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Structural Induction and Coinduction in a Fibrational Setting (1997)
| Venue: | Information and Computation |
| Citations: | 85 - 16 self |
BibTeX
@ARTICLE{Hermida97structuralinduction,
author = {Claudio Hermida and Bart Jacobs},
title = {Structural Induction and Coinduction in a Fibrational Setting},
journal = {Information and Computation},
year = {1997},
volume = {145},
pages = {107--152}
}
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Abstract
. We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = oe(X) where X may occur both positively and negatively in the type constructor oe. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunc...
Keyphrases
structural induction fibrational setting coinduction principle main result logical principle initial algebra basic result mixed induction coinduction principle terminal coalgebras sufficient criterion binary relation quotient type minimal solution induction principle type constructor oe categorical logic formulation alternative formulation
