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Summary: Set-based Operators and Fixed-Points by Support
Andreas Abel
April 22, 2004
In Abel and Altenkirch's article A Predicative Analysis of Structural Recur-
sion [AA02] one nds on page 34 a de nition of the semantics [[]] of a strictly
positive inductive type . This de nition (5.1) is itself a strictly-positive induc-
tive de nition on the meta-level. In the following we exhibit its correspondence
to the algorithmic construction of a least xed-point as found in Pierce's book
Types and Programming Languages [Pie02] on pages 290{295.
First we give a simpli ed version of the de nition restricted to a single type
variable.
De nition 5.1. For 2 Ty(X) and a closed type let
[[]] : P (Val ) ! P
Val ( )
be a monotone operator with a urelement relation
U Val Val ( )
such that for all v 2 Val ( ) and u 2 Val :
v 2 [[]](V ) u U v
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