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17
Towards a practical programming language based on dependent type theory
, 2007
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Dependently Typed Functional Programs and their Proofs
, 1999
"... Research in dependent type theories [ML71a] has, in the past, concentrated on its use in the presentation of theorems and theoremproving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs ..."
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Cited by 82 (13 self)
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Research in dependent type theories [ML71a] has, in the past, concentrated on its use in the presentation of theorems and theoremproving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs may readily be specified and established. In particular, it develops technology for programming with dependent inductive families of datatypes and proving those programs correct. It demonstrates the considerable advantage to be gained by indexing data structures with pertinent characteristic information whose soundness is ensured by typechecking, rather than human effort. Type theory traditionally presents safe and terminating computation on inductive datatypes by means of elimination rules which serve as induction principles and, via their associated reduction behaviour, recursion operators [Dyb91]. In the programming language arena, these appear somewhat cumbersome and give rise to unappealing code, complicated by the inevitable interaction between case analysis on dependent types and equational reasoning on their indices which must appear explicitly in the terms. Thierry Coquand’s proposal [Coq92] to equip type theory directly with the kind of
TYPE THEORY AND HOMOTOPY
"... The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy ..."
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Cited by 15 (0 self)
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The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy
Eliminating dependent pattern matching
 of Lecture Notes in Computer Science
, 2006
"... Abstract. This paper gives a reductionpreserving translation from Coquand’s dependent pattern matching [4] into a traditional type theory [11] with universes, inductive types and relations and the axiom K [22]. This translation serves as a proof of termination for structurally recursive pattern mat ..."
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Abstract. This paper gives a reductionpreserving translation from Coquand’s dependent pattern matching [4] into a traditional type theory [11] with universes, inductive types and relations and the axiom K [22]. This translation serves as a proof of termination for structurally recursive pattern matching programs, provides an implementable compilation technique in the style of functional programming languages, and demonstrates the equivalence with a more easily understood type theory. Dedicated to Professor Joseph Goguen on the occasion of his 65th birthday. 1
Axioms and (Counter)examples in Synthetic Domain Theory
 Annals of Pure and Applied Logic
, 1998
"... this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the p ..."
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Cited by 8 (7 self)
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this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the purposes of the axiomatic part of this paper, we believe that it would also be
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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Cited by 3 (2 self)
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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
The intrinsic topology of a MartinLöf universe
, 2012
"... Assuming the propositional axiom of extensionality, we show that a MartinLöf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a nontrivial, decidable, ext ..."
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Assuming the propositional axiom of extensionality, we show that a MartinLöf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a nontrivial, decidable, extensional property of the universe implies the weak limited principle of omniscience. This is a theorem in type theory. Without assuming extensionality, we deduce the following metatheorem: in intensional MartinLöf type theory with a universe, there is no closed term defining a nontrivial, decidable, extensional property of the universe. 1
Homotopy Models of Intensional Type Theory
"... We show how to provide a semantics for the identity type of intensional MartinLöf type theory using the machinery of abstract homotopy theory. One specific consequence is that the identity type is soundly modelled in ..."
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We show how to provide a semantics for the identity type of intensional MartinLöf type theory using the machinery of abstract homotopy theory. One specific consequence is that the identity type is soundly modelled in
CONSTRUCTING CATEGORIES AND SETOIDS OF SETOIDS IN TYPE THEORY
, 2013
"... Vol. 10(3:25)2014, pp. 1–14 www.lmcsonline.org ..."
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