ntop[0] := 0
ntop[x_] := s[ntop[x - 1]]

pton[0] := 0
pton[s[x_]] := pton[x] + 1

padd[0, x_] := x
padd[s[x_], y_] := s[padd[x, y]]

pmul[0,_] := 0
pmul[s[y_], x_] := padd[pmul[y, x], x]

psub[0, _] := 0
psub[x_, 0] := x
psub[s[x_], s[y_]] := psub[x, y]

bar[pp[0, x_]] := pp[0, x]
bar[pp[x_, 0]] := pp[x, 0]
bar[pp[s[x_], s[y_]]] := bar[pp[x, y]]

ntoz[z_] := pp[ntop[z], 0] /; z >= 0
ntoz[z_] := pp[0, ntop[-z]] /; z < 0

zton[pp[0, x_]] := -pton[x]
zton[pp[x_, 0]] := pton[x]
zton[x_] := zton[bar[x]]

pifz[0, x_, _] := x
pifz[_, _, y_] := y

pgt[x_, y_] := pifz[psub[x, y], y, x]

plt[x_, y_] := pifz[psub[x, y], x, y]

pgcm[0, x_] := x
pgcm[x_, y_] := pgcm[psub[pgt[x, y], plt[x, y]], plt[x, y]]

pdiv[0, _] := 0
pdiv[x_, y_] := pifz[psub[s[x],y],0,s[pdiv[psub[x, y], y]]]

prem[x_, y_] := psub[x, pmul[y, pdiv[x, y]]]

zminus[pp[x_,y_]] := pp[y,x]

zabs[pp[0,x_]] := pp[x,0]
zabs[z_] := z

ztop[pp[x_,0]] := x
ztop[pp[0,y_]] := y

ptoz[x_] := pp[x,0]

zadd[pp[x,y],pp[u,v]] := bar[pp[padd[x,u],padd[y,v]]]

zsub[x_,y_] := zadd[x,zminus[y]]

zmul[pp[x_,y_],pp[u_,v_]] := pp[padd[pmul[x,u],pmul[y,v]],padd[pmul[x,v],pmul[y,u]]]
zpmul[pp[x_,y_],z_] := pp[pmul[x,z],pmul[y,z]]

zpdiv[pp[x_,y_],z_] := pp[pdiv[x,z],pdiv[y,z]]

qbar[qq[x_,y_]] := qq[zpdiv[x,pgcm[ztop[x],y]],pdiv[y,pgcm[ztop[x],y]]]

ntoq[{x_,y_}] := qbar[qq[ntoz[x],ntop[y]]]
qton[qq[x_,y_]] := {zton[x],pton[y]}

qmul[qq[x_,y_],qq[u_,v_]] := qbar[qq[zmul[x,u],pmul[y,v]]]

qadd[qq[x_,y_],qq[u_,v_]] := qbar[qq[zadd[zpmul[x,v],zpmul[u,y]],pmul[y,v]]]