* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            dx(X) -> one()
            dx(a()) -> zero()
            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)
            dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
            dx(neg(ALPHA)) -> neg(dx(ALPHA))
            dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
            dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
        - Signature:
            {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            dx(X) -> one()
            dx(a()) -> zero()
            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)
            dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
            dx(neg(ALPHA)) -> neg(dx(ALPHA))
            dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
            dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
        - Signature:
            {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          dx(x){x -> div(x,y)} =
            dx(div(x,y)) ->^+ minus(div(dx(x),y),times(x,div(dx(y),exp(y,two()))))
              = C[dx(x) = dx(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            dx(X) -> one()
            dx(a()) -> zero()
            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)
            dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
            dx(neg(ALPHA)) -> neg(dx(ALPHA))
            dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
            dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
        - Signature:
            {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          dx#(X) -> c_1()
          dx#(a()) -> c_2()
          dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
          dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
          dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
          dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(X) -> c_1()
            dx#(a()) -> c_2()
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Strict TRS:
            dx(X) -> one()
            dx(a()) -> zero()
            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)
            dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
            dx(neg(ALPHA)) -> neg(dx(ALPHA))
            dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
            dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          dx#(X) -> c_1()
          dx#(a()) -> c_2()
          dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
          dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
          dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
          dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(X) -> c_1()
            dx#(a()) -> c_2()
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3,4,5,6,7,8,9}.
        Here rules are labelled as follows:
          1: dx#(X) -> c_1()
          2: dx#(a()) -> c_2()
          3: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
          4: dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
          5: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
          6: dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          7: dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          8: dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          9: dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(X) -> c_1()
            dx#(a()) -> c_2()
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(a()) -> c_2():9
             -->_1 dx#(a()) -> c_2():9
             -->_2 dx#(X) -> c_1():8
             -->_1 dx#(X) -> c_1():8
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          2:S:dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(a()) -> c_2():9
             -->_1 dx#(a()) -> c_2():9
             -->_2 dx#(X) -> c_1():8
             -->_1 dx#(X) -> c_1():8
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          3:S:dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(a()) -> c_2():9
             -->_1 dx#(X) -> c_1():8
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          4:S:dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(a()) -> c_2():9
             -->_1 dx#(a()) -> c_2():9
             -->_2 dx#(X) -> c_1():8
             -->_1 dx#(X) -> c_1():8
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          5:S:dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(a()) -> c_2():9
             -->_1 dx#(X) -> c_1():8
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          6:S:dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(a()) -> c_2():9
             -->_1 dx#(a()) -> c_2():9
             -->_2 dx#(X) -> c_1():8
             -->_1 dx#(X) -> c_1():8
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          7:S:dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(a()) -> c_2():9
             -->_1 dx#(a()) -> c_2():9
             -->_2 dx#(X) -> c_1():8
             -->_1 dx#(X) -> c_1():8
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          8:W:dx#(X) -> c_1()
             
          
          9:W:dx#(a()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: dx#(X) -> c_1()
          9: dx#(a()) -> c_2()
** Step 1.b:5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          4: dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          6: dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {dx#}
        TcT has computed the following interpretation:
              p(a) = [0]                   
            p(div) = [1] x1 + [1] x2 + [0] 
             p(dx) = [2]                   
            p(exp) = [1] x1 + [1] x2 + [0] 
             p(ln) = [1] x1 + [0]          
          p(minus) = [1] x1 + [1] x2 + [4] 
            p(neg) = [1] x1 + [0]          
            p(one) = [2]                   
           p(plus) = [1] x1 + [1] x2 + [4] 
          p(times) = [1] x1 + [1] x2 + [0] 
            p(two) = [1]                   
           p(zero) = [0]                   
            p(dx#) = [4] x1 + [0]          
            p(c_1) = [0]                   
            p(c_2) = [0]                   
            p(c_3) = [1] x1 + [1] x2 + [0] 
            p(c_4) = [1] x1 + [1] x2 + [0] 
            p(c_5) = [1] x1 + [0]          
            p(c_6) = [1] x1 + [1] x2 + [12]
            p(c_7) = [1] x1 + [0]          
            p(c_8) = [1] x1 + [1] x2 + [14]
            p(c_9) = [1] x1 + [1] x2 + [0] 
        
        Following rules are strictly oriented:
        dx#(minus(ALPHA,BETA)) = [4] ALPHA + [4] BETA + [16]
                               > [4] ALPHA + [4] BETA + [12]
                               = c_6(dx#(ALPHA),dx#(BETA))  
        
         dx#(plus(ALPHA,BETA)) = [4] ALPHA + [4] BETA + [16]
                               > [4] ALPHA + [4] BETA + [14]
                               = c_8(dx#(ALPHA),dx#(BETA))  
        
