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A Theory of Typed Coercions and its Applications
"... A number of important program rewriting scenarios can be recast as typedirected coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed ..."
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A number of important program rewriting scenarios can be recast as typedirected coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed code using gradual typing, and inlining code to enforce security policies such as access control and provenance tracking. In this paper we give a general theory of typedirected coercion insertion. We specifically explore the inherent tradeoff between expressiveness and ambiguity—the more powerful the strategy for generating coercions, the greater the possibility of several, semantically distinct rewritings for a given program. We consider increasingly powerful coercion generation strategies, work out example applications supported by the increased power (including those mentioned above), and identify the inherent ambiguity problems of each setting, along with various techniques to tame the ambiguities.
Mathematical Vernacular and Conceptual Wellformedness in Mathematical Language
 Proceedings of the 2nd Inter. Conf. on Logical Aspects of Computational Linguistics, LNCS/LNAI 1582
, 1998
"... . This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive developmen ..."
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. This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive development of mathematics with the support of the current theorem proving technology. The idea of semantic wellformedness in mathematical language is motivated with examples. A formal system based on a notion of conceptual category is then presented, showing how type checking supports our notion of wellformedness. The power of this system is then extended by incorporating a notion of subcategory, using ideas from a more general theory of coercive subtyping, which provides the mechanisms for modelling conventional abbreviations in mathematics. Finally, we outline how this formal work can be used in an implementation of MV. 1 Introduction By mathematical vernacular (MV), we mean a mathematical and n...
The algebraic hierarchy of the FTA Project
 Journal of Symbolic Computation, Special Issue on the Integration of Automated Reasoning and Computer Algebra Systems
, 2002
"... Abstract. We describe a framework for algebraic expressions for the proof assistant Coq. This framework has been developed as part of the FTA project in Nijmegen, in which a complete proof of the fundamental theorem of algebra has been formalized in Coq. The algebraic framework that is described her ..."
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Abstract. We describe a framework for algebraic expressions for the proof assistant Coq. This framework has been developed as part of the FTA project in Nijmegen, in which a complete proof of the fundamental theorem of algebra has been formalized in Coq. The algebraic framework that is described here is both abstract and structured. We apply a combination of record types, coercive subtyping and implicit arguments. The algebraic framework contains a full development of the real and complex numbers and of the rings of polynomials over these fields. The framework is constructive. It does not use anything apart from the Coq logic. The framework has been successfully used to formalize nontrivial mathematics as part of the FTA project.
PAL+: A LambdaFree Logical Framework
, 2000
"... A lambdafree logical framework takes parameterisation and definitions as the basic notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL + , is a logical framework for specification and implementation of type theor ..."
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A lambdafree logical framework takes parameterisation and definitions as the basic notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL + , is a logical framework for specification and implementation of type theories, such as MartinLof's type theory or UTT. As in MartinLof's logical framework [NPS90], computational rules can be introduced and are used to give meanings to the declared constants. However, PAL + only allows one to talk about the concepts that are intuitively in the object type theories: types and their objects, and families of types and families of objects of types. In particular, in PAL + , one cannot directly represent families of families of entities, which could be done in other logical frameworks by means of lambda abstraction. PAL + is in the spirit of de Bruijn's PAL for Automath [dB80]. Compared with PAL, PAL + allows one to represent parametric concepts such as famil...
Common nouns as types
 Logical Aspects of Computational Linguistics (LACL’2012). LNCS 7351
, 2012
"... Abstract. When modern type theories are employed for formal semantics, common nouns (CNs) are interpreted as types, not as predicates. Although this brings about some technical advantages, it is worthwhile to ask: what is special about CNs that merits them to be interpreted as types? We discuss the ..."
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Abstract. When modern type theories are employed for formal semantics, common nouns (CNs) are interpreted as types, not as predicates. Although this brings about some technical advantages, it is worthwhile to ask: what is special about CNs that merits them to be interpreted as types? We discuss the observation made by Geach that, unlike other lexical categories, CNs have criteria of identity, a component of meaning that makes it legitimate to compare, count and quantify. This is closely related to the notion of set (type) in constructive mathematics, where a set (type) is not given solely by specifying its objects, but together with an equality between its objects, and explains and justifies to some extent why types are used to interpret CNs in modern type theories. It is shown that, in order to faithfully interpret modified CNs as Σtypes so that the associated criteria of identity can be captured correctly, it is important to assume proof irrelevance in type theory. We shall also briefly discuss a proposal to interpret mass noun phrases as types in a uniform approach to the semantics of CNs. 1
Working with Mathematical Structures in Type Theory
"... Abstract. We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type familie ..."
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Abstract. We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unification and type reconstruction. We show how to exploit the previous quite common features to reduce the “syntactic ” gap between pen&paper and formalised algebra. However, to reach our goal we need to propose unification and type reconstruction heuristics that are slightly different from the ones usually implemented. We have implemented them in Matita. 1
Typetheoretical semantics with coercive subtyping
 Semantics and Linguistic Theory 20 (SALT20
, 2010
"... Abstract In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as functional subsets of entities as in Montague grammar. This brings about important advantages in linguistic interpretations but also leads to a limitation of expressive power because ..."
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Abstract In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as functional subsets of entities as in Montague grammar. This brings about important advantages in linguistic interpretations but also leads to a limitation of expressive power because there are fewer operations on types as compared with those on functional subsets. The theory of coercive subtyping adequately extends the modern type theories with a notion of subtyping and, as shown in this paper, plays a very useful role in making type theories more expressive for formal semantics. In particular, it gives a satisfactory treatment of the typetheoretic interpretation of modified common nouns and allows straightforward interpretations of interesting linguistic phenomena such as copredication, whose interpretations have been found difficult in a Montagovian setting. We shall also study some typetheoretic constructs that provide useful representational tools for formal lexical semantics, including how the socalled dottypes for representing logical polysemy may be expressed in a type theory with coercive subtyping.
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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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.
An Account of Natural Language Coordination in Type Theory with Coercive Subtyping
"... We discuss the semantics of NL coordination in modern type theories (MTTs) with coercive subtyping. The issue of conjoinable types is handled by means of a type universe of linguistic types. We discuss quantifier coordination, arguing that they should be allowed in principle and that the semantic i ..."
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We discuss the semantics of NL coordination in modern type theories (MTTs) with coercive subtyping. The issue of conjoinable types is handled by means of a type universe of linguistic types. We discuss quantifier coordination, arguing that they should be allowed in principle and that the semantic infelicity of some cases of quantifier coordination is due to the incompatible semantics of the relevant quantifiers. NonBoolean collective readings of conjunction are also discussed and, in particular, treated as involving the vectors of type Vec(A,n), an inductive family of types in an MTT. Lastly, the interaction between coordination and copredication is briefly discussed, showing that the proposed account of coordination and that of copredication by means of dottypes combine consistently as expected.
Dependent Coercions
, 1999
"... A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. A dependent coercion introduces a subtyping relation between a ty ..."
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A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. A dependent coercion introduces a subtyping relation between a type and a family of types in that an object of the type is mapped into one of the types in the family. We present the formal framework, discuss its metatheory, and consider applications such as its use in functional programming with dependent types. 1 Introduction Coercive subtyping, as studied in [Luo97, Luo99, JLS98], represents a new general approach to subtyping and inheritance in type theory. In particular, it provides a framework in which subtyping, inheritance, and abbreviation can be understood in dependent type theories where types are understood as consisting of canonical objects. In this paper, we extend the framework of coercive subtyping to introduce a notion of dependent coer...