TRS:
 {                 dx(X) -> one(),
                 dx(a()) -> zero(),
   dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)),
  dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))),
  dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)),
          dx(neg(ALPHA)) -> neg(dx(ALPHA)),
    dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two())))),
           dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA),
    dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one())), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))}
 RPO Product:
  Quasi-Precedence:
  empty
  
  Qed


TRS:
 {                 dx(X) -> one(),
                 dx(a()) -> zero(),
   dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)),
  dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))),
  dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)),
          dx(neg(ALPHA)) -> neg(dx(ALPHA)),
    dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two())))),
           dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA),
    dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one())), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))}
 Cdiprover:
  Interpretation class: quasisimplemixed
  Complexity bound: POLYTIME COMPUTABLE IF RPO-TERMINATING
  ln(X12) = + 1*X12 + 1
  two = + 0
  exp(X11, X10) = + 1*X10 + 1*X11 + 3
  div(X9, X8) = + 1*X8 + 1*X9 + 3
  neg(X7) = + 1*X7 + 1
  minus(X6, X5) = + 1*X5 + 1*X6 + 1
  times(X4, X3) = + 1*X3 + 1*X4 + 1
  plus(X2, X1) = + 1*X1 + 1*X2 + 1
  a = + 0
  zero = + 0
  dx(X0) = + 1*X0^2 + 0 + 2*X0
  one = + 0
  
  Qed