Results 1 
5 of
5
Logical Relations and Inductive/coinductive Types
, 1998
"... . We investigate a calculus with positive inductive and coinductive types , which we call ¯; , using logical relations. We show that parametric theories have the strong categorical properties, that the representable functors and natural transformations have the expected properties. Finally we ap ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
. We investigate a calculus with positive inductive and coinductive types , which we call ¯; , using logical relations. We show that parametric theories have the strong categorical properties, that the representable functors and natural transformations have the expected properties. Finally we apply the theory to show that terms of functorial type are almost canonical and that monotone inductive definitions can be reduced to positive in some cases. 1 Introduction We investigate a calculus with positive inductive and coinductive types, which we call ¯; , using logical relations. Here ¯ is used to construct the usual datatypes as initial algebras, where types are used for lazy types as terminal coalgebras. A calculus related to ¯; has for example been investigated by Geuvers [Geu92], but he does not consider nested ¯types. Loader introduces the strictly positive fragment (and only ¯types) in [Loa97] and shows that the theory of the PER model is maximal consistent. The restr...
A predicative strong normalisation proof for a λcalculus with interleaving inductive types
 TYPES FOR PROOF AND PROGRAMS, INTER40 A. ABEL AND T. ALTENKIRCH NATIONAL WORKSHOP, TYPES '99, SELECTED PAPERS. LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metaleve ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone
A Predicative Strong Normalisation Proof for a lambdaCalculus with Interleaving Inductive Types
, 2000
"... We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone operators on the metalevel. 1
A Finitary Subsystem of the Polymorphic lambdacalculus
"... We give a finitary normalisation proof for the restriction of system F where we quantify only over firstorder type. As an application, the functions representable in this fragment are exactly the ones provably total in Peano Arithmetic. This is inspired by the reduction of 1 1 comprehension to ind ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We give a finitary normalisation proof for the restriction of system F where we quantify only over firstorder type. As an application, the functions representable in this fragment are exactly the ones provably total in Peano Arithmetic. This is inspired by the reduction of 1 1 comprehension to inductive definitions presented in [Buch2] and this complements a result of [Leiv]. The argument uses a finitary model of a fragment of the system AF2 considered in [Kriv, Leiv].
A Predicative Strong Normalisation Proof for a
"... We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel ..."
Abstract
 Add to MetaCart
We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone operators on the metalevel.