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16
Containers  Constructing Strictly Positive Types
, 2004
"... ... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are t ..."
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Cited by 83 (28 self)
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... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are the notions of containers and container functors, introduced in Abbott, Altenkirch, and Ghani (2003a). These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in MartinLöf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e. natural transformations) and that, in the presence of Wtypes, all strictly positive types (including nested inductive and coinductive types) give rise to containers.
Wellfounded Trees and Dependent Polynomial Functors
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class ..."
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Cited by 39 (6 self)
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We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.
Fast and Loose Reasoning is Morally Correct
, 2006
"... Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to nontotal (partial) languages. We justify such reasoning. ..."
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Cited by 38 (1 self)
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Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to nontotal (partial) languages. We justify such reasoning.
First steps in synthetic guarded domain theory: stepindexing in the topos of trees
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Derivatives of containers
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2003
"... We are investigating McBride’s idea that the type of onehole contexts are the formal derivative of a functor from a categorical perspective. Exploiting our recent work on containers we are able to characterise derivatives by a universal property and show that the laws of calculus including a rule ..."
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Cited by 8 (4 self)
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We are investigating McBride’s idea that the type of onehole contexts are the formal derivative of a functor from a categorical perspective. Exploiting our recent work on containers we are able to characterise derivatives by a universal property and show that the laws of calculus including a rule for initial algebras as presented by McBride hold — hence the differentiable containers include those generated by polynomials and least fixpoints. Finally, we discuss abstract containers (i.e. quotients of containers) — this includes a container which plays the role of e x in calculus by being its own derivative.
Representing Nested Inductive Types Using Wtypes
"... We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive ..."
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Cited by 8 (4 self)
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We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive
Interactive programs and weakly final coalgebras (extended version
 Dependently typed programming, number 04381 in Dagstuhl Seminar Proceedings, 2004. Available via http://drops.dagstuhl.de/opus
"... GR/S30450/01. 2 A. Setzer, P. Hancock 1 Introduction According to MartinL"of [19]: "... I do not think that the search for logically ever more satisfactory high level programming languages can stop short of anything but ..."
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GR/S30450/01. 2 A. Setzer, P. Hancock 1 Introduction According to MartinL&quot;of [19]: &quot;... I do not think that the search for logically ever more satisfactory high level programming languages can stop short of anything but
Higher Inductive Types in Homotopy Type Theory
"... Homotopy Type Theory (HoTT) refers to the homotopical interpretation [1] of MartinLöf’s intensional, constructive type theory (MLTT) [5], together with several new principles motivated by that interpretation. Voevodsky’s Univalent Foundations program [6] is a conception for a new foundation for mat ..."
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Homotopy Type Theory (HoTT) refers to the homotopical interpretation [1] of MartinLöf’s intensional, constructive type theory (MLTT) [5], together with several new principles motivated by that interpretation. Voevodsky’s Univalent Foundations program [6] is a conception for a new foundation for mathematics, based on HoTT and implemented in a proof assistant like Coq [2]. Among the new principles to be added to MLTT are the Univalence Axiom [4], and the socalled higher inductive types (HITs), a new idea due to Lumsdaine and Shulman which allows for the introduction of some basic spaces and constructions from homotopy theory. For example, the ndimensional spheres S n can be implemented as HITs, in a way analogous to the implementation of the natural numbers as a conventional inductive type. Other examples include the unit interval; truncations, such as brackettypes [A]; and quotients by equivalent relations or groupoids. The combination of univalence and HITs is turning out to be a very powerful and workable system for the formalization of homotopy theory, with the recently given, formally verified proofs of some fundamental results, such as determinations of various of the homotopy groups of spheres by Brunerie and Licata. See [3] for much work in progress After briefly reviewing the foregoing developments, I will give an impredicative encoding of certain HITs on the basis of a new representation theorem, which states that every type of a particular kind is equivalent to its double dual in the space of coherent natural transformations. A realizability model is also provided, establishing the consistency of impredicative HoTT and its extension by HITs.