MAYBE

Problem:
 first(0(),X) -> nil()
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
 from(X) -> cons(X,n__from(s(X)))
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> n__from(X)
 activate(n__first(X1,X2)) -> first(X1,X2)
 activate(n__from(X)) -> from(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   activate#(n__first(X1,X2)) -> first#(X1,X2)
   activate#(n__from(X)) -> from#(X)
  TRS:
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
   from(X) -> cons(X,n__from(s(X)))
   first(X1,X2) -> n__first(X1,X2)
   from(X) -> n__from(X)
   activate(n__first(X1,X2)) -> first(X1,X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X
  Usable Rule Processor:
   DPs:
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    activate#(n__from(X)) -> from#(X)
   TRS:
    
   ADG Processor:
    DPs:
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     activate#(n__from(X)) -> from#(X)
    TRS:
     
    graph:
     activate#(n__first(X1,X2)) -> first#(X1,X2) ->
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     first#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(X)
    Restore Modifier:
     DPs:
      first#(s(X),cons(Y,Z)) -> activate#(Z)
      activate#(n__first(X1,X2)) -> first#(X1,X2)
      activate#(n__from(X)) -> from#(X)
     TRS:
      first(0(),X) -> nil()
      first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
      from(X) -> cons(X,n__from(s(X)))
      first(X1,X2) -> n__first(X1,X2)
      from(X) -> n__from(X)
      activate(n__first(X1,X2)) -> first(X1,X2)
      activate(n__from(X)) -> from(X)
      activate(X) -> X
     SCC Processor:
      #sccs: 1
      #rules: 2
      #arcs: 3/9
      DPs:
       activate#(n__first(X1,X2)) -> first#(X1,X2)
       first#(s(X),cons(Y,Z)) -> activate#(Z)
      TRS:
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       from(X) -> cons(X,n__from(s(X)))
       first(X1,X2) -> n__first(X1,X2)
       from(X) -> n__from(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(n__from(X)) -> from(X)
       activate(X) -> X
      Open