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Internal Type Theory
 Lecture Notes in Computer Science
, 1996
"... . We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with f ..."
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Cited by 38 (7 self)
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. We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in MartinLof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper [8] I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural development of what could be called "internal type theory", that is, the construction of models of (fragments of) type theory in type theory, and more generally, the formalization of the metatheory of type theory in type theory. The present paper is a first investigation of such an internal type theor...
Normalization and the Yoneda Embedding
"... this paper we describe a new, categorical approach to normalization in typed  ..."
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Cited by 22 (3 self)
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this paper we describe a new, categorical approach to normalization in typed 
Extracting a Proof of Coherence for Monoidal Categories from a Proof of Normalization for Monoids
 In TYPES
, 1995
"... . This paper studies the problem of coherence in category theory from a typetheoretic viewpoint. We first show how a CurryHoward interpretation of a formal proof of normalization for monoids almost directly yields a coherence proof for monoidal categories. Then we formalize this coherence proof in ..."
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. This paper studies the problem of coherence in category theory from a typetheoretic viewpoint. We first show how a CurryHoward interpretation of a formal proof of normalization for monoids almost directly yields a coherence proof for monoidal categories. Then we formalize this coherence proof in intensional intuitionistic type theory and show how it relies on explicit reasoning about proof objects for intensional equality. This formalization has been checked in the proof assistant ALF. 1 Introduction Mac Lane [18, pp.161165] proved a coherence theorem for monoidal categories. A basic ingredient in his proof is the normalization of object expressions. But it is only one ingredient and several others are needed too. Here we show that almost a whole proof of this coherence theorem is hidden in a CurryHoward interpretation of a proof of normalization for monoids. The second point of the paper is to contribute to the development of constructive category theory in the sense of Huet a...
A case study in machineassisted proofs: The Integers form an Integral Domain
, 1993
"... We present a formalization of the set Z of integers using MartinLof's type theory. In particular we focus on the task of proving that this set with the operations + and form an Integral Domain. The proofs are developed for an inductive definition of Z, but we also discuss what kind of proofs ..."
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We present a formalization of the set Z of integers using MartinLof's type theory. In particular we focus on the task of proving that this set with the operations + and form an Integral Domain. The proofs are developed for an inductive definition of Z, but we also discuss what kind of proofs could be obtained for a formulation where the set is defined as a quotient. The differences between both approaches when one is interested in regarding the computational meaning of proofs are pointed out. In order to better reason about the proofs of the properties following from the postulates of an integral domain, an abstract formalization of this algebraic system is also proposed. With this, we aimed at not just being able to formally reflect the derivation of the properties independently of the concrete representation we were interested in, but also to translate these results to every algebraic structure satisfying those postulates. Keywords and phrases: integers, type theory, integral dom...
Implementing a Category of Sets in ALF
, 1994
"... Peter Aczel [1] and Gerard Huet [8] have implemented the category of sets in LEGO and Coq respectively. Here we show an implementation of the category of sets in ALF [2], a proof assistant based on MartinLöf's logical framework (or theory of logical types) [10]. We used WINDOW ALF. This syste ..."
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Cited by 4 (1 self)
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Peter Aczel [1] and Gerard Huet [8] have implemented the category of sets in LEGO and Coq respectively. Here we show an implementation of the category of sets in ALF [2], a proof assistant based on MartinLöf's logical framework (or theory of logical types) [10]. We used WINDOW ALF. This system allows one to manipulate the proof term in order to refine it until it is complete. Some facilities are provided which show the term in a readable way (special symbols for constants, infix use of symbols, hiding of arguments, etc). What is presented below is, unfortunately, not what is shown on the screen, but the source code for the type checker and the window interface. Thus, for instance, no arguments are hidden and lambda is used instead of . We refer to the introduction to the ALF chapter of the library for further information. We have essentially followed Peter Aczel's development. But we ha
Dependent Record Types, Subtyping and Proof Reutilization
"... . We present an example of formalization of systems of algebras using an extension of MartinLof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all the features of the extended theory that we consider relevant f ..."
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Cited by 4 (1 self)
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. We present an example of formalization of systems of algebras using an extension of MartinLof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all 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 MartinLof'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 MartinLof's theory of types, from now on referred to as the logical framework, has been presented in [15, 7]. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A...
Experiments in Formalizing Basic Category Theory in Higher Order Logic and Set Theory
, 1995
"... this paper is the product category, defined by ..."
Proofrelevance of families of setoids and identity in type theory
, 2010
"... Families of types are fundamental objects in MartinLöf type theory. When extending the notion of setoid (type with an equivalence relation) to families of setoids, a choice between proofrelevant or proofirrelevant indexing appears. It is shown that a family of types may be canonically extended to ..."
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Families of types are fundamental objects in MartinLöf type theory. When extending the notion of setoid (type with an equivalence relation) to families of setoids, a choice between proofrelevant or proofirrelevant indexing appears. It is shown that a family of types may be canonically extended to a proofrelevant family of setoids via the identity types, but that such a family is in general proofirrelevant if, and only if, the proofobjects of identity types are unique. A similar result is shown for fibre representations of families. The ubiquitous role of proofirrelevant families is discussed. 1