A minimalist two-level foundation for constructive mathematics

Maietti, Maria Emilia

doi:10.1016/j.apal.2009.01.006

Cited by 59 publications

(166 citation statements)

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“…Other examples come from theories used in the formalization of constructive mathematics: the category of total setoidsà la Bishop and functional relations based on the Minimalist Type Theory in [Mai09], which coincides with the exact completion Ex G mtt where the doctrine G mtt is defined as in [MR13b], or the category of total setoidsà la Bishop and functional relations based on the Calculus of Constructions [Coq90], which coincides with the exact completion Ex G CoC where the doctrine G CoC is constructed from the Calculus of Constructions as G mtt in [MR13b], and it forms a topos as mentioned in [BCP03].…”

Section: It Is Immediate To Check Thatmentioning

confidence: 99%

Triposes, exact completions, and Hilbert's ε-operator

Maietti¹,

Pasquali²,

Rosolini³

2017

Tbilisi Math. J.

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Triposes were introduced as presentations of toposes by J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. They introduced a construction that, from a tripos P : C op G G Pos, produces an elementary topos TP in such a way that the fibration of the subobjects of the topos TP is freely obtained from P . One can also construct the "smallest" elementary doctrine made of subobjects which fully extends P , more precisely the free full comprehensive doctrine with comprehensive diagonals P cx : PrdP op G G Pos on P . The base category has finite limits and embeds into the topos TP via a functor K: PrdP G G TP determined by the universal property of P cx and which preserves finite limits. Hence it extends to an exact functor K ex : (PrdP ) ex/lex G G TP from the exact completion of PrdP .We characterize the triposes P for which the functor K ex is an equivalence as those P equipped with a so-called ε-operator. We also show that the tripos-to-topos construction need not preserve ε-operators by producing counterexamples from localic triposes constructed from wellordered sets. A characterization of the tripos-to-topos construction as a completion to an exact category is instrumental for the results in the paper and we derived it as a consequence of a more general characterization of an exact completion related to Lawvere's hyperdoctrines.

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Section: It Is Immediate To Check Thatmentioning

confidence: 99%

Triposes, exact completions, and Hilbert's ε-operator

Maietti¹,

Pasquali²,

Rosolini³

2017

Tbilisi Math. J.

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“…When developing our theorems we assume to work in the extensional set theory of the two-level minimalist foundation in [10]. This was designed according to the principles given in [11].…”

Section: Some Remarks On Foundationsmentioning

confidence: 99%

“…Furthermore, by working in the minimalist foundation introduced in [10], we prove that the powercollection of all subsets of a set is the free overlap algebra join-generated from the set. Then, we observe that we can present atomic set-based overlap algebras simply as suitable formal topologies, thus providing a predicative characterization of discrete locales within the language of formal topology, instead of using the reacher language of overlap algebras.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, Sambin worked in the minimalist foundation introduced in [10,11] as a common core among the most relevant foundation for constructive mathematics. Since this foundation is predicative, in there one speaks of powercollections and not of powersets, since the power of a set, even that of a singleton, is never a set predicatively.…”

Section: Introductionmentioning

confidence: 99%

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Constructive version of Boolean algebra

Ciraulo

Maietti

Toto

2012

Logic Journal of IGPL

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The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we first show some properties concerning overlap algebras: we prove that the notion of overlap morphism corresponds classically to that of map preserving arbitrary joins; we provide a description of atomic set-based overlap algebras in the language of formal topology, thus giving a predicative characterization of discrete locales; we show that the power-collection of a set is the free overlap algebra join-generated from the set.Then, we generalize the concept of overlap algebra and overlap morphism in various ways to provide constructive versions of the category of Boolean algebras with maps preserving arbitrary existing joins.

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“…In [6], a two-level theory to formalize constructive mathematics is presented, developing ideas already outlined in [7]. One level is given by an intensional type theory, called Minimal Type Theory.…”

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confidence: 99%

Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita

Asperti

Maietti²,

Coen

et al. 2011

Lecture Notes in Computer Science

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The project entitled "Formalization of Formal Topology by means of the interactive theorem prover Matita" is an official bilateral project between the Universities of Padova and Bologna, funded by the former, active ¡¡¡¡¡¡¡ .mine from march 2008 until august 2010. The project brought together and exploited the synergic collaboration of two communities of researchers, both centered around constructive type theory: ======= from March 2008 until August 2010. The project aimed to bring together and exploit the synergic collaboration of two communities of researchers, both centered around constructive type theory: ¿¿¿¿¿¿¿ .r3432 on one side the Logic Group at the University of Padova, focused on developing formal, pointfree topology within a constructive and predicative framework; on the other side, the Helm group at the University of Bologna, developing the Matita Interactive Theorem Prover [2], a young proof assistant based on the Calculus of Inductive Constructions as its logical foundation. The idea of the project was to formalize and check the new approach to formal topology being developed in Padova by means of Matita, with the aim on one side to assess the truly foundational nature of the theoretical framework (i.e. its reduction to notions so elementary to be easily understood by an automatic device), and on the other to drive the development of Matita, testing the tool on a non trivial set of mathematical results, and addressing from an original theoretical perspective some key problems of constructive interactive proving (general recursion, extensionality, quotients, . . .).The project is a rare example of a significant collaboration between mathematicians and computer scientists in handling mathematical knowledge.It is worth to emphasize that the interest in the formalization from the mathematical perspective is not in the automatic verification of the results (on which mathematicians are already largely confident with) but in the phenomenological goal to investigate the most natural way to organize a new foundational framework in a coherent set of interconnected components, their mutual relations and dependencies, our interaction with these representations, and their influence on the concrete mathematical experience (see [1]). Formalization is neither a goal nor a technique, but first and foremost a methodology.The formalization work was mainly focused on Overlap Algebras [4,3], new algebraic structures designed to ease reasoning about subsets within intuitionistic logic. The main result checked in Matita [8] is the embedding of a suitable

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A minimalist two-level foundation for constructive mathematics

Cited by 59 publications

References 41 publications

Triposes, exact completions, and Hilbert's ε-operator

Triposes, exact completions, and Hilbert's ε-operator

Constructive version of Boolean algebra

Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita

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