P. Martin-Lof. Philosophical Implications of Type Theory., 1987. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th. Privately circulated notes.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of types indexed by objects of type ff then ff fi is also a type. The application of an object f of this latter type yields an object fa of type fia, if a is an object of type ff. The understanding of propositions as inductively defined by their introduction rules, as explained and justified in [Mar87], allows us to grasp propositions as sets, and thereby, their proofs as elements of those sets. There is, in principle, no formal distinction in the language of the theory between the type of sets and the type of propositions. Further, in the presence of families of types, this interpretation of ....

....form the function type (x : ff)fi, where all occurrences of x in fi become bound. Abstraction then is introduced as an operation of object formation. This is the corresponding rule Gamma; x:ff b : fi Gamma [x]b : x : ff)fi The stipulation for the formation of a context Gamma; x:ff in [Mar87, NPS89, Mar92], for instance, requires that Gamma is a context, ff is a type under the context Gamma and, further, that the variable x has not already been declared in Gamma. This last restriction 5 is proper of systems of proof rules where an assumption, x:ff say, may be introduced such that the type ff ....

P. Martin-Lof. Philosophical Implications of Type Theory., 1987. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th. Privately circulated notes.

....The algorithm has been proven sound with respect to the latter calculus. We include a discussion on that proof in the present work. 1. Introduction The subject of this paper is the specification and implementation of a proof checker for an extension of Martin Lof s theory of logical types [Mar87] with dependent record types and subtyping. The original formulation of Martin Lof s theory of types, from now on referred to as the logical framework, has been presented in [NPS89, CNSvS94, Tas97] The system of types that this calculus embodies are the type Set (the type of inductively defined ....

....of types indexed by objects of type ff then ff fi is also a type. The application of an object f of this latter type yields an object fa of type fia, if a is an object of type ff. The understanding of propositions as inductively defined by their introduction rules, as explained and justified in [Mar87], allows us to grasp propositions as sets, and thereby, their proofs as elements of those sets. There is, in principle, no formal distinction in the language of the theory between the type of sets and the type of propositions. Further, in the presence of families of types, this interpretation of ....

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P. Martin-Lof. Philosophical Implications of Type Theory., 1987. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th. Privately circulated notes.

....the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of Martin Lof s theory of logical types [13] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of Martin Lof s theory of types, from now on referred to as the logical framework, has been presented in [14, 17, 8] The ....

....fi is a family of types over the type ff then ff fi is also a type. The application of an object f of this latter type yields an object fa of type fia, if a is an object of type ff. The understanding of propositions as inductively defined by their introduction rules, explained and justified in [13], allows to grasp propositions as sets, and thereby, their proofs as elements of those sets. There is no formal distinction in the language of the theory between the type of sets and the type of propositions. Further, in the presence of families of types, this interpretation of propositions can be ....

P. Martin-Lof. Philosophical Implications of Type Theory., 1987. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th. Privately circulated notes.

....the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of Martin Lof s theory of logical types [14] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of Martin Lof s theory of types, from now on referred to as the logical framework, has been presented in [15, 7] The system ....

....whole code involved in the formalization but rather concentrate on the fragments that we consider the most interesting ones. Finally, in last section we discuss related work. 2. Representation of systems of algebras in type theory We shall use an extension of Martin Lof s theory of logical types [14] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. A brief description of the main features of the original and the extended theory are given below. For a more comprehensive presentation we refer to ....

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Per Martin-Lof. Philosophical Implications of Type Theory. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th., 1987. Privately circulated notes.

....of the example. We also provide code of the formalization as accepted by a type checker implemented for the extended theory. Introduction. Our starting point, to which we refer hereafter as type theory, is the formulation of Martin Lof s set theory using the theory of types as logical framework [12, 13, 8]. In type theory it is possible to form families of types on a given type, and thus types can be formed by applying such families to individuals. Having families of types allows to introduce dependent function types, that is, types of functions whose output type depends on individuals of the input ....

Martin-Lof P. Philosophical Implications of Type Theory. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th., 1987. Privately circulated notes.

....work proposal is defined by their common interest to work together in at least one of the areas proposed. There will be people working in each of these areas in the Plata region. 1. Introduction We consider the formulation of Martin Lof s set theory using the theory of types as logical framework [ML87, NPS90, CNSvS94] This has been extended in [BT96] with dependent record types and subtyping. It is this extended theory to which we refer hereafter as type theory. Type theory is originally introduced as a formal language for constructive mathematics and can also be seen as a functional ....

Per Martin-Lof. Philosophical Implications of Type Theory. Privately circulated notes., 1987. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di Firenze, Florence, March 15th. - May 15th., 1987.

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