MAYBE

Problem:
 minus(n__0(),Y) -> 0()
 minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
 geq(X,n__0()) -> true()
 geq(n__0(),n__s(Y)) -> false()
 geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
 div(0(),n__s(Y)) -> 0()
 div(s(X),n__s(Y)) ->
 if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
 if(true(),X,Y) -> activate(X)
 if(false(),X,Y) -> activate(Y)
 0() -> n__0()
 s(X) -> n__s(X)
 div(X1,X2) -> n__div(X1,X2)
 minus(X1,X2) -> n__minus(X1,X2)
 activate(n__0()) -> 0()
 activate(n__s(X)) -> s(activate(X))
 activate(n__div(X1,X2)) -> div(activate(X1),X2)
 activate(n__minus(X1,X2)) -> minus(X1,X2)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   minus#(n__0(),Y) -> 0#()
   minus#(n__s(X),n__s(Y)) -> activate#(Y)
   minus#(n__s(X),n__s(Y)) -> activate#(X)
   minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
   geq#(n__s(X),n__s(Y)) -> activate#(Y)
   geq#(n__s(X),n__s(Y)) -> activate#(X)
   geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
   div#(s(X),n__s(Y)) -> activate#(Y)
   div#(s(X),n__s(Y)) -> geq#(X,activate(Y))
   div#(s(X),n__s(Y)) ->
   if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
   if#(true(),X,Y) -> activate#(X)
   if#(false(),X,Y) -> activate#(Y)
   activate#(n__0()) -> 0#()
   activate#(n__s(X)) -> activate#(X)
   activate#(n__s(X)) -> s#(activate(X))
   activate#(n__div(X1,X2)) -> activate#(X1)
   activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
   activate#(n__minus(X1,X2)) -> minus#(X1,X2)
  TRS:
   minus(n__0(),Y) -> 0()
   minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
   geq(X,n__0()) -> true()
   geq(n__0(),n__s(Y)) -> false()
   geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
   div(0(),n__s(Y)) -> 0()
   div(s(X),n__s(Y)) ->
   if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
   if(true(),X,Y) -> activate(X)
   if(false(),X,Y) -> activate(Y)
   0() -> n__0()
   s(X) -> n__s(X)
   div(X1,X2) -> n__div(X1,X2)
   minus(X1,X2) -> n__minus(X1,X2)
   activate(n__0()) -> 0()
   activate(n__s(X)) -> s(activate(X))
   activate(n__div(X1,X2)) -> div(activate(X1),X2)
   activate(n__minus(X1,X2)) -> minus(X1,X2)
   activate(X) -> X
  ADG Processor:
   DPs:
    minus#(n__0(),Y) -> 0#()
    minus#(n__s(X),n__s(Y)) -> activate#(Y)
    minus#(n__s(X),n__s(Y)) -> activate#(X)
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
    geq#(n__s(X),n__s(Y)) -> activate#(Y)
    geq#(n__s(X),n__s(Y)) -> activate#(X)
    geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
    div#(s(X),n__s(Y)) -> activate#(Y)
    div#(s(X),n__s(Y)) -> geq#(X,activate(Y))
    div#(s(X),n__s(Y)) ->
    if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
    if#(true(),X,Y) -> activate#(X)
    if#(false(),X,Y) -> activate#(Y)
    activate#(n__0()) -> 0#()
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__div(X1,X2)) -> activate#(X1)
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
   TRS:
    minus(n__0(),Y) -> 0()
    minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
    geq(X,n__0()) -> true()
    geq(n__0(),n__s(Y)) -> false()
    geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
    div(0(),n__s(Y)) -> 0()
    div(s(X),n__s(Y)) ->
    if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
    if(true(),X,Y) -> activate(X)
    if(false(),X,Y) -> activate(Y)
    0() -> n__0()
    s(X) -> n__s(X)
    div(X1,X2) -> n__div(X1,X2)
    minus(X1,X2) -> n__minus(X1,X2)
    activate(n__0()) -> 0()
    activate(n__s(X)) -> s(activate(X))
    activate(n__div(X1,X2)) -> div(activate(X1),X2)
    activate(n__minus(X1,X2)) -> minus(X1,X2)
    activate(X) -> X
   graph:
    if#(false(),X,Y) -> activate#(Y) -> activate#(n__0()) -> 0#()
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__s(X)) -> activate#(X)
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__s(X)) -> s#(activate(X))
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    if#(true(),X,Y) -> activate#(X) -> activate#(n__0()) -> 0#()
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    div#(s(X),n__s(Y)) ->
    if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0()) ->
    if#(true(),X,Y) -> activate#(X)
    div#(s(X),n__s(Y)) ->
    if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0()) ->
    if#(false(),X,Y) -> activate#(Y)
    div#(s(X),n__s(Y)) -> geq#(X,activate(Y)) ->
    geq#(n__s(X),n__s(Y)) -> activate#(Y)
    div#(s(X),n__s(Y)) -> geq#(X,activate(Y)) ->
    geq#(n__s(X),n__s(Y)) -> activate#(X)
    div#(s(X),n__s(Y)) -> geq#(X,activate(Y)) ->
    geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
    div#(s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__0()) -> 0#()
    div#(s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__s(X)) -> activate#(X)
    div#(s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__s(X)) -> s#(activate(X))
    div#(s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    div#(s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    div#(s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y)) ->
    geq#(n__s(X),n__s(Y)) -> activate#(Y)
    geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y)) ->
    geq#(n__s(X),n__s(Y)) -> activate#(X)
    geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y)) ->
    geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
    geq#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__0()) -> 0#()
    geq#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    geq#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    geq#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    geq#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    geq#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    geq#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__0()) -> 0#()
    geq#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__s(X)) -> activate#(X)
    geq#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__s(X)) -> s#(activate(X))
