  Abstract interpretation-based static analyses rely on abstract domains of
program properties, such as intervals or congruences for integer variables.
Galois connections (GCs) between posets provide the most widespread and useful
formal tool for mathematically specifying abstract domains. Recently, Darais
and Van Horn [2016] put forward a notion of constructive Galois connection for
unordered sets (rather than posets), which allows to define abstract domains in
a so-called mechanized and calculational proof style and therefore enables the
use of proof assistants like Coq and Agda for automatically extracting verified
algorithms of static analysis. We show here that constructive GCs are
isomorphic, in a precise and comprehensive meaning including sound abstract
functions, to so-called partitioning GCs--an already known class of GCs which
allows to cast standard set partitions as an abstract domain. Darais and Van
Horn [2016] also provide a notion of constructive GC for posets, which we prove
to be isomorphic to plain GCs and therefore lose their constructive attribute.
Drawing on these findings, we put forward and advocate the use of purely
partitioning GCs, a novel class of constructive abstract domains for a
mechanized approach to abstract interpretation. We show that this class of
abstract domains allows us to represent a set partition with more flexibility
while retaining a constructive approach to Galois connections.
