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A lambda calculus model of MartinLöf's theory of types with explicit substitution
 In this thesis
, 1997
"... This paper presents a proofirrelevant model of MartinLof's theory of types with explicit substitution; that is, a model in the style of [Smi88], in which types are interpreted as truth values and objects (or proofs) are irrelevant. The fundamental difference here is the need to cope with a fo ..."
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This paper presents a proofirrelevant model of MartinLof's theory of types with explicit substitution; that is, a model in the style of [Smi88], in which types are interpreted as truth values and objects (or proofs) are irrelevant. The fundamental difference here is the need to cope with a formal system which in addition to types has sets and substitutions. This difference leads us to a whole reformulation of the model which consists in defining an interpretation in terms of the untyped lambda calculus. From this interpretation the proofirrelevant model is obtained as a particular instance. Finally, the paper outlines the definition of a realizability model which is also obtained as a particular instance. Keywords: type theory, explicit substitution, models of type theory, proofirrelevant model, realizability model. Contents 1 Introduction 1 2 Type theory 2 2.1 Syntax : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 3 A lambda calculus model 8 3.1 Semantic...
EM + Ext − + ACint is equivalent to ACext
, 2004
"... It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality princip ..."
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It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality principle (Ext−), which is provable in MartinLöf’s extensional type theory. In particular, EM ⇔ ACext holds in extensional type theory. The following is the principle ACint of intensional choice: if A, B are sets and R a relation such that (∀x: A)(∃y: B)R(x, y) is true, then there is a function f: A → B such that (∀x: A)R(x, f(x)) is true. It is provable in MartinLöf’s type theory [8, p. 50]. It follows from ACint that surjective functions have right inverses: If =B is an equivalence relation on B and f: A → B, we say that f is surjective if (∀y: B)(∃x: A)(y =B f(x)) is true. With R(y, x) def = (y =B f(x)), surjectivity
Propositional Functions and Families of Types
 In Workshop on Programming Logic
, 1989
"... Introduction In order to capture some of the programmers errors, several computer languages, like Pascal and ML, are equipped with a type system. Using the CurryHoward interpretation of propositions as types [3, 8], or as we shall say here, propositions as sets, a type system can be made strong en ..."
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Introduction In order to capture some of the programmers errors, several computer languages, like Pascal and ML, are equipped with a type system. Using the CurryHoward interpretation of propositions as types [3, 8], or as we shall say here, propositions as sets, a type system can be made strong enough to be used to specify the task a program is supposed to do. This is one of the basis for MartinLof's suggestion in [11] to use his formulation of type theory for programming; his ideas are exploited in [14] and there are several computer implementations of type theory [4, 16]. Similar ideas are also behind Coquand and Huet's calculus of constructions [2]. The idea of propositions as sets is closely related to the intuitionistic explanations of the logical constants given by Heyting [7]. In MartinLof's type theory, the interpretation of propositions as sets is fundamental since the notions of proposition and set are identical. So a logical constant is definitionally equal to th
On the Degeneracy of ΣTypes in Presence of Computational Classical Logic
"... Abstract. We show that a minimal dependent type theory based on Σtypes and equality is degenerated in presence of computational classical logic. By computational classical logic is meant a classical logic derived from a control operator equipped with reduction rules similar to the ones of Felleisen’ ..."
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Abstract. We show that a minimal dependent type theory based on Σtypes and equality is degenerated in presence of computational classical logic. By computational classical logic is meant a classical logic derived from a control operator equipped with reduction rules similar to the ones of Felleisen’s C or Parigot’s µ operators. As a consequence, formalisms such as MartinLöf’s type theory or the (Setpredicative variant of the) Calculus of Inductive Constructions are inconsistent in presence of computational classical logic. Besides, an analysis of the role of the ηrule for control operators through a settheoretic model of computational classical logic is given.
U.U.D.M. Report 2008:42 Setoids and universes
"... Abstract. Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disj ..."
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Abstract. Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disjoint sums. It is shown that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably nonderivable. 1.
Constructive notions of set Part I Sets in Martin–Löf type theory
"... This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences ..."
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This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences with the notion of set as codified in the classical theory Zermelo–Fraenkel. In the current article we illustrate in some detail the notion of set as expressed in Martin–Löf type theory and present the essential characters of this theory. In a second article we shall explore a distinct notion of set, as arising in the context of intuitionistic versions of Zermelo–Fraenkel set theory. The theory we shall analyse there is Aczel’s CZF (Constructive Zermelo– Fraenkel) and we shall supplement its exposition by a succinct account of Aczel’s interpretation of CZF in type theory. This will enable us to compare the two notions in a more precise sense. 1.