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Simple unificationbased type inference for GADTs
, 2006
"... Generalized algebraic data types (GADTs), sometimes known as “guarded recursive data types ” or “firstclass phantom types”, are a simple but powerful generalization of the data types of Haskell and ML. Recent works have given compelling examples of the utility of GADTs, although type inference is k ..."
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Cited by 191 (36 self)
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Generalized algebraic data types (GADTs), sometimes known as “guarded recursive data types ” or “firstclass phantom types”, are a simple but powerful generalization of the data types of Haskell and ML. Recent works have given compelling examples of the utility of GADTs, although type inference is known to be difficult. Our contribution is to show how to exploit programmersupplied type annotations to make the type inference task almost embarrassingly easy. Our main technical innovation is wobbly types, which express in a declarative way the uncertainty caused by the incremental nature of typical typeinference algorithms.
Pattern Matching with Dependent Types.
 Electronic Proceedings of the Third Annual BRA Workshop on Logical Frameworks (Bastad, Sweden),
, 1992
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Inductive Families
 Formal Aspects of Computing
, 1997
"... A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets ar ..."
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Cited by 77 (13 self)
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A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalization to type theory of MartinLof's theory of iterated inductive definitions in predicate logic. Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalized induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion. The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applic...
A General Formulation of Simultaneous InductiveRecursive Definitions in Type Theory
 Journal of Symbolic Logic
, 1998
"... The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by re ..."
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Cited by 77 (9 self)
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The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive definition which is implicit in MartinLöf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous inductionrecursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model ...
The Theory of LEGO  A Proof Checker for the Extended Calculus of Constructions
, 1994
"... LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO ..."
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Cited by 72 (10 self)
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LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the metatheory of LEGO's type systems leading to a machinechecked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...
Generic programming within dependently typed programming
 In Generic Programming, 2003. Proceedings of the IFIP TC2 Working Conference on Generic Programming, Schloss Dagstuhl
, 2003
"... Abstract We show how higher kinded generic programming can be represented faithfully within a dependently typed programming system. This development has been implemented using the Oleg system. The present work can be seen as evidence for our thesis that extensions of type systems can be done by prog ..."
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Cited by 66 (8 self)
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Abstract We show how higher kinded generic programming can be represented faithfully within a dependently typed programming system. This development has been implemented using the Oleg system. The present work can be seen as evidence for our thesis that extensions of type systems can be done by programming within a dependently typed language, using data as codes for types. 1.
Tagless Staged Interpreters for Typed Languages
 In the International Conference on Functional Programming (ICFP ’02
, 2002
"... Multistage programming languages provide a convenient notation for explicitly staging programs. Staging a definitional interpreter for a domain specific language is one way of deriving an implementation that is both readable and efficient. In an untyped setting, staging an interpreter "removes ..."
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Cited by 56 (13 self)
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Multistage programming languages provide a convenient notation for explicitly staging programs. Staging a definitional interpreter for a domain specific language is one way of deriving an implementation that is both readable and efficient. In an untyped setting, staging an interpreter "removes a complete layer of interpretive overhead", just like partial evaluation. In a typed setting however, HindleyMilner type systems do not allow us to exploit typing information in the language being interpreted. In practice, this can have a slowdown cost factor of three or more times.
A finite axiomatization of inductiverecursive definitions
 Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science
, 1999
"... Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an in ..."
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Cited by 51 (15 self)
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Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductiverecursive definitions. We prove consistency by constructing a settheoretic model which makes use of one Mahlo cardinal. 1
Intuitionistic Model Constructions and Normalization Proofs
, 1998
"... We investigate semantical normalization proofs for typed combinatory logic and weak calculus. One builds a model and a function `quote' which inverts the interpretation function. A normalization function is then obtained by composing quote with the interpretation function. Our models are just ..."
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Cited by 50 (7 self)
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We investigate semantical normalization proofs for typed combinatory logic and weak calculus. One builds a model and a function `quote' which inverts the interpretation function. A normalization function is then obtained by composing quote with the interpretation function. Our models are just like the intended model, except that the function space includes a syntactic component as well as a semantic one. We call this a `glued' model because of its similarity with the glueing construction in category theory. Other basic type constructors are interpreted as in the intended model. In this way we can also treat inductively defined types such as natural numbers and Brouwer ordinals. We also discuss how to formalize terms, and show how one model construction can be used to yield normalization proofs for two different typed calculi  one with explicit and one with implicit substitution. The proofs are formalized using MartinLof's type theory as a meta language and mechanized using the A...