MAYBE

Problem:
 and(true(),X) -> activate(X)
 and(false(),Y) -> false()
 if(true(),X,Y) -> activate(X)
 if(false(),X,Y) -> activate(Y)
 add(0(),X) -> activate(X)
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 first(0(),X) -> nil()
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 add(X1,X2) -> n__add(X1,X2)
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   and#(true(),X) -> activate#(X)
   if#(true(),X,Y) -> activate#(X)
   if#(false(),X,Y) -> activate#(Y)
   add#(0(),X) -> activate#(X)
   add#(s(X),Y) -> activate#(Y)
   add#(s(X),Y) -> activate#(X)
   add#(s(X),Y) -> s#(n__add(activate(X),activate(Y)))
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   first#(s(X),cons(Y,Z)) -> activate#(X)
   first#(s(X),cons(Y,Z)) -> activate#(Y)
   from#(X) -> activate#(X)
   activate#(n__add(X1,X2)) -> activate#(X1)
   activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
   activate#(n__first(X1,X2)) -> activate#(X2)
   activate#(n__first(X1,X2)) -> activate#(X1)
   activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
   activate#(n__from(X)) -> from#(X)
   activate#(n__s(X)) -> s#(X)
  TRS:
   and(true(),X) -> activate(X)
   and(false(),Y) -> false()
   if(true(),X,Y) -> activate(X)
   if(false(),X,Y) -> activate(Y)
   add(0(),X) -> activate(X)
   add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
   from(X) -> cons(activate(X),n__from(n__s(activate(X))))
   add(X1,X2) -> n__add(X1,X2)
   first(X1,X2) -> n__first(X1,X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__add(X1,X2)) -> add(activate(X1),X2)
   activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
   activate(n__from(X)) -> from(X)
   activate(n__s(X)) -> s(X)
   activate(X) -> X
  Usable Rule Processor:
   DPs:
    and#(true(),X) -> activate#(X)
    if#(true(),X,Y) -> activate#(X)
    if#(false(),X,Y) -> activate#(Y)
    add#(0(),X) -> activate#(X)
    add#(s(X),Y) -> activate#(Y)
    add#(s(X),Y) -> activate#(X)
    add#(s(X),Y) -> s#(n__add(activate(X),activate(Y)))
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    first#(s(X),cons(Y,Z)) -> activate#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Y)
    from#(X) -> activate#(X)
    activate#(n__add(X1,X2)) -> activate#(X1)
    activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__from(X)) -> from#(X)
    activate#(n__s(X)) -> s#(X)
   TRS:
    f23(x,y) -> x
    f23(x,y) -> y
    activate(n__add(X1,X2)) -> add(activate(X1),X2)
    activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
    activate(n__from(X)) -> from(X)
    activate(n__s(X)) -> s(X)
    activate(X) -> X
    add(0(),X) -> activate(X)
    add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
    add(X1,X2) -> n__add(X1,X2)
    first(0(),X) -> nil()
    first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
    first(X1,X2) -> n__first(X1,X2)
    from(X) -> cons(activate(X),n__from(n__s(activate(X))))
    from(X) -> n__from(X)
    s(X) -> n__s(X)
   EDG Processor:
    DPs:
     and#(true(),X) -> activate#(X)
     if#(true(),X,Y) -> activate#(X)
     if#(false(),X,Y) -> activate#(Y)
     add#(0(),X) -> activate#(X)
     add#(s(X),Y) -> activate#(Y)
     add#(s(X),Y) -> activate#(X)
     add#(s(X),Y) -> s#(n__add(activate(X),activate(Y)))
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     first#(s(X),cons(Y,Z)) -> activate#(X)
     first#(s(X),cons(Y,Z)) -> activate#(Y)
     from#(X) -> activate#(X)
     activate#(n__add(X1,X2)) -> activate#(X1)
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     activate#(n__first(X1,X2)) -> activate#(X2)
     activate#(n__first(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     activate#(n__from(X)) -> from#(X)
     activate#(n__s(X)) -> s#(X)
    TRS:
     f23(x,y) -> x
     f23(x,y) -> y
     activate(n__add(X1,X2)) -> add(activate(X1),X2)
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(n__from(X)) -> from(X)
     activate(n__s(X)) -> s(X)
     activate(X) -> X
     add(0(),X) -> activate(X)
     add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
     add(X1,X2) -> n__add(X1,X2)
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
     first(X1,X2) -> n__first(X1,X2)
     from(X) -> cons(activate(X),n__from(n__s(activate(X))))
     from(X) -> n__from(X)
     s(X) -> n__s(X)
    graph:
     from#(X) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1)
     from#(X) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     from#(X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     from#(X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     from#(X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     from#(X) -> activate#(X) -> activate#(n__from(X)) -> from#(X)
     from#(X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__from(X)) -> from#(X)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__s(X)) -> s#(X)
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__from(X)) -> from#(X)
     first#(s(X),cons(Y,Z)) -> activate#(Y) ->
     activate#(n__s(X)) -> s#(X)
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__from(X)) -> from#(X)
     first#(s(X),cons(Y,Z)) -> activate#(X) ->
     activate#(n__s(X)) -> s#(X)
     add#(s(X),Y) -> activate#(Y) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     add#(s(X),Y) -> activate#(Y) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     add#(s(X),Y) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     add#(s(X),Y) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     add#(s(X),Y) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     add#(s(X),Y) -> activate#(Y) -> activate#(n__from(X)) -> from#(X)
     add#(s(X),Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(X)
