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
  TDG 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
   graph:
    from#(X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    from#(X) -> activate#(X) -> activate#(n__from(X)) -> from#(X)
    from#(X) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(X1,X2)
    from#(X) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(X)
    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__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Y) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Y) ->
    activate#(n__from(X)) -> from#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Y) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Y) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__from(X)) -> from#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    add#(s(X),Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(X)
    add#(s(X),Y) -> activate#(Y) -> activate#(n__from(X)) -> from#(X)
    add#(s(X),Y) -> activate#(Y) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    add#(s(X),Y) -> activate#(Y) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    add#(s(X),Y) -> activate#(X) -> activate#(n__from(X)) -> from#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    add#(0(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    add#(0(),X) -> activate#(X) -> activate#(n__from(X)) -> from#(X)
    add#(0(),X) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    add#(0(),X) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    if#(false(),X,Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(X)
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__from(X)) -> from#(X)
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    if#(true(),X,Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__from(X)) -> from#(X)
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    if#(true(),X,Y) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    activate#(n__from(X)) -> from#(X) ->
    from#(X) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(Y)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> s#(n__add(activate(X),activate(Y)))
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> activate#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> activate#(Y)
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(0(),X) -> activate#(X)
    and#(true(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    and#(true(),X) -> activate#(X) ->
    activate#(n__from(X)) -> from#(X)
    and#(true(),X) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    and#(true(),X) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
   SCC Processor:
    #sccs: 1
    #rules: 10
    #arcs: 48/225
    DPs:
     from#(X) -> activate#(X)
     activate#(n__add(X1,X2)) -> add#(X1,X2)
     add#(0(),X) -> activate#(X)
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     activate#(n__from(X)) -> from#(X)
     first#(s(X),cons(Y,Z)) -> activate#(X)
     first#(s(X),cons(Y,Z)) -> activate#(Y)
     add#(s(X),Y) -> activate#(Y)
     add#(s(X),Y) -> activate#(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
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [from#](x0) = 1/2x0 + 1/2,
      
      [first#](x0, x1) = 1/2x0 + x1 + 1/2,
      
      [add#](x0, x1) = 1/2x0 + 1/2x1 + 1/2,
      
      [activate#](x0) = 1/2x0,
      
      [n__from](x0) = x0 + 2,
      
      [n__s](x0) = x0,
      
      [from](x0) = x0 + 2,
      
      [n__first](x0, x1) = x0 + 2x1 + 2,
      
      [cons](x0, x1) = 1/2x0 + 1/2x1,
      
      [nil] = 0,
      
      [first](x0, x1) = x0 + 2x1 + 2,
      
      [n__add](x0, x1) = x0 + 2x1 + 2,
      
      [s](x0) = x0,
      
      [add](x0, x1) = x0 + 2x1 + 2,
      
      [0] = 0,
      
      [if](x0, x1, x2) = 1/2x0 + 2x1 + x2,
      
      [false] = 1,
      
      [activate](x0) = x0,
      
      [and](x0, x1) = 2x0 + 2x1 + 7/2,
      
      [true] = 3
     orientation:
      from#(X) = 1/2X + 1/2 >= 1/2X = activate#(X)
      
      activate#(n__add(X1,X2)) = 1/2X1 + X2 + 1 >= 1/2X1 + 1/2X2 + 1/2 = add#(X1,X2)
      
      add#(0(),X) = 1/2X + 1/2 >= 1/2X = activate#(X)
      
      activate#(n__first(X1,X2)) = 1/2X1 + X2 + 1 >= 1/2X1 + X2 + 1/2 = first#(X1,X2)
      
      first#(s(X),cons(Y,Z)) = 1/2X + 1/2Y + 1/2Z + 1/2 >= 1/2Z = activate#(Z)
      
      activate#(n__from(X)) = 1/2X + 1 >= 1/2X + 1/2 = from#(X)
      
      first#(s(X),cons(Y,Z)) = 1/2X + 1/2Y + 1/2Z + 1/2 >= 1/2X = activate#(X)
      
      first#(s(X),cons(Y,Z)) = 1/2X + 1/2Y + 1/2Z + 1/2 >= 1/2Y = activate#(Y)
      
      add#(s(X),Y) = 1/2X + 1/2Y + 1/2 >= 1/2Y = activate#(Y)
      
      add#(s(X),Y) = 1/2X + 1/2Y + 1/2 >= 1/2X = activate#(X)
      
      and(true(),X) = 2X + 19/2 >= X = activate(X)
      
      and(false(),Y) = 2Y + 11/2 >= 1 = false()
      
      if(true(),X,Y) = 2X + Y + 3/2 >= X = activate(X)
      
      if(false(),X,Y) = 2X + Y + 1/2 >= Y = activate(Y)
      
      add(0(),X) = 2X + 2 >= X = activate(X)
      
      add(s(X),Y) = X + 2Y + 2 >= X + 2Y + 2 = s(n__add(activate(X),activate(Y)))
      
      first(0(),X) = 2X + 2 >= 0 = nil()
      
      first(s(X),cons(Y,Z)) = X + Y + Z + 2 >= 1/2X + 1/2Y + Z + 1 = cons(activate(Y),n__first(activate(X),activate(Z)))
      
      from(X) = X + 2 >= X + 1 = cons(activate(X),n__from(n__s(activate(X))))
      
      add(X1,X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = n__add(X1,X2)
      
      first(X1,X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = n__first(X1,X2)
      
      from(X) = X + 2 >= X + 2 = n__from(X)
      
      s(X) = X >= X = n__s(X)
      
      activate(n__add(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = add(X1,X2)
      
      activate(n__first(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = first(X1,X2)
      
      activate(n__from(X)) = X + 2 >= X + 2 = from(X)
      
      activate(n__s(X)) = X >= X = s(X)
      
      activate(X) = X >= X = X
     problem:
      DPs:
       
      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
     Qed