MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict 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(exp(ALPHA, BETA)) ->
    plus(times(BETA, times(exp(ALPHA, minus(BETA, one())), dx(ALPHA))),
         times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))
  , dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following weak dependency pairs:

Strict DPs:
  { dx^#(X) -> c_1()
  , dx^#(a()) -> c_2()
  , dx^#(plus(ALPHA, BETA)) -> c_3(dx^#(ALPHA), dx^#(BETA))
  , dx^#(times(ALPHA, BETA)) -> c_4(dx^#(ALPHA), dx^#(BETA))
  , dx^#(minus(ALPHA, BETA)) -> c_5(dx^#(ALPHA), dx^#(BETA))
  , dx^#(neg(ALPHA)) -> c_6(dx^#(ALPHA))
  , dx^#(div(ALPHA, BETA)) -> c_7(dx^#(ALPHA), dx^#(BETA))
  , dx^#(exp(ALPHA, BETA)) -> c_8(dx^#(ALPHA), dx^#(BETA))
  , dx^#(ln(ALPHA)) -> c_9(dx^#(ALPHA)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { dx^#(X) -> c_1()
  , dx^#(a()) -> c_2()
  , dx^#(plus(ALPHA, BETA)) -> c_3(dx^#(ALPHA), dx^#(BETA))
  , dx^#(times(ALPHA, BETA)) -> c_4(dx^#(ALPHA), dx^#(BETA))
  , dx^#(minus(ALPHA, BETA)) -> c_5(dx^#(ALPHA), dx^#(BETA))
  , dx^#(neg(ALPHA)) -> c_6(dx^#(ALPHA))
  , dx^#(div(ALPHA, BETA)) -> c_7(dx^#(ALPHA), dx^#(BETA))
  , dx^#(exp(ALPHA, BETA)) -> c_8(dx^#(ALPHA), dx^#(BETA))
  , dx^#(ln(ALPHA)) -> c_9(dx^#(ALPHA)) }
Strict 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(exp(ALPHA, BETA)) ->
    plus(times(BETA, times(exp(ALPHA, minus(BETA, one())), dx(ALPHA))),
         times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))
  , dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { dx^#(X) -> c_1()
  , dx^#(a()) -> c_2()
  , dx^#(plus(ALPHA, BETA)) -> c_3(dx^#(ALPHA), dx^#(BETA))
  , dx^#(times(ALPHA, BETA)) -> c_4(dx^#(ALPHA), dx^#(BETA))
  , dx^#(minus(ALPHA, BETA)) -> c_5(dx^#(ALPHA), dx^#(BETA))
  , dx^#(neg(ALPHA)) -> c_6(dx^#(ALPHA))
  , dx^#(div(ALPHA, BETA)) -> c_7(dx^#(ALPHA), dx^#(BETA))
  , dx^#(exp(ALPHA, BETA)) -> c_8(dx^#(ALPHA), dx^#(BETA))
  , dx^#(ln(ALPHA)) -> c_9(dx^#(ALPHA)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2},
  Uargs(c_6) = {1}, Uargs(c_7) = {1, 2}, Uargs(c_8) = {1, 2},
  Uargs(c_9) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

              [a] = [1]                  
                                         
   [plus](x1, x2) = [1] x1 + [1] x2 + [1]
                                         
  [times](x1, x2) = [1] x1 + [1] x2 + [1]
                                         
  [minus](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
        [neg](x1) = [1] x1 + [2]         
                                         
    [div](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
    [exp](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
         [ln](x1) = [1] x1 + [2]         
                                         
       [dx^#](x1) = [1]                  
                                         
            [c_1] = [0]                  
                                         
            [c_2] = [0]                  
                                         
    [c_3](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
    [c_4](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
    [c_5](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
        [c_6](x1) = [1] x1 + [0]         
                                         
    [c_7](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
    [c_8](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
        [c_9](x1) = [1] x1 + [0]         

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { dx^#(plus(ALPHA, BETA)) -> c_3(dx^#(ALPHA), dx^#(BETA))
  , dx^#(times(ALPHA, BETA)) -> c_4(dx^#(ALPHA), dx^#(BETA))
  , dx^#(minus(ALPHA, BETA)) -> c_5(dx^#(ALPHA), dx^#(BETA))
  , dx^#(neg(ALPHA)) -> c_6(dx^#(ALPHA))
  , dx^#(div(ALPHA, BETA)) -> c_7(dx^#(ALPHA), dx^#(BETA))
  , dx^#(exp(ALPHA, BETA)) -> c_8(dx^#(ALPHA), dx^#(BETA))
  , dx^#(ln(ALPHA)) -> c_9(dx^#(ALPHA)) }
Weak DPs:
  { dx^#(X) -> c_1()
  , dx^#(a()) -> c_2() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ dx^#(X) -> c_1()
, dx^#(a()) -> c_2() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { dx^#(plus(ALPHA, BETA)) -> c_3(dx^#(ALPHA), dx^#(BETA))
  , dx^#(times(ALPHA, BETA)) -> c_4(dx^#(ALPHA), dx^#(BETA))
  , dx^#(minus(ALPHA, BETA)) -> c_5(dx^#(ALPHA), dx^#(BETA))
  , dx^#(neg(ALPHA)) -> c_6(dx^#(ALPHA))
  , dx^#(div(ALPHA, BETA)) -> c_7(dx^#(ALPHA), dx^#(BETA))
  , dx^#(exp(ALPHA, BETA)) -> c_8(dx^#(ALPHA), dx^#(BETA))
  , dx^#(ln(ALPHA)) -> c_9(dx^#(ALPHA)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..