     add#(s(X),Y) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     add#(s(X),Y) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     add#(s(X),Y) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     add#(s(X),Y) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     add#(s(X),Y) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     add#(s(X),Y) -> activate#(X) -> activate#(n__from(X)) -> from#(X)
     add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
     add#(0(),X) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     add#(0(),X) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     add#(0(),X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     add#(0(),X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     add#(0(),X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     add#(0(),X) -> activate#(X) -> activate#(n__from(X)) -> from#(X)
     add#(0(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__from(X)) -> from#(X)
     if#(false(),X,Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(X)
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__from(X)) -> from#(X)
     if#(true(),X,Y) -> activate#(X) ->
     activate#(n__s(X)) -> s#(X)
     activate#(n__from(X)) -> from#(X) ->
     from#(X) -> activate#(X)
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) ->
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) ->
     first#(s(X),cons(Y,Z)) -> activate#(X)
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) ->
     first#(s(X),cons(Y,Z)) -> activate#(Y)
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__from(X)) -> from#(X)
     activate#(n__first(X1,X2)) -> activate#(X2) ->
     activate#(n__s(X)) -> s#(X)
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__from(X)) -> from#(X)
     activate#(n__first(X1,X2)) -> activate#(X1) ->
     activate#(n__s(X)) -> s#(X)
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2) ->
     add#(0(),X) -> activate#(X)
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2) ->
     add#(s(X),Y) -> activate#(Y)
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2) ->
     add#(s(X),Y) -> activate#(X)
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2) ->
     add#(s(X),Y) -> s#(n__add(activate(X),activate(Y)))
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__from(X)) -> from#(X)
     activate#(n__add(X1,X2)) -> activate#(X1) ->
     activate#(n__s(X)) -> s#(X)
     and#(true(),X) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> activate#(X1)
     and#(true(),X) -> activate#(X) ->
     activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
     and#(true(),X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X2)
     and#(true(),X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> activate#(X1)
     and#(true(),X) -> activate#(X) ->
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     and#(true(),X) -> activate#(X) ->
     activate#(n__from(X)) -> from#(X)
     and#(true(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    Restore Modifier:
     DPs:
      and#(true(),X) -> activate#(X)
      if#(true(),X,Y) -> activate#(X)
      if#(false(),X,Y) -> activate#(Y)
      add#(0(),X) -> activate#(X)
      add#(s(X),Y) -> activate#(Y)
      add#(s(X),Y) -> activate#(X)
      add#(s(X),Y) -> s#(n__add(activate(X),activate(Y)))
      first#(s(X),cons(Y,Z)) -> activate#(Z)
      first#(s(X),cons(Y,Z)) -> activate#(X)
      first#(s(X),cons(Y,Z)) -> activate#(Y)
      from#(X) -> activate#(X)
      activate#(n__add(X1,X2)) -> activate#(X1)
      activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
      activate#(n__first(X1,X2)) -> activate#(X2)
      activate#(n__first(X1,X2)) -> activate#(X1)
      activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
      activate#(n__from(X)) -> from#(X)
      activate#(n__s(X)) -> s#(X)
     TRS:
      and(true(),X) -> activate(X)
      and(false(),Y) -> false()
      if(true(),X,Y) -> activate(X)
      if(false(),X,Y) -> activate(Y)
      add(0(),X) -> activate(X)
      add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
      first(0(),X) -> nil()
      first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
      from(X) -> cons(activate(X),n__from(n__s(activate(X))))
      add(X1,X2) -> n__add(X1,X2)
      first(X1,X2) -> n__first(X1,X2)
      from(X) -> n__from(X)
      s(X) -> n__s(X)
      activate(n__add(X1,X2)) -> add(activate(X1),X2)
      activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
      activate(n__from(X)) -> from(X)
      activate(n__s(X)) -> s(X)
      activate(X) -> X
     SCC Processor:
      #sccs: 1
      #rules: 13
      #arcs: 99/324
      DPs:
       from#(X) -> activate#(X)
       activate#(n__from(X)) -> from#(X)
       activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
       first#(s(X),cons(Y,Z)) -> activate#(Y)
       activate#(n__first(X1,X2)) -> activate#(X1)
       activate#(n__first(X1,X2)) -> activate#(X2)
       activate#(n__add(X1,X2)) -> add#(activate(X1),X2)
       add#(s(X),Y) -> activate#(X)
       activate#(n__add(X1,X2)) -> activate#(X1)
       add#(s(X),Y) -> activate#(Y)
       add#(0(),X) -> activate#(X)
       first#(s(X),cons(Y,Z)) -> activate#(X)
       first#(s(X),cons(Y,Z)) -> activate#(Z)
      TRS:
       and(true(),X) -> activate(X)
       and(false(),Y) -> false()
       if(true(),X,Y) -> activate(X)
       if(false(),X,Y) -> activate(Y)
       add(0(),X) -> activate(X)
       add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
       from(X) -> cons(activate(X),n__from(n__s(activate(X))))
       add(X1,X2) -> n__add(X1,X2)
       first(X1,X2) -> n__first(X1,X2)
       from(X) -> n__from(X)
       s(X) -> n__s(X)
       activate(n__add(X1,X2)) -> add(activate(X1),X2)
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(n__from(X)) -> from(X)
       activate(n__s(X)) -> s(X)
       activate(X) -> X
      Open