Question: Buchberger function same as Basis function then how to make it as solver function?

Buchberger function same as Basis function then how to make it as solver function?

is it the newly added y(y-1) be used in solver to find x ?

i find eliminate without y*(y-1) same solution as adding y*(y-1)

why premas and Reduce are different for calculating remainder h?

where is wrong in commented L and S polynomials ? LCM wrong?

with(Groebner):
with(ListTools):
with(combinat):

premas := proc(P,A, PRing)
local R, i, x;
R := P;
i := nops(A);
while i <> 0 do
 x := MainVariable(A[i], PRing);
 R := prem(R, A[i],x);
 i := i - 1:
od:
return R:
end proc:

Buchberger := proc(Fparam, PRing)
#Fparam := [x*y-x, -y+x^2]:
F := MakeUnique(Fparam):
GG := choose(F, 2):
G := F:
while GG <> [] do
f := GG[1][1]:
g := GG[1][2]:
GG := [seq(GG[i], i=2..nops(GG))]:
#L := lcm(TrailingTerm(f, PRing)[2],TrailingTerm(g, PRing)[2]):
#S := simplify(L/LeadingTerm(f, PRing)[2]*f - L/LeadingTerm(g, PRing)[2]*g);
S := SPolynomial(f, g, PRing);
#print("S",S);
#print("G",G);
# S reduce to G modulo h
#h := premas(S, G, PolynomialRing([x, y, z]));
h := Reduce(S, G, PRing, 'h');
#print("h",h);
if h = 0 then
h := []:
end if:
if h <> [] then
 GG := [op(GG), [SelectFirst(G), h]]:
 G := [op(G), h]:
end if:
od:
return G:
end proc:
sol := Buchberger([x*y-x, -y+x^2], plex(x, y));
factor(sol[nops(sol)]);
eliminate(sol,[x,y]);
eliminate([x*y-x, -y+x^2],[x,y]);
sol := Basis([x*y-x, -y+x^2], plex(x, y));
solve([x*y-x, -y+x^2]);