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Packaging mathematical structures
 THEOREM PROVING IN HIGHER ORDER LOGICS 5674
, 2009
"... This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated ..."
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Cited by 19 (5 self)
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This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.
The Matita Interactive Theorem Prover
"... Abstract. Matita is an interactive theorem prover being developed by the Helm team at the University of Bologna. Its stable version 0.5.x may be downloaded at ..."
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Cited by 8 (6 self)
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Abstract. Matita is an interactive theorem prover being developed by the Helm team at the University of Bologna. Its stable version 0.5.x may be downloaded at
A constructive and formal proof of Lebesgue's Dominated Convergence Theorem in the interactive theorem prover Matita
, 2008
"... We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [SZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in ..."
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Cited by 7 (4 self)
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We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [SZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort.
Manifest fields and module mechanisms in intensional type theory
 In TYPES 08
, 2009
"... Abstract. Manifest fields in a type of modules are shown to be expressible in intensional type theory without strong extensional equality rules. These intensional manifest fields are made available with the help of coercive subtyping. It is shown that, for both Σtypes and dependent record types, th ..."
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Cited by 5 (3 self)
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Abstract. Manifest fields in a type of modules are shown to be expressible in intensional type theory without strong extensional equality rules. These intensional manifest fields are made available with the help of coercive subtyping. It is shown that, for both Σtypes and dependent record types, the withclause for expressing manifest fields can be introduced by means of the intensional manifest fields. This provides not only a higherorder module mechanism with MLstyle sharing, but a powerful modelling mechanism in formalisation and verification of OOstyle program modules. 1
Hints in unification
"... Abstract. Several mechanisms such as Canonical Structures [14], Type Classes [16,13], or Pullbacks [10] have been recently introduced with the aim to improve the power and flexibility of the type inference algorithm for interactive theorem provers. We claim that all these mechanisms are particular i ..."
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Cited by 2 (1 self)
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Abstract. Several mechanisms such as Canonical Structures [14], Type Classes [16,13], or Pullbacks [10] have been recently introduced with the aim to improve the power and flexibility of the type inference algorithm for interactive theorem provers. We claim that all these mechanisms are particular instances of a simpler and more general technique, just consisting in providing suitable hints to the unification procedure underlying type inference. This allows a simple, modular and not intrusive implementation of all the above mentioned techniques, opening at the same time innovative and unexpected perspectives on its possible applications. 1
Intensional Manifest Fields in Module Types
, 2010
"... A manifest field in a type of modules is a field whose expected data is not only of a certain type but the same as a specific object of that type. All of the previous approaches to manifest fields in type theory are based on some extensional notions of computational equality. In this paper, we show ..."
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A manifest field in a type of modules is a field whose expected data is not only of a certain type but the same as a specific object of that type. All of the previous approaches to manifest fields in type theory are based on some extensional notions of computational equality. In this paper, we show that this is unnecessary: manifest fields are expressible in intensional type theories without extensional equality rules. These intensional manifest fields are made available with the help of coercive subtyping. It is shown that, for both Σtypes and dependent record types, the withclause for expressing manifest fields can be introduced by means of the intensional manifest fields. This provides an internal mechanism in intensional type theories to express definitional entries in module types, which has useful applications including, for example, the representation of higherorder modules with MLstyle sharing.
devant le jury composé de:
, 2006
"... dans l’école doctorale de Mathématiques, Sciences et Technologies de ..."
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<inria00368403v2>
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Packaging mathematical structures