P. Martin-Lof. An Intuitionistic Theory of Types: Predicative Part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theorem [Vel94] However, in [Fri97] the type theoretic fan theorem is only mentioned and the proof is omitted. The importance of the fan theorem justi es this more extended presentation. Type theory here means Martin L of s type theory, of which there exist di erent formulations (for example, [Mar75], Mar84] NPS90] and [Tas97] The exposition here should suit all of them. The proof of the fan theorem presented here has been written down in full detail with the assistance of the proof editor ALF [Mag94] which is an implementation of the formulation of type theory given in [Tas97] The rest ....

P. Martin-Lof. An Intuitionistic Theory of Types: Predicative Part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium

....Much later, Coquand and Huet combined both calculi, resulting in the Calculus of Constructions [13] Making use of impredicativity, data structures could be encoded in this calculus, but these encodings were far too complex to be used by nonspecialists. A different approach was taken by Martin Lof [32,33], whose theory was based on the notion of inductive definition, originating in Godel s system T [25] Coquand and Paulin Mohring later incorporated a similar notion to the Calculus of Constructions under the name of inductive type [14] But despite their legitimate success, inductive types are not ....

P. Martin-Lof. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Proceedings of the 73' Logic Colloquium, volume 80 of Studies in Logic and the Foundations of Mathematics. NorthHolland, 1975.

....conception of logical truth, according to which AC is neither a principle of logic, nor even compatible with reasoning that eschews EM. According to some other conceptions, however, AC is a logical principle and EM is not. Notable examples are the type theories of Tait [7] 8] and Martin Lof [5], as is informally the logic underlying Bishop s constructive analysis [1] as noted in [5] Such systems of logic evidently cannot be modeled in topoi in the standard way. However, Seely [6] has shown how to model a range of such type theories in locally cartesian closed (LCC) categories ....

....compatible with reasoning that eschews EM. According to some other conceptions, however, AC is a logical principle and EM is not. Notable examples are the type theories of Tait [7] 8] and Martin Lof [5] as is informally the logic underlying Bishop s constructive analysis [1] as noted in [5]) Such systems of logic evidently cannot be modeled in topoi in the standard way. However, Seely [6] has shown how to model a range of such type theories in locally cartesian closed (LCC) categories (the source of this idea is Lawvere [4] The type theories considered by Seely, which are closely ....

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Martin-Lof, P.: 1973, "An Intuitionistic Theory of Types: Predicative Part," Logic Colloquium '73, Bristol, ed. H. E. Rose & J. C. Sheperdson, North-Holland, Amsterdam, 73--118.

....is just fun(Month; m:Day) where Day is a family of twelve types. And (x:maxday[x] is a term in it. 11 3 Formalizing Type Theory Nuprl presents a particular formalization of type theory. There are several others [17, 22, 23, 24, 26, 42] At the core are ideas from deBruijn [17] and Martin Lof [34] extended with the notions of inductive types and direct computation which give Nuprl its unique architecture. The formalization comes in two parts. First we lay down the structure of terms and then evaluation. terms evaluation This is the functional programming core of the theory, on top of ....

....purpose. But then it must belong to a type. We might try attaining closure by postulating. Type 2 Type. But this causes many troubles. It allows us to inhabit every type [21] and permits nonterminating functions in the function spaces [28] So following the traditions of predicative type theory [45, 34, 19, 20], we introduce a hierarchy of types. Following [35] we call them universes. They are denoted U 1 ; U 2 ; and we have that U i 2 U i 1 and if A 2 U i then A 2 U j for i j. Logic is introduced into type theory by taking propositions as types. This is well explained elsewhere, e.g. 12, ....

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P. Martin-Lof. An intuitionistic theory of types: predicative part. In Logic Colloquium '73, pages 73--118. North-Holland, Amsterdam, 1973.

