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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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We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
The Open Calculus of Constructions: An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving
"... The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational t ..."
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The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higherorder style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a firstorder semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.
Formalizing categorical models of type theory in type theory
 In International Workshop on Logical Frameworks and MetaLanguages: Theory and Practice
, 2007
"... This note is about work in progress on the topic of “internal type theory ” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of de ..."
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This note is about work in progress on the topic of “internal type theory ” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of dependent type theory. We discuss how to formalize the notion of category with families inside type theory and how to build initial categories with families. Initial categories with families will be term models which play the role of canonical syntax for dependent type theory. We also discuss the formalization of the result that categories with finite limits give rise to categories with families. This yields a typetheoretic perspective on Curien’s work on “substitution up to isomorphism”. Our formalization is being carried out in the proof assistant Agda 2 developed at Chalmers. 1
Proof Theory of MartinLof Type Theory  An
 Mathematiques et Sciences Humaines, 42 année, n o 165:59 – 99
, 2004
"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsisten ..."
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We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of MartinLof type theory with Wtype and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of MartinLof type theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.
Generic programming with dependent types
 Spring School on Datatype Generic Programming
, 2006
"... In these lecture notes we give an overview of recent research on the relationship ..."
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In these lecture notes we give an overview of recent research on the relationship
Outrageous but meaningful coincidences: Dependent typesafe syntax and evaluation
 In ACM SIGPLAN Workshop on Genetic Programming (WGP’10
, 2010
"... Tagless interpreters for welltyped terms in some object language are a standard example of the power and benefit of precise indexing in types, whether with dependent types, or generalized algebraic datatypes. The key is to reflect object language types as indices (however they may be constituted) f ..."
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Tagless interpreters for welltyped terms in some object language are a standard example of the power and benefit of precise indexing in types, whether with dependent types, or generalized algebraic datatypes. The key is to reflect object language types as indices (however they may be constituted) for the term datatype in the host language, so that host type coincidence ensures object type coincidence. Whilst this technique is widespread for simply typed object languages, dependent types have proved a tougher nut with nontrivial computation in type equality. In their typesafe representations, Danielsson [2006] and Chapman [2009] succeed in capturing the equality rules, but at the cost of representing equality derivations explicitly within terms. This article delivers a typesafe representation for a dependently typed object language, dubbed KIPLING, whose computational type equality just appropriates that of its host, Agda. The KIPLING interpreter example is not merely de rigeur— it is key to the construction. At the heart of the technique is that key component of generic programming, the universe. 1.
Partial recursive functions in MartinLöf Type Theory
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
"... Abstract. In this article we revisit the approach by Bove and Capretta for formulating partial recursive functions in MartinLöf Type Theory by indexed inductiverecursive definitions. We will show that all inductiverecursive definitions used there can be replaced by inductive definitions. However, ..."
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Abstract. In this article we revisit the approach by Bove and Capretta for formulating partial recursive functions in MartinLöf Type Theory by indexed inductiverecursive definitions. We will show that all inductiverecursive definitions used there can be replaced by inductive definitions. However, this encoding results in an additional technical overhead. In order to obtain directly executable partial recursive functions, we introduce restrictions on the indexed inductiverecursive definitions used. Then we introduce a data type of partial recursive functions. This allows to define higher order partial recursive functions like the map functional, which depend on other partial recursive functions. This data type will be based on the closed formalisation of indexed inductiverecursive definitions introduced by Dybjer and the author. All elements of this data type will represent partial recursive functions, and the set of partial recursive functions will be closed under the standard operations for forming partial recursive functions, and under the total functions. Keywords: MartinLöf type theory, computability theory, recursion theory, Kleene index, Kleene brackets, partial recursive functions, inductiverecursive definitions, indexed inductionrecursion.
A data type of partial recursive functions in MartinLöf Type Theory. 35pp, submitted
, 2007
"... In this article we investigate how to represent partialrecursive functions in MartinLöf’s type theory. Our representation will be based on the approach by Bove and Capretta, which makes use of indexed inductiverecursive definitions (IIRD). We will show how to restrict the IIRD used so that we obt ..."
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In this article we investigate how to represent partialrecursive functions in MartinLöf’s type theory. Our representation will be based on the approach by Bove and Capretta, which makes use of indexed inductiverecursive definitions (IIRD). We will show how to restrict the IIRD used so that we obtain directly executable partial recursive functions, Then we introduce a data type of partial recursive functions. We show how to evaluate elements of this data type inside MartinLöf’s type theory, and that therefore the functions defined by this data type are in fact partialrecursive. The data type formulates a very general schema for defining functions recursively in dependent type theory. The initial version of this data type, for which we introduce an induction principle, needs to be expanded, in order to obtain closure under composition. We will obtain two versions of this expanded data type, and prove that they define the same set of partialrecursive functions. Both versions will be large types. Next we prove a Kleenestyle normal form theorem. Using it we will show how to obtain a data type of partial recursive functions which is a small set. Finally, we show how to define selfevaluation as a partial recursive function. We obtain a correct version of this evaluation function, which not only computes recursively a result, but as well a proof that the result is correct. Keywords: MartinLöf type theory, computability theory, recursion theory, Kleene index, Kleene brackets, Kleene’s normal form theorem, partial recursive functions, inductiverecursive definitions, indexed inductionrecursion, selfevaluation. 1