inductive denition, we automatically get principles of recursion and induction associated wit... |

13 |
Judicaël Courant, Yann Coscoy, David Delahaye, Daniel de Rauglaudre, Jean-Christophe Fillâtre, Eduardo Giménez, Hugo Herbelin, Gérard Huet, Henri Laulhère
- Barras, Boutin, et al.
- 1998
(Show Context)
Citation Context ... form of set theory (for example in ZF). This paper presents a development of such tools for generic algebraic reasoning, which has been completely formalized in the Coq proof development system (see =-=[3-=-]). We want to enable the users of such tools to easily dene their own algebraic structures, manipulate objects and reason about them in a way that is not too far from ordinary mathematical practice. ... |

13 |
Elimination of extensionality in Martin-Löf type theory, in: Henk Barendregt, Tobias Nipkow (Eds
- Hofmann
- 1993
(Show Context)
Citation Context ...), but the version of (intensional) type theory implemented in Coq does not. Nevertheless a model of extensional type theory inside intensional type theory has been constructed by Martin Hofmann (see =-=[1-=-0]). We use a variant of this model, which has already been implemented by Huet and Sabi in [13] and used by Pottier. The elements of a type are build up using some constructors, and elements of a typ... |

10 |
Properties of Typing Systems
- Stefanova
- 1999
(Show Context)
Citation Context ...xtended Calculus of Constructions (see [15]) everything could be formalized in a weaker system. Any Pure Type System that is at least as expressive as λP ω (see [2]) endowed with inductive types (see =-=[22]-=-), or Martin-Löf’s Type Theory with at least two universes (see [16], [17] or [19]) is enough. We assume that we have two universes of types ∗ s for sets and ∗ p for propositions (Set and Prop in the ... |

9 |
Using re to build ecient and certi decision procedures
- Boutin
- 1997
(Show Context)
Citation Context ...can be used to program tactics inside the system (see [12]). An application of this re ection mechanism to algebra was developed by Samuel Boutin in Coq for the simplication of ring expressions (see [=-=5-=-]). In the present work the need to parameterize the construction of the syntactic level on the type of signatures posed an additional problem. A very general type construction similar to Martin-Lof's... |

6 |
Inductively de types
- Coquand, Paulin
- 1990
(Show Context)
Citation Context ...esponding subtree; this is done by giving a function h : (C b) ! (W B C). (h c1) (h c2) (h c3) (h c4) c1 c2 c3 c4 (C b) b Formally we can dene (W B C) in the Calculus of Inductive Constructions (see [=-=7-=-] and [9]) as the inductive type (W B C) with one constructor sup : (b : B)((C b) ! (W B C)) ! (W B C). As for any inductive denition, we automatically get principles of recursion and induction associ... |

5 | Proving and computing: a certified version of the buchberger’s algorithm
- Théry
- 1997
(Show Context)
Citation Context ...algebraic objects can be automatically manipulated inside a proof checker. This can be done through the use of certified versions of algorithms borrowed from Computer Algebra, as was done by Théry in =-=[23]-=- and by Coquand and Persson in [8] for Buchberger’s algorithm. The files of the implementation are available via the Internet at the site http://www.cs.kun.nl/˜venanzio/universal algebra.html. Type Th... |

3 | Notes towards a formalisation of Constructive Galois Theory
- Aczel
(Show Context)
Citation Context ...Algebra as a general tool to dene algebraic structures. Previous work on Algebra in Type Theory was done by Paul Jackson using the proof system Nuprl (see [14]), by Peter Aczel on Galois Theory (see [1]) and by Huet and Sabi on Category Theory (see [13]). A large class of algebraic structures has been developed in Coq by Loc Pottier. Another aim is the use of a two level approach to the derivati... |

2 | Integrated development of algebra in type theory. Preliminary version
- Coquand, Persson
- 1998
(Show Context)
Citation Context ...lly manipulated inside a proof checker. This can be done through the use of certied versions of algorithms borrowed from Computer Algebra, as was done by Thery in [23] and by Coquand and Persson in [8=-=]-=- for Buchberger's algorithm. Thesles of the implementation are available via the Internet at the site http://www.cs.kun.nl/~venanzio/universal algebra.html. Type Theory and Coq. The work presented her... |