    geq#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    geq#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    geq#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2) ->
    div#(s(X),n__s(Y)) -> activate#(Y)
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2) ->
    div#(s(X),n__s(Y)) -> geq#(X,activate(Y))
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2) ->
    div#(s(X),n__s(Y)) ->
    if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
    activate#(n__div(X1,X2)) -> activate#(X1) ->
    activate#(n__0()) -> 0#()
    activate#(n__div(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__div(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__div(X1,X2)) -> activate#(X1) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    activate#(n__div(X1,X2)) -> activate#(X1) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    activate#(n__div(X1,X2)) -> activate#(X1) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    activate#(n__minus(X1,X2)) -> minus#(X1,X2) ->
    minus#(n__0(),Y) -> 0#()
    activate#(n__minus(X1,X2)) -> minus#(X1,X2) ->
    minus#(n__s(X),n__s(Y)) -> activate#(Y)
    activate#(n__minus(X1,X2)) -> minus#(X1,X2) ->
    minus#(n__s(X),n__s(Y)) -> activate#(X)
    activate#(n__minus(X1,X2)) -> minus#(X1,X2) ->
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__0()) -> 0#()
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    minus#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__0()) -> 0#()
    minus#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    minus#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    minus#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    minus#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    minus#(n__s(X),n__s(Y)) -> activate#(X) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    minus#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__0()) -> 0#()
    minus#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__s(X)) -> activate#(X)
    minus#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__s(X)) -> s#(activate(X))
    minus#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> activate#(X1)
    minus#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
    minus#(n__s(X),n__s(Y)) -> activate#(Y) ->
    activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y)) ->
    minus#(n__0(),Y) -> 0#()
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y)) ->
    minus#(n__s(X),n__s(Y)) -> activate#(Y)
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y)) ->
    minus#(n__s(X),n__s(Y)) -> activate#(X)
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y)) ->
    minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
   Restore Modifier:
    DPs:
     minus#(n__0(),Y) -> 0#()
     minus#(n__s(X),n__s(Y)) -> activate#(Y)
     minus#(n__s(X),n__s(Y)) -> activate#(X)
     minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
     geq#(n__s(X),n__s(Y)) -> activate#(Y)
     geq#(n__s(X),n__s(Y)) -> activate#(X)
     geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
     div#(s(X),n__s(Y)) -> activate#(Y)
     div#(s(X),n__s(Y)) -> geq#(X,activate(Y))
     div#(s(X),n__s(Y)) ->
     if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
     if#(true(),X,Y) -> activate#(X)
     if#(false(),X,Y) -> activate#(Y)
     activate#(n__0()) -> 0#()
     activate#(n__s(X)) -> activate#(X)
     activate#(n__s(X)) -> s#(activate(X))
     activate#(n__div(X1,X2)) -> activate#(X1)
     activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
     activate#(n__minus(X1,X2)) -> minus#(X1,X2)
    TRS:
     minus(n__0(),Y) -> 0()
     minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
     geq(X,n__0()) -> true()
     geq(n__0(),n__s(Y)) -> false()
     geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
     div(0(),n__s(Y)) -> 0()
     div(s(X),n__s(Y)) ->
     if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
     if(true(),X,Y) -> activate(X)
     if(false(),X,Y) -> activate(Y)
     0() -> n__0()
     s(X) -> n__s(X)
     div(X1,X2) -> n__div(X1,X2)
     minus(X1,X2) -> n__minus(X1,X2)
     activate(n__0()) -> 0()
     activate(n__s(X)) -> s(activate(X))
     activate(n__div(X1,X2)) -> div(activate(X1),X2)
     activate(n__minus(X1,X2)) -> minus(X1,X2)
     activate(X) -> X
    SCC Processor:
     #sccs: 1
     #rules: 15
     #arcs: 73/324
     DPs:
      if#(false(),X,Y) -> activate#(Y)
      activate#(n__minus(X1,X2)) -> minus#(X1,X2)
      minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
      minus#(n__s(X),n__s(Y)) -> activate#(X)
      activate#(n__div(X1,X2)) -> div#(activate(X1),X2)
      div#(s(X),n__s(Y)) ->
      if#(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
      if#(true(),X,Y) -> activate#(X)
      activate#(n__div(X1,X2)) -> activate#(X1)
      activate#(n__s(X)) -> activate#(X)
      div#(s(X),n__s(Y)) -> geq#(X,activate(Y))
      geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
      geq#(n__s(X),n__s(Y)) -> activate#(X)
      geq#(n__s(X),n__s(Y)) -> activate#(Y)
      div#(s(X),n__s(Y)) -> activate#(Y)
      minus#(n__s(X),n__s(Y)) -> activate#(Y)
     TRS:
      minus(n__0(),Y) -> 0()
      minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
      geq(X,n__0()) -> true()
      geq(n__0(),n__s(Y)) -> false()
      geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
      div(0(),n__s(Y)) -> 0()
      div(s(X),n__s(Y)) ->
      if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
      if(true(),X,Y) -> activate(X)
      if(false(),X,Y) -> activate(Y)
      0() -> n__0()
      s(X) -> n__s(X)
      div(X1,X2) -> n__div(X1,X2)
      minus(X1,X2) -> n__minus(X1,X2)
      activate(n__0()) -> 0()
      activate(n__s(X)) -> s(activate(X))
      activate(n__div(X1,X2)) -> div(activate(X1),X2)
      activate(n__minus(X1,X2)) -> minus(X1,X2)
      activate(X) -> X
     Open