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D.2.3.14 sort

Procedure from library general.lib (see general_lib).

sort(id[,v,o,n]); id = ideal/module/intvec/list
sort may be called with 1, 2 or 3 arguments in the following way:
sort(id[,v,n]); v=intvec of positive integers, n=integer,
sort(id[,o,n]); o=string (any allowed ordstr of a ring), n=integer

a list l of two elements:
        l[1]: object of same type as input but sorted in the following way:
           - if id=ideal/module: generators of id are sorted w.r.t. intvec v
             (id[v[1]] becomes 1-st, id[v[2]]  2-nd element, etc.). If no v is
             present, id is sorted w.r.t. ordering o (if o is given) or w.r.t.
             actual monomial ordering (if no o is given):
             NOTE: generators with SMALLER(!) leading term come FIRST
             (e.g. sort(id); sorts backwards to actual monomial ordering)
           - if id=list or intvec: sorted w.r.t. < (indep. of other arguments)
           - if n!=0 the ordering is inverse, i.e. w.r.t. v(size(v)..1)
             default: n=0
         l[2]: intvec, describing the permutation of the input (hence l[2]=v
             if v is given (with positive integers))

If v is given id may be any simply indexed object (e.g. any list or string); if v[i]<0 and i<=size(id) v[i] is set internally to i; entries of v must be pairwise distinct to get a permutation id. Zero generators of ideal/module are deleted
If 'o' is given, the input is sorted by considering leading terms w.r.t. the new ring ordering given by 'o'

LIB "general.lib";
ring r0 = 0,(x,y,z,t),lp;
ideal i = x3,z3,xyz;
sort(i);            //sorts using lex ordering, smaller polys come first
==> [1]:
==>    _[1]=z3
==>    _[2]=xyz
==>    _[3]=x3
==> [2]:
==>    2,3,1
==> [1]:
==>    _[1]=xyz
==>    _[2]=z3
==>    _[3]=x3
==> [2]:
==>    3,2,1
sort(i,"ls")[1];     //sort w.r.t. negative lex ordering
==> _[1]=x3
==> _[2]=xyz
==> _[3]=z3
intvec v =1,10..5,2..4;v;
==> 1,10,9,8,7,6,5,2,3,4
sort(v)[1];          // sort v lexicographically
==> 1,2,3,4,5,6,7,8,9,10
sort(v,"Dp",1)[1];   // sort v w.r.t (total sum, reverse lex)
==> 1,2,3,4,5,6,7,8,9,10
// Note that in general: lead(sort(M)) != sort(lead(M)), e.g:
module M = [0, 1, 1, 0], [1, 0, 0, 1]; M;
==> M[1]=gen(3)+gen(2)
==> M[2]=gen(4)+gen(1)
sort(lead(M), "c, dp")[1];
==> _[1]=gen(4)
==> _[2]=gen(3)
lead(sort(M, "c, dp")[1]);
==> _[1]=gen(3)
==> _[2]=gen(4)
// In order to sort M wrt a NEW ordering by considering OLD leading
// terms use one of the following equivalent commands:
module( M[ sort(lead(M), "c,dp")[2] ] );
==> _[1]=gen(4)+gen(1)
==> _[2]=gen(3)+gen(2)
sort( M, sort(lead(M), "c,dp")[2] )[1];
==> _[1]=gen(4)+gen(1)
==> _[2]=gen(3)+gen(2)
// BUG: Please, don't use this sort for integer vectors or lists
// with them if there can be negative integers!
// TODO: for some HiWi
==> -3,-2,-1,0,1,2,3
sort(list(-v, v))[1];
==> [1]:
==>    -1,-10,-9,-8,-7,-6,-5,-2,-3,-4
==> [2]:
==>    1,10,9,8,7,6,5,2,3,4