Bengt Nordstr#m, Kent Petersson, and Jan M. Smith. Programming in Martin-L#f's Type Theory. An Introduction. Oxford University Press, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from programs satisfying specications, is to dene the pure program directly and then prove properties of this program. In type theory, programs and proofs can be identied, so the problem of proof checking coincides with the problem of type checking. In Martin L#f s monomorphic 1 type theory [NPS90], the typing relation is a decidable property. Clearly, this is a desirable property for the meta logic of a proof development system, since it is then possible to construct a decision procedure for proof 1 or explicitly polymorphic 1.2. INTERACTIVE PROOF ASSISTANTS FOR TYPE THEORY 3 checking. ....

....: x i Gamma1 . In the version of type theory with explicit substitution, here referred to as the substitution calculus, there are several more judgement forms, since contexts and substitutions are completely formalised as well. Here we will use the notation of type theory as it is presented in [NPS90], since it is this notation which is used in ALF. 2.1. TYPE AND OBJECT FORMATION 11 In later chapters we will convert to the notation of the substitution calculus, since there it is more convenient for our purposes. 2.1.1 Type formation There are two ways of forming ground types, and function ....

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Bengt Nordstr#m, Kent Petersson, and Jan M. Smith. Programming in Martin-L#f's Type Theory. An Introduction. Oxford University Press, 1990.

....logical properties. This is, in fact, easy since Cayenne types can, through the CurryHoward isomorphism, encode predicate calculus as types, see gure 1. Terms of the dioeerent types correspond to the proof of the corresponding properties. This is all well known from constructive type theory [NPS90] and well before that [How80] Predicate calculus Cayenne type Absurd (or any empty type) any non empty type x y Either x y x y Pair x y 8x 2 A:P (x) x: A) P (x) 9x 2 A:P (x) x: A; y: P (x) data Absurd = data Pair x y = pair x y data Either x y = Left x Right y Fig. 1. ....

Bengt Nordstr#m, Kent Petersson, and Jan M. Smith. Programming in Martin-L#f's Type Theory. An Introduction. Oxford University Press, 1990.

....of the Type System for Exp : 8 ii 1 Introduction This paper deals with the application of constructive type theory to the theory of programming languages. By constructive type theory we understand rst and foremost Martin L#f s theory of logical types (see [NPS90]) which is conceived as a formal language in which to carry out constructive mathematics. However, constructive type theory can also be viewed as a programming language. In type theory, we represent theorems as types and proofs of the theorems as objects of the corresponding types. In general ....

....mainly for those who already have some knowledge of type theory, and in particular of Martin L#f s type theory, we present in this section a brief introduction to this theory to make the following sections more readable. For a more complete introduction to the subject, the reader can refer to [CNSvS94, NPS90]. Martin L#f s type theory has a basic type and two type formers. The basic type is the type of sets, which we write Set. For each set S, the elements of S form a type. Given a type a and a family b of types over a, we can construct the function type from a to b. We write a a for ia is an ....

B. Nordstr#m, K. Petersson, and J. M. Smith. Programming in Martin-L#f's Type Theory. An Introduction. Oxford University Press, 1990.

....we use the proof editor ALF and its pattern matching facility. 1 Introduction This paper deals with the application of constructive type theory to the theory of programming languages. By constructive type theory we understand rst and foremost Martin L#f s theory of logical types (see [NPS90]) which is conceived as a formal language in which to carry out constructive mathematics. However, constructive type theory can also be viewed as a programming language. In type theory, we represent theorems as types and proofs of the theorems as objects of the corresponding types. When a theorem ....

....Martin L#f s type theory. 3.1 Brief Introduction to Martin L#f s Type Theory Below, we explain the basic concepts of Martin L#f s type theory to make the following sections more readable. We write a a for ia is an object of type aj. For a more complete introduction to the subject, refer to [NPS90]. Sets : The only basic type is the type of sets. Sets are inductively de ned. In other words, a set is determined by the rules that construct its elements. We write Set to refer to the type of sets. Elements of Sets : For each set S, the elements of S form a type called El(S) However, for ....

B. Nordstr#m, K. Petersson, and J. M. Smith. Programming in MartinL #f's Type Theory. An Introduction. Oxford University Press, 1990.

....logic in it. Section 5 introduces a number of dioeerent sets and the nal section give a short description of ALF, an implementation of the type theory of this chapter. Although self contained, this chapter can be seen as complement to our book, Programming in Type Theory. An Introduction [33], in that Martin L#f s Type Theory 3 we here give a presentation of Martin L#f s monomorphic type theory in which there are two basic levels, that of types and that of sets. The book is mainly concerned with a polymorphic formulation where instead of a level of types there is a theory of ....

Bengt Nordstr#m, Kent Petersson, and Jan M. Smith. Programming in Martin-L#f's Type Theory. An Introduction. Oxford University Press, 1990.

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