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44
Constructing strictly positive families
 In The Australasian Theory Symposium (CATS2007
, 2007
"... We present an inductive definition of a universe containing codes for strictly positive families (SPFs) such as vectors or simply typed lambda terms. This construction extends the usual definition of inductive strictly positive types as given in previous joint work with McBride. We relate this to In ..."
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We present an inductive definition of a universe containing codes for strictly positive families (SPFs) such as vectors or simply typed lambda terms. This construction extends the usual definition of inductive strictly positive types as given in previous joint work with McBride. We relate this to Indexed Containers, which were recently proposed in joint work with Ghani, Hancock and McBride. We demonstrate by example how dependent types can be encoded in this universe and give examples for generic programs.
Why dependent types matter
 In preparation, http://www.epig.org/downloads/ydtm.pdf
, 2005
"... We exhibit the rationale behind the design of Epigram, a dependently typed programming language and interactive program development system, using refinements of a well known program—merge sort—as a running example. We discuss its relationship with other proposals to introduce aspects of dependent ty ..."
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We exhibit the rationale behind the design of Epigram, a dependently typed programming language and interactive program development system, using refinements of a well known program—merge sort—as a running example. We discuss its relationship with other proposals to introduce aspects of dependent types into functional programming languages and sketch some topics for further work in this area. 1.
A few constructions on constructors
 Types for Proofs and Programs
, 2005
"... Abstract. We present four constructions for standard equipment which can be generated for every inductive datatype: case analysis, structural recursion, no confusion, acyclicity. Our constructions follow a twolevel approach—they require less work than the standard techniques which inspired them [11 ..."
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Abstract. We present four constructions for standard equipment which can be generated for every inductive datatype: case analysis, structural recursion, no confusion, acyclicity. Our constructions follow a twolevel approach—they require less work than the standard techniques which inspired them [11, 8]. Moreover, given a suitably heterogeneous notion of equality, they extend without difficulty to inductive families of datatypes. These constructions are vital components of the translation from dependently typed programs in pattern matching style [7] to the equivalent programs expressed in terms of induction principles [21] and as such play a crucial behindthescenes rôle in Epigram [25]. 1
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
Typebased termination of generic programs
 Science of Computer Programming
, 2007
"... Instances of a polytypic or generic program for a concrete recursive type often exhibit a recursion scheme that is derived from the recursion scheme of the instantiation type. In practice, the programs obtained from a generic program are usually terminating, but the proof of termination cannot be ca ..."
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Instances of a polytypic or generic program for a concrete recursive type often exhibit a recursion scheme that is derived from the recursion scheme of the instantiation type. In practice, the programs obtained from a generic program are usually terminating, but the proof of termination cannot be carried out with traditional methods as term orderings alone, since termination often crucially relies on the program type. In this article, it is demonstrated that typebased termination using sized types handles such programs very well. A framework for sized polytypic programming is developed which ensures (typebased) termination of all instances. 1
Proof theory of MartinLöf type theory. An overview
 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 equiconsistent ..."
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Cited by 4 (2 self)
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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 MartinLöf type theory with Wtype and one microscopic universe containing only two finite sets in carried out. Then we look at the analysis MartinLöf 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.
When is a type refinement an inductive type
 In FOSSACS, volume 6604 of Lecture Notes in Computer Science
, 2011
"... Abstract. Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the Nindexed type of vectors ref ..."
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Abstract. Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the Nindexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purposespecific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations often store redundant information about data and their refinements. This paper shows how to generically derive inductive characterisations of refinements of inductive types, and argues that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. These characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined. 1
TypeSafe Diff for Families of Datatypes
"... The UNIX diff program finds the difference between two text files using a classic algorithm for determining the longest common subsequence; however, when working with structured input (e.g. program code), we often want to find the difference between treelike data (e.g. the abstract syntax tree). In ..."
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The UNIX diff program finds the difference between two text files using a classic algorithm for determining the longest common subsequence; however, when working with structured input (e.g. program code), we often want to find the difference between treelike data (e.g. the abstract syntax tree). In a functional programming language such as Haskell, we can represent this data with a family of (mutually recursive) datatypes. In this paper, we describe a functional, datatypegeneric implementation of diff (and the associated program patch). Our approach requires advanced type system features to preserve type safety; therefore, we present the code in Agda, a dependentlytyped language wellsuited to datatypegeneric programming. In order to establish the usefulness of our work, we show that its efficiency can be improved with memoization and that it can also be defined in Haskell.
A monadic formalization of ML5
 In Prepreceedings of Workshop on Logical Frameworks and Metalanguages: Theory and Practice
, 2010
"... ML5 is a programming language for spatially distributed computing, based on a CurryHoward correspondence with the modal logic S5. However, the ML5 programming language differs from the logic in several ways. In this paper, we give a semantic embedding of ML5 into the dependently typed programming l ..."
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ML5 is a programming language for spatially distributed computing, based on a CurryHoward correspondence with the modal logic S5. However, the ML5 programming language differs from the logic in several ways. In this paper, we give a semantic embedding of ML5 into the dependently typed programming language Agda, which both explains these discrepancies between ML5 and S5 and suggests some simplifications and generalizations of the language. Our embedding translates ML5 into a slightly different logic: intuitionistic S5 extended with a lax modality that encapsulates effectful computations in a monad. Rather than formalizing lax S5 as a proof theory, we embed it as a universe within the the dependently typed host language, with the universe elimination given by implementing the modal logic’s Kripke semantics. 1