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Abstract
This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs’ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobs’ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.
Original language  English 

Title of host publication  Computer Science Logic 
Subtitle of host publication  24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, Brno, Czech Republic, August 2327, 2010. Proceedings 
Editors  Anuj Dawar , Helmut Veith 
Publisher  Springer 
Pages  336350 
Number of pages  15 
Volume  6247 
ISBN (Print)  9783642152047 
DOIs  
Publication status  Published  2010 
Publication series
Name  Lecture Notes In Computer Science 

Publisher  Springer 
Volume  6247 
ISSN (Print)  03029743 
Keywords
 fibrational Induction Rules
 initial algebras
 computer science
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 1 Finished

Categorical Foundations of Indexed Programming
Johann, P. & Ghani, N.
EPSRC (Engineering and Physical Sciences Research Council)
1/07/09 → 31/12/12
Project: Research