....to extend the ML style type system and reconstruction algorithm to support the typing of variable length arrays and other related types that depend on computed values. The types we use are first order dependent products, introduced by Martin Lof as a Cartesian product of a family of types [Mar75] in an article on intuitionistic type theory. The fundamental assumption that makes dependent type reconstruction possible is that we can only express terminating programs. We will simplify our study by specifically investigating first order product types dependent on natural numbers. The ....

.... out to be closely related to excursions by Tait and Martin Lof into the theory T 1 of lambda calculus with infinite sequences of terms and types [Tai65, Mar72b] The way we have formulated sequences of types as primitive recursive formulas follows Martin Lof s early intensional type theory [Mar75] Our work belongs with the family of research on languages without general recursion. Under this paradigm, all programs terminate and programs are constructed with a variety of powerful systems of well founded recursion. Such languages have the property of strong normalization. There is a ....

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Per Martin-Lof. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium '73, pages 73--118. North Holland, 1975.

....the type in inhabited by the single element denoted by the term Axiom. x2T is a membership term. It is an encoding for the equality term x=x2T. It denotes nothing if T is not a type or if x is not in T and is inhabited by the single term Axiom otherwise. Like the related type theory of Marin Lof [17] or the type theory of Whitehead and Russell s Principia Mathematica, the Nuprl s type theory is a predicative type theory supporting an unbounded cumulative hierarchy of type universes. Every universe is itself a type and every type is an element of some universe. Ufig denotes the type universe ....

Per. Martin--Lof. An intuitionistic theory of types: predicative part. In Logic Colloquium '73., pages 73--118. Amsterdam:North-Holland, 1973.

....to provide good methodology and tools which can be used to apply the useful principles of software development like separation of concerns and divide and conquer and to guarantee the correctness of programs with respect to their specifications. Type theories (e.g. Martin Lof s type theory [ML75, ML84] the Automath type theory [dB80] Nuprl s type theory [C 86] and Coquand Huet s calculus of constructions [CH88] were mainly developed for foundation and formalization of mathematics. Since the work by Martin Lof, it has become known that type theories can also provide basic 1 ....

.... with in this paper is the Extended Calculus of Constructions (ECC) Luo89, Luo90a] As a formal system, ECC extends the calculus of constructions [CH88] with predicative type universes and Sigma types (strong sum) it may also be seen as an extension of Martin Lof s type theory with universes [ML75] by an impredicative universe (higher order logic) However, different from Martin Lof s type theory and the calculus of constructions, the incorporation of both an impredicative universe and predicative universes enhances a conceptual distinction between the notion of logical formulae ....

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P. Martin-Lof. An intuitionistic theory of types: predicative part. In H.Rose and J.C.Shepherdson, editors, Logic Colloquium'73, 1975.

....such consideration is the representation of mathematical structures such as algebras, automata, and ordered sets. It is by now widely recognized [36, 10] that the appropriate type theoretic representation of mathematical structures is as elements of strong sum types 1 introduced by Martin Lof [35, 36, 37] and Howard [24] Strong sums have also been used to model modularity constructs in programming languages [33, 41, 2, 4] Unfortunately, strong sums are, in a sense, incompatible with impredicativity [12, 23, 41] As a result, it is necessary to extend the calculus with a level of types, and to ....

....operations of the calculus: the formation of products and strong sums indexed by a type of that universe level. Cumulative hierarchies of this kind arise in many formal systems for mathematics; they arise in various guises in Principia Mathematica [44, 47] and in many contemporary type theories [35, 36, 37, 8, 9, 10]. Universe hierarchies are tedious to use in practice. Many workers have attempted to avoid the complications of such a hierarchy by assuming that there is a type of all types [34, 2, 38, 4] This assumption destroys the normalization property of the calculus [35, 38, 25] As a result, every type ....

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Per Martin-Lof. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium '73, volume 80 of Studies in Logic and the Foundations of Mathematics, pages 73--118. North-Holland, 1975.

