If Random is not a list of compound terms, use Key = 0. The list elements must be numerical terms.
If Random is a list of compound terms, then the sort will be according to the Keyth argument of the list elements. The Keyth argument of each list element must be a numeric term.
In all cases, Order specifies whether the list is sorted into ascending (<, =<) or descending (>, >=) order and whether duplicates are to be retained (=<, >=) or eliminated (<, >). The way to remember the Order argument is that it is the relation which holds between adjacent elements in the result.
The sort is done according to numerical ordering as opposed to sort/4 which uses the standard ordering of terms. See compare/3 for this standard ordering. In particular for numeric terms of different type, e.g. integers and floats, the numerical and standard orderings differ: 1 < 2.0 but 2.0 @< 1. Additionally the ordering of bounded reals differs. While the standard ordering treats a bounded real as a compound term and orders them by lower bound and then upper bound, numerical ordering treats them as true intervals. As a consequence the order of overlapping intervals is undefined: 1.0__1.1 @< 1.0__1.2 while no numerical order is defined. In such cases an arithmetic exception is thrown. This can have unexpected consequences: care must be taken when sorting a list containing both rationals and bounded reals. While integers and floats are converted to zero-width intervals for the purposes of comparison, rationals are converted to small intervals guaranteed to contain the rational, e.g X is breal(1_1) gives X=0.99999999999999989__1.0000000000000002 and thus no order is defined between 1_1 and 1.0__1.0.
Success: number_sort(0,<,[3,1,6,7,2],S). (gives S=[1,2,3,6,7]). number_sort(0,=<,[1,1.0,1_1,3,2,3,4,1],S). (gives S=[1,1.0,1_1,1,2,3,3,4]). number_sort(0,=<,[1,1.0,1.0__1.0,3,2,3,4,1],S). (gives S=[1,1.0,1.0__1.0,1,2,3,3,4]). number_sort(2,<,[f(a,3),h(b,1)],S). (gives S=[h(2,1),f(1,3)]). number_sort([2,1],=<,[f(3,a(2)),f(1,a(1)),f(0,a(3)),f(1,a(4))],S). (gives S=[f(1,a(1)),f(3,a(2)),f(0,a(3)),f(1,a(4))]). Fail: number_sort(0,<,[2,1,3,4],[2,1,3,4]). Error: number_sort(0,>,[1,3,N],S). (Error 4). number_sort(0,>,[q,1,3,a,e],S). (Error 5). number_sort(1,<,[f(1),f(3),5],S). (Error 5). number_sort(1.0,<,[f(1),f(3),f(5)],S). (Error 5). number_sort(1,<,[f(a,3),h(b,1)],S). (Error 5). number_sort(2,<,[f(1,2),g(3,1),f(5)],S). (Error 6). number_sort(0,<,[1,0.9__1.1],S). (Error 20). number_sort(0,=<,[1_1,1.0__1.0],S). (Error 20).