this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through is a slight strengthening of the Stripping property (also called Generation). This property says, for example, that if \Gamma ` v:T:M : U has a derivation D, then one can find a subderivation of

. We give an interpretation of quotient types within in a dependent type theory with an impredicative universe of propositions (Calculus of Constructions). In the model, type dependency arises only at the propositional level, therefore universes and large eliminations cannot be interpreted. In exchange, the model is much simpler and more intuitive than the one proposed by the author in [10]. Moreover, we interpret a choice operator for quotient types that, under certain restrictions, allows one to recover a representative from an equivalence class. Since the model is constructed syntactically, the interpretation function from the syntax with quotient types to the model gives rise to a procedure which eliminates quotient types by replacing propositional equality by equality relations defined by induction on the type structure ("book equalities"). 1 Introduction Intensional type theories like the Calculus of Constructions have been proposed as a framework in which to formalise mathemati...

The smallest transitive relation ! on well-typed normal terms such that if t is a strict subterm of u then t ! u and if T is the normal form of the type of t and the term t is not a sort then T ! t is well-founded in the type systems of the cube. Thus every term admits a j-long normal form. Introduction In this paper we prove that the smallest transitive relation ! on well-typed normal terms such that ffl if t is a strict subterm of u then t ! u, ffl if T is the normal form of the type of t and the term t is not a sort then T ! t is well-founded in the type systems of the cube [1]. This result is proved using the notion of marked terms introduced by de Vrijer [6]. A motivation for this theorem is to define the j-long form of a normal term in these type systems. In simply typed -calculus, to define the j-long form of a normal term we first define the j-long form of a variable x of type P 1 ! ::: ! P n ! P (P atomic) as the term [y 1 : P 1 ]:::[y n : P n ](x y 0 1 ::: y 0 n ) w...

We study the cube of type assignment systems, as introduced in [13], and confront it with Barendregt’s typed λ-cube [4]. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for the systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one.

We present a dependent-type system for a #-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

Pure Type Systems (PTS fi s) provide a parametric framework for typed-calculi `a la Church [1, 2, 10, 11]. One important aspect of PTS fi s is to feature a definitional equality based on fi-conversion. In some instances however, one desires a stronger

this paper we are chieAEy concerned with the specications U \Gamma , U and . The denitions are taken from e.g. [1]

We present a theory of dependent types with explicit substitutions. We follow a meta-theoretical approach where open expressions ---expressions with meta-variables--- are first-class objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

CÉSAR MUÑOZ∗ Abstract. We present a dependent-type system for a λ-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.