.... abstracted over the data structures they operate on has roots in some of our earlier work [10] but has been most succesfully promoted by Johan Jeuring [2, 3] Martin Lof pioneered the use of dependent type systems which have now become standard in many types systems for programming logics [5] and we incorporate them into the terminating parts of our language. Nelson investigates the issue of type inference for dependent types [7, 8] 9 Conclusion Programming languages have always embodied a conceptual distinction between their compile and link time aspects and their run time ....

Per Martin-Lof. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium '73, pages 73--118. North Holland, 1975.

....term representing it. Our approach is strongly inspired by two early papers by Martin Lof, where he emphasized the importance of intuitionistic abstractions on the meta level and the notion of definitional equality [19] and proved normalization for his type theory by using a model construction [20]. We pursue these ideas further. In particular we wish to emphasize the following aspects of our normalization proofs: ffl They are expressed as properties of normalization algorithms, rather than as the usual 89 propositions referring to binary reduction relations. ffl The normalization ....

....model from normal terms and meanings, and use it for normalization of two different versions of the calculus. One of these has explicit substitutions and is similar to the oe calculus of Abadi, Cardelli, Curien, and L evy [1] The other is a nameless variant of the calculus used by Martin Lof [20]. In an accompanying paper, a similar technique is used by Catarina Coquand [6] for normalization in simply typed calculus with full reduction. In this case a Kripke model is used for the non standard semantics. 2 Type theory and ALF 2.1 Martin Lof s type theory The formal meta language is ....

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P. Martin-Lof. An intuitionistic theory of types: Predicative part. In Logic Colloquium `73, pages 73--118. North-Holland, 1975.

....and therefore also the theory of inductiverecursive definitions, have proof theoretical strength of at least Rathjen s theory KPM. 1 Introduction Induction recursion is a powerful definition method in intuitionistic type theory in the sense of Scott ( Constructive Validity ) 30] and Martin Lof [16, 17, 18]. The first occurrence of formal induction recursion is Martin Lof s definition of a universe a la Tarski [18] which consists of a set U 0 of codes for small sets together with a decoding function T 0 which maps a code to the small set it denotes. U 0 is inductively generated at the same time as ....

....elements of U 0 , and the introduction rules for U 0 refer to T 0 . It is called universe a la Tarski because of the similarity with Tarski s truth definition: U 0 is a generalized syntax of formulas and T 0 maps each formula to its meaning . In earlier formulations of Martin Lof type theory [16, 17] Department of Mathematics and Computing Science, Chalmers University of Technology. Email: peterd cs.chalmers.se y Department of Mathematics, Uppsala University. Email: setzer math.uu.se 1 universes are formulated a la Russell , where there is no syntactic distinction between an element ....

P. Martin-Lof. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium `73, pages 73--118. North-Holland, 1975.

....84] and abstract types [Mitchell 85] In all these languages, phases can be distinguished syntactically: there are separate syntactic sorts of type expressions and value expressions. Phase distinction are however lost when moving to languages like Pebble [Burstall 84a] based on dependent types [Martin Lf 73] A dependent type is a type which may depend upon the value of some expression. This dependency directly causes phase mixing, unless one is careful to distinguish between compile time values (e.g. given by constant expressions in Pascal) and run time values. The loss of phase distinctions ....

P.Martin-Lf, An intuitionistic theory of types: predicative part, in Logic Colloquium III, F.Rose, J.Sheperdson ed. pp 73-118, North-Holland, 1973.

....carried out in many type systems, for example [Cardelli 84] applied it to first order l calculus, Cardelli Wegner 85] to second order l calculus, and [Wand 87] to polymorphic l calculus with implicit typing. Here we apply it to a more general type system derived from intuitionistic type theory [Martin Lf 73] Cardelli 86] A novel notion of power types, analogous to powersets, is introduced to model subtyping in such a system. Combined with type quantifiers, power types can express bounded type quantification, leading to parametric inheritance and partiallyabstract data types. One of our types ....

P.Martin-Lf, An intuitionistic theory of types: predicative part, in Logic Colloquium III, F.Rose, J.Sheperdson Eds., pp 73118, North-Holland, 1973.

