MAYBE

Problem:
 division(x,y) -> div(x,y,0())
 div(x,y,z) -> if(lt(x,y),x,y,inc(z))
 if(true(),x,y,z) -> z
 if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 lt(x,0()) -> false()
 lt(0(),s(y)) -> true()
 lt(s(x),s(y)) -> lt(x,y)
 inc(0()) -> s(0())
 inc(s(x)) -> s(inc(x))

Proof:
 DP Processor:
  DPs:
   division#(x,y) -> div#(x,y,0())
   div#(x,y,z) -> inc#(z)
   div#(x,y,z) -> lt#(x,y)
   div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
   if#(false(),x,s(y),z) -> minus#(x,s(y))
   if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z)
   minus#(s(x),s(y)) -> minus#(x,y)
   lt#(s(x),s(y)) -> lt#(x,y)
   inc#(s(x)) -> inc#(x)
  TRS:
   division(x,y) -> div(x,y,0())
   div(x,y,z) -> if(lt(x,y),x,y,inc(z))
   if(true(),x,y,z) -> z
   if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   lt(x,0()) -> false()
   lt(0(),s(y)) -> true()
   lt(s(x),s(y)) -> lt(x,y)
   inc(0()) -> s(0())
   inc(s(x)) -> s(inc(x))
  Usable Rule Processor:
   DPs:
    division#(x,y) -> div#(x,y,0())
    div#(x,y,z) -> inc#(z)
    div#(x,y,z) -> lt#(x,y)
    div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
    if#(false(),x,s(y),z) -> minus#(x,s(y))
    if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z)
    minus#(s(x),s(y)) -> minus#(x,y)
    lt#(s(x),s(y)) -> lt#(x,y)
    inc#(s(x)) -> inc#(x)
   TRS:
    inc(0()) -> s(0())
    inc(s(x)) -> s(inc(x))
    lt(x,0()) -> false()
    lt(0(),s(y)) -> true()
    lt(s(x),s(y)) -> lt(x,y)
    minus(s(x),s(y)) -> minus(x,y)
    minus(x,0()) -> x
   Open