        
        Following rules are (at-least) weakly oriented:
          dx#(div(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [0]
                               >= [4] ALPHA + [4] BETA + [0]
                               =  c_3(dx#(ALPHA),dx#(BETA)) 
        
          dx#(exp(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [0]
                               >= [4] ALPHA + [4] BETA + [0]
                               =  c_4(dx#(ALPHA),dx#(BETA)) 
        
                dx#(ln(ALPHA)) =  [4] ALPHA + [0]           
                               >= [4] ALPHA + [0]           
                               =  c_5(dx#(ALPHA))           
        
               dx#(neg(ALPHA)) =  [4] ALPHA + [0]           
                               >= [4] ALPHA + [0]           
                               =  c_7(dx#(ALPHA))           
        
        dx#(times(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [0]
                               >= [4] ALPHA + [4] BETA + [0]
                               =  c_9(dx#(ALPHA),dx#(BETA)) 
        
*** Step 1.b:5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 1.b:5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {dx#}
        TcT has computed the following interpretation:
              p(a) = [2]                  
            p(div) = [1] x1 + [1] x2 + [2]
             p(dx) = [1]                  
            p(exp) = [1] x1 + [1] x2 + [0]
             p(ln) = [1] x1 + [0]         
          p(minus) = [1] x1 + [1] x2 + [1]
            p(neg) = [1] x1 + [0]         
            p(one) = [0]                  
           p(plus) = [1] x1 + [1] x2 + [0]
          p(times) = [1] x1 + [1] x2 + [0]
            p(two) = [1]                  
           p(zero) = [1]                  
            p(dx#) = [8] x1 + [0]         
            p(c_1) = [0]                  
            p(c_2) = [1]                  
            p(c_3) = [1] x1 + [1] x2 + [1]
            p(c_4) = [1] x1 + [1] x2 + [0]
            p(c_5) = [1] x1 + [0]         
            p(c_6) = [1] x1 + [1] x2 + [5]
            p(c_7) = [1] x1 + [0]         
            p(c_8) = [1] x1 + [1] x2 + [0]
            p(c_9) = [1] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
        dx#(div(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                             > [8] ALPHA + [8] BETA + [1] 
                             = c_3(dx#(ALPHA),dx#(BETA))  
        
        
        Following rules are (at-least) weakly oriented:
          dx#(exp(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_4(dx#(ALPHA),dx#(BETA)) 
        
                dx#(ln(ALPHA)) =  [8] ALPHA + [0]           
                               >= [8] ALPHA + [0]           
                               =  c_5(dx#(ALPHA))           
        
        dx#(minus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [8]
                               >= [8] ALPHA + [8] BETA + [5]
                               =  c_6(dx#(ALPHA),dx#(BETA)) 
        
               dx#(neg(ALPHA)) =  [8] ALPHA + [0]           
                               >= [8] ALPHA + [0]           
                               =  c_7(dx#(ALPHA))           
        
         dx#(plus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_8(dx#(ALPHA),dx#(BETA)) 
        
        dx#(times(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_9(dx#(ALPHA),dx#(BETA)) 
        
**** Step 1.b:5.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:5.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          4: dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {dx#}
        TcT has computed the following interpretation:
              p(a) = [1]                   
            p(div) = [1] x1 + [1] x2 + [0] 
             p(dx) = [0]                   
            p(exp) = [1] x1 + [1] x2 + [0] 
             p(ln) = [1] x1 + [0]          
          p(minus) = [1] x1 + [1] x2 + [0] 
            p(neg) = [1] x1 + [0]          
            p(one) = [0]                   
           p(plus) = [1] x1 + [1] x2 + [0] 
          p(times) = [1] x1 + [1] x2 + [2] 
            p(two) = [0]                   
           p(zero) = [0]                   
            p(dx#) = [8] x1 + [0]          
            p(c_1) = [2]                   
            p(c_2) = [1]                   
            p(c_3) = [1] x1 + [1] x2 + [0] 
            p(c_4) = [1] x1 + [1] x2 + [0] 
            p(c_5) = [1] x1 + [0]          
            p(c_6) = [1] x1 + [1] x2 + [0] 
            p(c_7) = [1] x1 + [0]          
            p(c_8) = [1] x1 + [1] x2 + [0] 
            p(c_9) = [1] x1 + [1] x2 + [11]
        
        Following rules are strictly oriented:
        dx#(times(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                               > [8] ALPHA + [8] BETA + [11]
                               = c_9(dx#(ALPHA),dx#(BETA))  
        
