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Indexed induction and coinduction, fibrationally
 In CALCO
, 2011
"... Abstract. This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Her ..."
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Abstract. This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs ’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U: E → B, a fibration with base B/I that models indexing by I and is also a CCU (QCE). 1
Bisimulation and Apartness in Coalgebraic Specification
, 1995
"... . A first basic fact in algebra (or, in algebraic specification) is the existence of free algebras, as suitable sets of terms. In coalgebraic specification, terminal coalgebras are known to exist in Sets (and in other categories) via the standard limit construction. Here we characterize these termin ..."
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. A first basic fact in algebra (or, in algebraic specification) is the existence of free algebras, as suitable sets of terms. In coalgebraic specification, terminal coalgebras are known to exist in Sets (and in other categories) via the standard limit construction. Here we characterize these terminal coalgebras as sets of "trees of observations". It is a standard result that elements of (the carrier of) a coalgebra are bisimilar (i.e. indistinguishable via the coalgebra operations) if and only if they have the same interpretation in the terminal coalgebra. This now becomes: if and only if they have the same tree of observations. Instead of putting emphasis on bisimulationwhich is a rather evasive notionwe consider its negation, which we write as #, and call "apartness ". It behaves like apartness in constructive mathematics. Indeed, the big advantage of apartness over bisimulation is that it can be established in a finite number of steps. It is a positive notion. Finally we show...
INDEXED INDUCTION AND COINDUCTION, FIBRATIONALLY
, 2012
"... Vol. 9(3:6)2013, pp. 1–31 www.lmcsonline.org ..."
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