YES(?,ELEMENTARY)

'epo* (timeout of 60.0 seconds)'
--------------------------------
Answer:           YES(?,ELEMENTARY)
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  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))))}

Proof Output:    
  Strict Rules in Predicative Notation:
   {  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;))))}
  Safe Mapping:
   safe(dx) = {}, safe(one) = {}, safe(a) = {}, safe(zero) = {},
   safe(plus) = {1, 2}, safe(times) = {1, 2}, safe(minus) = {1, 2},
   safe(neg) = {1}, safe(div) = {1, 2}, safe(exp) = {1, 2},
   safe(two) = {}, safe(ln) = {1}
  Argument Permutation:
   mu(dx) = [1]
  Precedence:
   dx ~ dx