....that if we appeal to the Church Rosser theorem, we can prove the following result: Corollary 4.9. Every element of NF(A;B) is nf(g) for some g. 5. Historical Background and Related Work Our method is an example of normalization by intuitionistic model construction, a method going back to (Martin Lof 1975a; Martin Lof 1975b) The general idea is to prove normalization by first interpreting a term in a suitable model and then map this interpretation back to the normal form of the term. By working in an intuitionistic framework one ensures that the normalization function thus obtained is an ....

....to the Church Rosser theorem, we can prove the following result: Corollary 4.9. Every element of NF(A;B) is nf(g) for some g. 5. Historical Background and Related Work Our method is an example of normalization by intuitionistic model construction, a method going back to (Martin Lof 1975a; Martin Lof 1975b) The general idea is to prove normalization by first interpreting a term in a suitable model and then map this interpretation back to the normal form of the term. By working in an intuitionistic framework one ensures that the normalization function thus obtained is an algorithm. Martin Lof s ....

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P. Martin-Lof, An Intuitionistic Theory of Types: Predicative Part, in Logic Colloquium `73, H. E. Rose and J. C. Shepherdson, eds., North-Holland, 1975, pp. 73-118.

....HOAS recognizes these common patterns and exploits the scoping mechanism of the meta language (which already has all these tools) to implement them. Higher order abstract syntax goes back to Church s [3] original definition of the calculus and Martin Lof s work on intuitionistic type theory [15]. Huet and Lang [12] noticed that higherorderness was a valuable technique for improving the abstraction level of syntax. Pfenning and Elliott [23] introduced the name higher order abstract syntax and generalized their work. Highorder abstract syntax is the central representation technique in ....

P. Martin-Lof. Intuitionistic theory of types: Predicative part. In sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 73--118. North-Holland, 1975.

....Type Theory of LF The LF type theory is a predicative, dependently typed calculus, closely related to the Pi fragment of AUT PI [55] a language belonging to the AUTOMATH family. LF can also be fruitfully compared to several other systems 4 such as AUT QE [55] Martin Lof s early type theories [28], Huet and Coquand s Calculus of Constructions [10] and Meyer and Reinhold s [33] Some of these comparisons are only superficial; others are more substantial if carried out under an appropriate notational transliteration. The variant of the LF type theory that we shall discuss in greatest ....

....of our choice of definitional equality is the diamond property for parallel reduction: Proposition 2.1 (Diamond Property) If U U 0 and U U 00 , then there exists V such that U 0 V and U 00 V . 2 This result can be readily established by adapting the method of Tait and Martin Lof [28, 42, 53] to our system. It follows that satisfies the ChurchRosser property: Corollary 2.2 (Church Rosser Property) If U U 0 and U U 00 , then there exists V such that U 0 V and U 00 V . 2 It is noteworthy that the Church Rosser property holds for our notion of reduction ....

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Martin-L of, P. An intuitionistic theory of types: Predicative part. In Logic Colloquium '73 (1975), H. E. Rose and J. C. Shepherdson, Eds., vol. 80 of Studies in Logic and the Foundations of Mathematics, NorthHolland, pp. 73--118.

....Now dual to products (as Lawvere knows) are disjoint sums which must be used for the interpretation of the existential quantifier (cf. Kreisel Goodman) These sums were not employed by de Bruijn, but it would be easy to add them to his system. Later Martin Lof introduced the eliminators split [24] and funsplit [26] On the logical side we first have Heyting s and Kolmogorov s explanations of the logical constants in terms of proofs and problems respectively. For example, according to Heyting, a proof of 8x:P [x] is a function that given an arbitrary element a returns a proof of P [a] and ....

P. Martin-Lof. An intuitionistic theory of types: Predicative part. In Logic Colloquium `73, pages 73--118. North-Holland, 1975.

