There seem to be problems with the various orderings.
As a partial order is reflexive the basic operation in PartialOrder is "<=" and
hence there are categorial implementations for >, >= and < there.
for the category OrderedSet for the total (linear) orderings in FriCAS we have
"<" as the basic operation. This gives confusion what to implement in a domain
of OrderedSet.
E.g. I defined a FiniteSet using as first parameter Symbol and a second
parameter a List of Symbols using as the linear ordering the one from the
occurence in the List - and implemented <= using smaller? - of course a
mistake, as smaller? is really < which strange results for min and max. But
the setting confused.
I suggest the following:
stay on "<=" as basic operations also for OrderedSet.
It suffices to have the following categorial implementation, all the others are
inherited from PartialOrder
max(x, y) ==
x <= y => y
x
min(x, y) ==
x <= y => x
y
and drop all the other, which are inherited anyway.
Perhaps one could think about an Attribute linearilyOrdered or totallyOrdered
)abbrev category COMPAR Comparable
++ Description:
++ The class of set equipped with possibly unnatural linear order
++ (needed for technical reasons).
Comparable() : Category == SetCategory with
--operations
smaller? : (%, %) -> Boolean
++ smaller?(x, y) is a strict total ordering on the elements of the set.
PartialOrder() : Category == with
"<": (%,%) -> Boolean
++ x < y is a less than test.
">": (%, %) -> Boolean
++ x > y is a greater than test.
">=": (%, %) -> Boolean
++ x >= y is a greater than or equal test.
"<=": (%, %) -> Boolean
++ x <= y is a less than or equal test.
add
x >= y == y <= x
x > y == y < x
x < y == x <= y and not(y <= x)
OrderedSet() : Category == Join(Comparable, PartialOrder) with
--operations
max : (%, %) -> %
++ max(x,y) returns the maximum of x and y relative to "<".
min : (%, %) -> %
++ min(x,y) returns the minimum of x and y relative to "<".
add
-- transitional definitions
smaller?(x, y) == x < y
--declarations
x, y : %
--definitions
-- These really ought to become some sort of macro
max(x, y) ==
x > y => x
y
min(x, y) ==
x > y => y
x
((x : %) > (y : %)) : Boolean == y < x
((x : %) >= (y : %)) : Boolean == not (x < y)
((x : %) <= (y : %)) : Boolean == not (y < x)
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