MAYBE

Problem:
 and(true(),X) -> X
 and(false(),Y) -> false()
 if(true(),X,Y) -> X
 if(false(),X,Y) -> Y
 add(0(),X) -> X
 add(s(X),Y) -> s(add(X,Y))
 first(0(),X) -> nil()
 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
 from(X) -> cons(X,from(s(X)))

Proof:
 DP Processor:
  DPs:
   add#(s(X),Y) -> add#(X,Y)
   first#(s(X),cons(Y,Z)) -> first#(X,Z)
   from#(X) -> from#(s(X))
  TRS:
   and(true(),X) -> X
   and(false(),Y) -> false()
   if(true(),X,Y) -> X
   if(false(),X,Y) -> Y
   add(0(),X) -> X
   add(s(X),Y) -> s(add(X,Y))
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
   from(X) -> cons(X,from(s(X)))
  Usable Rule Processor:
   DPs:
    add#(s(X),Y) -> add#(X,Y)
    first#(s(X),cons(Y,Z)) -> first#(X,Z)
    from#(X) -> from#(s(X))
   TRS:
    
   ADG Processor:
    DPs:
     add#(s(X),Y) -> add#(X,Y)
     first#(s(X),cons(Y,Z)) -> first#(X,Z)
     from#(X) -> from#(s(X))
    TRS:
     
    graph:
     from#(X) -> from#(s(X)) -> from#(X) -> from#(s(X))
     first#(s(X),cons(Y,Z)) -> first#(X,Z) ->
     first#(s(X),cons(Y,Z)) -> first#(X,Z)
     add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> add#(X,Y)
    Restore Modifier:
     DPs:
      add#(s(X),Y) -> add#(X,Y)
      first#(s(X),cons(Y,Z)) -> first#(X,Z)
      from#(X) -> from#(s(X))
     TRS:
      and(true(),X) -> X
      and(false(),Y) -> false()
      if(true(),X,Y) -> X
      if(false(),X,Y) -> Y
      add(0(),X) -> X
      add(s(X),Y) -> s(add(X,Y))
      first(0(),X) -> nil()
      first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
      from(X) -> cons(X,from(s(X)))
     SCC Processor:
      #sccs: 3
      #rules: 3
      #arcs: 3/9
      DPs:
       add#(s(X),Y) -> add#(X,Y)
      TRS:
       and(true(),X) -> X
       and(false(),Y) -> false()
       if(true(),X,Y) -> X
       if(false(),X,Y) -> Y
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
       from(X) -> cons(X,from(s(X)))
      Open
      
      DPs:
       first#(s(X),cons(Y,Z)) -> first#(X,Z)
      TRS:
       and(true(),X) -> X
       and(false(),Y) -> false()
       if(true(),X,Y) -> X
       if(false(),X,Y) -> Y
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
       from(X) -> cons(X,from(s(X)))
      Open
      
      DPs:
       from#(X) -> from#(s(X))
      TRS:
       and(true(),X) -> X
       and(false(),Y) -> false()
       if(true(),X,Y) -> X
       if(false(),X,Y) -> Y
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
       from(X) -> cons(X,from(s(X)))
      Open