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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | )abbrev category ORDRING OrderedRing
++ Description:
++ Ordered sets which are also rings, that is, domains where the ring
++ operations are compatible with the ordering.
++
++ Axiom\br
++ \tab{5}\spad{0<a and b<c => ab< ac}
OrderedRing() : Category == SIG where
SIG ==> Join(OrderedAbelianGroup,Ring,Monoid) with
positive? : % -> Boolean
++ positive?(x) tests whether x is strictly greater than 0.
negative? : % -> Boolean
++ negative?(x) tests whether x is strictly less than 0.
sign : % -> Integer
++ sign(x) is 1 if x is positive, -1 if x is negative,
++ 0 if x equals 0.
abs : % -> %
++ abs(x) returns the absolute value of x.
add
positive? x == x>0
negative? x == x<0
sign x ==
positive? x => 1
negative? x => -1
zero? x => 0
error "x satisfies neither positive?, negative? or zero?"
abs x ==
positive? x => x
negative? x => -x
zero? x => 0
error "x satisfies neither positive?, negative? or zero?"
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