V. Capretta. Universal algebra in type theory. In Y. Bertot et al., editors, Proc. TPHOLs '99, volume 1690 of LNCS, pages 131--148. Springer-Verlag, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a map ar : F N which assigns its arity to each function symbol. Denition 1. The type Signature of signatures is dened by Signature : Class = ffun : Set; ar : fun Ng Our representation is the standard one but there are alternatives formalisations that only focus on nite signatures, see e.g. [12]. Each signature has an associated notion of algebra. Recall that an algebra for a signature S = F; ar) consists of a set A, called the carrier of the algebra, and of a function f : A ar(f) A for every f 2 F . Denition 2. The type Algebra of algebras over a signature is dened by Algebra[S ....

V. Capretta. Universal algebra in type theory. In Y. Bertot, G. Dowek, A. Hirschowitz, C. Paulin, and L. Th#ry, editors, Proceedings of TPHOL'99, volume 1690 of Lecture Notes in Computer Science, pages 131148. Springer-Verlag, 1999.

....a map ar : F N which assigns its arity to each function symbol. Denition 1. The type Signature of signatures is dened by Signature : Class = ffun : Set; ar : fun Ng Our representation is the standard one but there are alternatives formalisations that only focus on nite signatures, see e.g. [14]. Each signature has an associated notion of algebra. Recall that an algebra for a signature S = F; ar) consists of a set A, called the carrier of the algebra, and of a function f : A ar(f) A for every f 2 F . To formalise algebras, we therefore need to formalise sets and n ary functions. ....

V. Capretta. Universal algebra in type theory. In Y. Bertot, G. Dowek, A. Hirschowitz, C. Paulin, and L. Th#ry, editors, Proceedings of TPHOL'99, volume 1690 of Lecture Notes in Computer Science, pages 131148. Springer-Verlag, 1999.

....maintainability. In fact the problematic in theorem provers is very close to the one in programming languages. Many formalisations involving standard concepts in logic have been obtained within a logical framework: for example, universal algebra (and standard constructions over algebras) in CoQ [1] completeness proof for Kripke semantics for intuitionistic minimal propositional logic in ALF [3] basic rst order model theory in HOL [6] temporal logic in CoQ [7] basic logic programming theory (including uni cation [9] soundness and completeness of SLD resolution) in CoQ [10] ....

....in ALF [13] All these developments share some common de nitions and properties which have been formalised in each of these developments. Of course, some of these formalisations aim to provide a generic tool for others works. For example, the formalisation of universal algebra, done in [1], provides a tool for generic algebraic reasoning and the proof of the existence of an isomorphism between three inference system for rst order logic, done in [12] allows to guarantee that meta logical provability properties about one of them would also hold in relation to the others (properties ....

V. Capretta. Universal algebra in type theory. In Y. Bertot, G. Dowek, A. Hirshowitz, C. Paulin, and L. Thery, editors, 12th International Conference on Theorem Proving in Higher Order Logics TPHOLs'99, volume 1609 of Lecture Notes in Computer Science, pages 131148. Springer Verlag, 1999.

....of constructors of Tn and their types are de ned by recursion on n. Families of this kind have not only theoretical interest. They arise in the course of formalization of mathematics in a proof tool. I rst encountered them when I was working on the formalization of Universal Algebra in Coq (see [6] and [7] The family of term algebras on the type of signatures is one of them. The type of single sorted signatures is Sig : List(N) Given a signature : a 1 ; an ] the type of terms over is de ned by Term : t 11 : Term t 1a1 : Term (f 1 t 11 t 1a1 ) Term ....

....: Sig directly with an inductive de nition, because the number and arity of the constructors depend on the signature : They are not xed for the whole family. The situation is even more complicated when we consider many sorted signatures, which require families of mutual inductive types. In [6] we used Martin L of s W types to solve this instance of the problem. Here we formulate the general problem, we show that W types still provide a good model, but also propose a better solution (which, however, requires an extension of type theory) You can see the details of its application to ....

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Venanzio Capretta. Universal algebra in type theory. In Yves Bertot, Gilles Dowek, Andre Hirschowits, Christine Paulin, and Laurent Thery, editors, Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs '99, volume 1690 of LNCS, pages 131-148. Springer-Verlag, 1999.

....setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1 Introduction Proof development systems such as Agda [19] Coq [10] and Lego [36] rely on powerful type systems and have been successfully used in the formalisation of mathematics [4, 7, 8, 12, 14 16, 40, 42]. Nevertheless, their underlying type theories Martin L of s intensional type theory [39] and the Calculus of Inductive Constructions [46] fail to support extensional concepts such as quotients and subsets, which play a fundamental role in mathematics. While signi cant e orts have been devoted to ....

V. Capretta. Universal algebra in type theory. In Y. Bertot, G. Dowek, A. Hirschowitz, C. Paulin, and L. Thry, editors, Proceedings of TPHOL'99, volume 1690 of Lecture Notes in Computer Science, pages 131-148. SpringerVerlag, 1999.

....a proof and the size of the resulting proof term. This is done by replacing derivations on semantic objects with computations on syntactic ones. The general approach consists in formalizing in the most general way the semantic structures as algebras, and the syntactic ones as term algebras. In [4] we saw a development of Universal Algebra in Type Theory, implemented using the proof assistant Coq [1] We use now that formalization to develop equational logic inside Type Theory. We follow the presentation by Meinke and Tucker in the chapter on Universal Algebra of the Handbook of Logic in ....

....using the proof assistant Coq [1] We use now that formalization to develop equational logic inside Type Theory. We follow the presentation by Meinke and Tucker in the chapter on Universal Algebra of the Handbook of Logic in Computer Science [8] In x2 we summarize the notions and the results of [4]. Since the de nition of term algebra given in that work is not very manageable, in x3 we use a di erent formulation based on a less general but but lighter construction. The semantics of terms is de ned by a term evaluation function, that takes as parameter an assignment of values in an algebra ....

[Article contains additional citation context not shown here]

Venanzio Capretta. Universal algebra in type theory. In Yves Bertot, Gilleds Dowek, Andre Hirschowits, Christine Paulin, and Laurent Thery, editors, Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs '99, volume 1690 of LNCS, pages 131-148. Springer, 1999.

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V. Capretta. Universal algebra in type theory. In Y. Bertot et al., editors, Proc. TPHOLs '99, volume 1690 of LNCS, pages 131--148. Springer-Verlag, 1999.

No context found.

V. Capretta. Universal algebra in type theory. In Y. Bertot et al., editors, Proc. TPHOLs '99, volume 1690 of LNCS, pages 131-148. Springer-Verlag, 1999.

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