> // WRITING FUNCTIONS IN SINGULAR. PROVING GEOMETRIC STATEMENTS LOCALLY. 
> proc Milnor(poly h){  return(vdim(std(jacob(h))));};
> ring R =0, (x,y), ds;
> poly h = x5+y2+2y3+y4; 
> Milnor(h);
4
> jacob(h);
_[1]=5x4
_[2]=2y+6y2+4y3
> std(_);
_[1]=y
_[2]=x4
> ideal J = jacob(h);
> LIB "zeroset.lib";
> Factor(J[2]);
[1]:
   _[1]=2
   _[2]=y
   _[3]=1+y
   _[4]=1+2y
[2]:
   1,1,1,1
> poly h'=subst(h,y,y-1); // move (0,-1) to the origin
> Milnor(h');
4
> poly h'=subst(h,y,y-1/2);
// ** redefining h' **
> Milnor(h');
4
> J;
J[1]=5x4
J[2]=2y+6y2+4y3
> ring S = 0, (x,y), dp;
> J;
   ? `J` is undefined
   ? error occurred in STDIN line 22: `J;`
> ideal J'=fetch(R,J);
> vdim(groebner(J'));
12

> // milnor and tjurina are functions in "sing.lib"
> LIB "sing.lib";
> setring R;
> milnor(h);
4
> 
. // Tjurina number
. tjurina(h);
4
> h';
1/16-1/2y2+y4+x5
> tjurina(h');
0


// STAIRCASES for various weights
> ring R=0, (x,y,z), ws(1,1,1);
> poly s1 = x3-yz;
> poly s2 = y3-xz;
> poly s3 = z3-xy;
> ideal I = s1,s2,s3;
> kbase(std(I));
_[1]=z4
_[2]=z3
_[3]=z2
_[4]=z
_[5]=y3
_[6]=y2
_[7]=y
_[8]=x3
_[9]=x2
_[10]=x
_[11]=1
> 

// DIAGONAL OF A PARALLELOGRAM
> ring R=0,(a,b,c,d,u,v),dp;
// ** redefining R **
> // relations in the parallelogam 
> ideal I = a-u-v, ad-bc, d*(v-u)-(c-u)*b;
> // relations to prove
> poly g1 = a2-2ac-2bd+b2;
> poly g2 = 2cu-2cv-2bd-u2+v2+b2;

> // try to prove globally
> ideal gbrad = groebner(radical(I));
> reduce(g1,gbrad);
b2-2bd-2cu+u2-2cv+2uv+v2
> reduce(g2,gbrad);
b2-2bd+2cu-u2-2cv+v2

> // prove locally: notice that the statement 
> // obiously holds for the unit square  
> ring r=0,(c,d,a,b,u,v),(dp(2),ds(4));
> ideal I'=fetch(R,I);
> I' = subst(I', a, a+1);
> I' = subst(I', b, b+1);
> I' = subst(I', u, u+1);
> poly g1'=fetch(R,g1);
> g1' = subst(g1', a, a+1);
> g1' = subst(g1', b, b+1);
> g1' = subst(g1', u, u+1);
> poly g2'=fetch(R,g2);
> g2' = subst(g2', a, a+1);
> g2' = subst(g2', b, b+1);
> g2' = subst(g2', u, u+1);
> ideal stb = std(I');
> reduce(g1',stb);
0
> reduce(g2',stb);
0
>