....The following is a list of published papers on intuitionistic type theory and its precursor the intuitionistic theory of iterated inductive definitions. ffl Hauptsatz for the Intuitionistic Theory of Iterated Inductive Definitions [17] ffl An Intuitionistic Theory of Types: Predicative Part [19]. ffl About Models for Intuitionistic Type Theories and The Notion of Definitional Equality [18] ffl Constructive Mathematics and Computer Programming [20] 2 Some Historical Roots The importance of trying to put conceptual ideas in their historical context cannot be overestimated. Let us ....

....Now dual to products (as Lawvere knows) are disjoint sums which must be used for the interpretation of the existential quantifier (cf. Kreisel Goodman) These sums were not employed by de Bruijn, but it would be easy to add them to his system. Later Martin Lof introduced the eliminators split [19] and funsplit [22] On the logical side we first have Heyting s and Kolmogov s explanations of the logical constants in terms of proofs and problems respectively. For example, according to Heyting, a proof of 8x:P [x] is a function that given an arbitrary element a returns a proof of P [a] and a ....

[Article contains additional citation context not shown here]

P. Martin-Lof. An intuitionistic theory of types: Predicative part. In Logic Colloquium `73, pages 73--118. North-Holland, 1975.

....the dependencies by allowing the specification for the output to depend on more than one input value (in some pre defined manner) We have implemented an object calculus using the very dependent function types in the Nuprl system. Although we use the Nuprl type theory (based on Martin Lof s [19, 20] type theory) and the Nuprl terminology in this paper, our results should carry over to other proof systems that use dependent types, including Coq [11] lego [18] Alf [9] and pvs [10] We present the following results: ffl a dependent record calculus, forming the foundation for a theory of ....

Per Martin-Lof. An intuitionistic theory of types: Predicative part. In Logic Colloquium '73, pages 73--118. North--Holland, 1975.

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P. Martin-Lof. An intuitionistic theory of types: predicative part. In H.E. Rose and J. Sheperdson, editors, Logic Colloquium '73, pages 73 -- 118, North-Holland, Amsterdam, 1975.

.... [40] in which the truth predicate is not de nable) he used a hierarchy of languages which could elegantly be explained via the notion of orders as is shown in [19] Similarly, when Martin L of s impredicative type theory was shown to su er from the paradox, he moved to the predicative version in [26] and has since, built layers of universes that again could be elegantly explained by orders (see for example, page 84 of [26] Also, 31] provides a treatment of trans nite orders as universes, 25] discusses predicative universes in the Calculus of Constructions [7] 8] introduces the ....

.... the notion of orders as is shown in [19] Similarly, when Martin L of s impredicative type theory was shown to su er from the paradox, he moved to the predicative version in [26] and has since, built layers of universes that again could be elegantly explained by orders (see for example, page 84 of [26]) Also, 31] provides a treatment of trans nite orders as universes, 25] discusses predicative universes in the Calculus of Constructions [7] 8] introduces the generalised Calculus of Constructions CC which includes a cumulative hierarchy of universes, 13] studies type checking and ....

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P. Martin-Lof. An intuitionistic theory of types: predicative part. In H.E. Rose and J. Shepherdson, editors, logic Colloquium '73. North Holland, 1975.

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Per Martin-Lof. An intuitionistic theory of types: Predicative part. In Logic Colloquium '73, pages 73--118. North-Holland, Amsterdam, 1973.

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P. Martin-Lo"f, An intuitionistic theory of types: predicative part, Logic Colloquium 73, H. Rose and J. Shepherdson, Eds., North-Holland, 1974, pp. 73-118.

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P. Martin-Lof. An intuitionistic theory of types: predicative part. In H.E. Rose and J. Sheperdson, editors, Logic Colloquium '73, pages 73 -- 118, Amsterdam, 1975. NorthHolland.

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P. Martin-Lof. An Intuitionistic Theory of Types: Predicative Part. Logic Colloquium `73, 1975,p. 73-118, eds. H. E. Rose and J. C. Shepherdson, North-Holland.

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