YES

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(X1,X2)
 activate(n__first(X1,X2)) -> first(X1,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)) -> add#(X1,X2)
   activate#(n__first(X1,X2)) -> first#(X1,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(X1,X2)
   activate(n__first(X1,X2)) -> first(X1,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)) -> add#(X1,X2)
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    activate#(n__from(X)) -> from#(X)
    activate#(n__s(X)) -> s#(X)
   TRS:
    activate(n__add(X1,X2)) -> add(X1,X2)
    activate(n__first(X1,X2)) -> first(X1,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)
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     
    interpretation:
     [from#](x0) = 1x0,
     
     [first#](x0, x1) = 1x0 + 1x1,
     
     [s#](x0) = 0,
     
     [add#](x0, x1) = 4x0 + 4x1,
     
     [if#](x0, x1, x2) = x0 + 8x1 + 1x2 + 0,
     
     [activate#](x0) = x0,
     
     [and#](x0, x1) = 3x0 + 8x1 + 0,
     
     [n__from](x0) = 2x0,
     
     [n__s](x0) = 4x0 + 1,
     
     [from](x0) = x0 + 2,
     
     [n__first](x0, x1) = 2x0 + 4x1 + 1,
     
     [cons](x0, x1) = 4x0 + x1,
     
     [nil] = 1,
     
     [first](x0, x1) = 5x0 + 1x1 + 8,
     
     [n__add](x0, x1) = 5x0 + 5x1 + 0,
     
     [s](x0) = x0 + 1,
     
     [add](x0, x1) = x0 + x1 + 0,
     
     [0] = 1,
     
     [false] = 2,
     
     [activate](x0) = x0 + 0,
     
     [true] = 3
    orientation:
     and#(true(),X) = 8X + 6 >= X = activate#(X)
     
     if#(true(),X,Y) = 8X + 1Y + 3 >= X = activate#(X)
     
     if#(false(),X,Y) = 8X + 1Y + 2 >= Y = activate#(Y)
     
     add#(0(),X) = 4X + 5 >= X = activate#(X)
     
     add#(s(X),Y) = 4X + 4Y + 5 >= Y = activate#(Y)
     
     add#(s(X),Y) = 4X + 4Y + 5 >= X = activate#(X)
     
     add#(s(X),Y) = 4X + 4Y + 5 >= 0 = s#(n__add(activate(X),activate(Y)))
     
     first#(s(X),cons(Y,Z)) = 1X + 5Y + 1Z + 2 >= Z = activate#(Z)
     
     first#(s(X),cons(Y,Z)) = 1X + 5Y + 1Z + 2 >= X = activate#(X)
     
     first#(s(X),cons(Y,Z)) = 1X + 5Y + 1Z + 2 >= Y = activate#(Y)
     
     from#(X) = 1X >= X = activate#(X)
     
     activate#(n__add(X1,X2)) = 5X1 + 5X2 + 0 >= 4X1 + 4X2 = add#(X1,X2)
     
     activate#(n__first(X1,X2)) = 2X1 + 4X2 + 1 >= 1X1 + 1X2 = first#(X1,X2)
     
     activate#(n__from(X)) = 2X >= 1X = from#(X)
     
     activate#(n__s(X)) = 4X + 1 >= 0 = s#(X)
     
     activate(n__add(X1,X2)) = 5X1 + 5X2 + 0 >= X1 + X2 + 0 = add(X1,X2)
     
     activate(n__first(X1,X2)) = 2X1 + 4X2 + 1 >= 5X1 + 1X2 + 8 = first(X1,X2)
     
     activate(n__from(X)) = 2X + 0 >= X + 2 = from(X)
     
     activate(n__s(X)) = 4X + 1 >= X + 1 = s(X)
     
     activate(X) = X + 0 >= X = X
     
     add(0(),X) = X + 1 >= X + 0 = activate(X)
     
     add(s(X),Y) = X + Y + 1 >= 5X + 5Y + 5 = s(n__add(activate(X),activate(Y)))
     
     add(X1,X2) = X1 + X2 + 0 >= 5X1 + 5X2 + 0 = n__add(X1,X2)
     
     first(0(),X) = 1X + 8 >= 1 = nil()
     
     first(s(X),cons(Y,Z)) = 5X + 5Y + 1Z + 8 >= 2X + 4Y + 4Z + 4 = cons(activate(Y),n__first(activate(X),activate(Z)))
     
     first(X1,X2) = 5X1 + 1X2 + 8 >= 2X1 + 4X2 + 1 = n__first(X1,X2)
     
     from(X) = X + 2 >= 6X + 6 = cons(activate(X),n__from(n__s(activate(X))))
     
     from(X) = X + 2 >= 2X = n__from(X)
     
     s(X) = X + 1 >= 4X + 1 = n__s(X)
    problem:
     DPs:
      
     TRS:
      activate(n__add(X1,X2)) -> add(X1,X2)
      activate(n__first(X1,X2)) -> first(X1,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)
    Qed