        
        Following rules are (at-least) weakly oriented:
          dx#(div(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_3(dx#(ALPHA),dx#(BETA)) 
        
          dx#(exp(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_4(dx#(ALPHA),dx#(BETA)) 
        
                dx#(ln(ALPHA)) =  [8] ALPHA + [0]           
                               >= [8] ALPHA + [0]           
                               =  c_5(dx#(ALPHA))           
        
        dx#(minus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_6(dx#(ALPHA),dx#(BETA)) 
        
               dx#(neg(ALPHA)) =  [8] ALPHA + [0]           
                               >= [8] ALPHA + [0]           
                               =  c_7(dx#(ALPHA))           
        
         dx#(plus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_8(dx#(ALPHA),dx#(BETA)) 
        
***** Step 1.b:5.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:5.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
          
        The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {dx#}
        TcT has computed the following interpretation:
              p(a) = [1]                   
            p(div) = [1] x1 + [1] x2 + [10]
             p(dx) = [1] x1 + [1]          
            p(exp) = [1] x1 + [1] x2 + [2] 
             p(ln) = [1] x1 + [0]          
          p(minus) = [1] x1 + [1] x2 + [6] 
            p(neg) = [1] x1 + [0]          
            p(one) = [0]                   
           p(plus) = [1] x1 + [1] x2 + [13]
          p(times) = [1] x1 + [1] x2 + [8] 
            p(two) = [1]                   
           p(zero) = [2]                   
            p(dx#) = [1] x1 + [0]          
            p(c_1) = [1]                   
            p(c_2) = [0]                   
            p(c_3) = [1] x1 + [1] x2 + [10]
            p(c_4) = [1] x1 + [1] x2 + [1] 
            p(c_5) = [1] x1 + [0]          
            p(c_6) = [1] x1 + [1] x2 + [4] 
            p(c_7) = [1] x1 + [0]          
            p(c_8) = [1] x1 + [1] x2 + [12]
            p(c_9) = [1] x1 + [1] x2 + [8] 
        
        Following rules are strictly oriented:
        dx#(exp(ALPHA,BETA)) = [1] ALPHA + [1] BETA + [2]
                             > [1] ALPHA + [1] BETA + [1]
                             = c_4(dx#(ALPHA),dx#(BETA)) 
        
        
        Following rules are (at-least) weakly oriented:
          dx#(div(ALPHA,BETA)) =  [1] ALPHA + [1] BETA + [10]
                               >= [1] ALPHA + [1] BETA + [10]
                               =  c_3(dx#(ALPHA),dx#(BETA))  
        
                dx#(ln(ALPHA)) =  [1] ALPHA + [0]            
                               >= [1] ALPHA + [0]            
                               =  c_5(dx#(ALPHA))            
        
        dx#(minus(ALPHA,BETA)) =  [1] ALPHA + [1] BETA + [6] 
                               >= [1] ALPHA + [1] BETA + [4] 
                               =  c_6(dx#(ALPHA),dx#(BETA))  
        
               dx#(neg(ALPHA)) =  [1] ALPHA + [0]            
                               >= [1] ALPHA + [0]            
                               =  c_7(dx#(ALPHA))            
        
         dx#(plus(ALPHA,BETA)) =  [1] ALPHA + [1] BETA + [13]
                               >= [1] ALPHA + [1] BETA + [12]
                               =  c_8(dx#(ALPHA),dx#(BETA))  
        
        dx#(times(ALPHA,BETA)) =  [1] ALPHA + [1] BETA + [8] 
                               >= [1] ALPHA + [1] BETA + [8] 
                               =  c_9(dx#(ALPHA),dx#(BETA))  
        
****** Step 1.b:5.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:5.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
          
        The strictly oriented rules are moved into the weak component.
******* Step 1.b:5.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {dx#}
        TcT has computed the following interpretation:
              p(a) = [0]                   
            p(div) = [1] x1 + [1] x2 + [2] 
             p(dx) = [1] x1 + [0]          
            p(exp) = [1] x1 + [1] x2 + [8] 
             p(ln) = [1] x1 + [1]          
          p(minus) = [1] x1 + [1] x2 + [10]
            p(neg) = [1] x1 + [0]          
            p(one) = [1]                   
           p(plus) = [1] x1 + [1] x2 + [6] 
          p(times) = [1] x1 + [1] x2 + [3] 
            p(two) = [0]                   
           p(zero) = [1]                   
            p(dx#) = [2] x1 + [2]          
            p(c_1) = [1]                   
            p(c_2) = [0]                   
            p(c_3) = [1] x1 + [1] x2 + [2] 
            p(c_4) = [1] x1 + [1] x2 + [9] 
            p(c_5) = [1] x1 + [0]          
            p(c_6) = [1] x1 + [1] x2 + [15]
            p(c_7) = [1] x1 + [0]          
            p(c_8) = [1] x1 + [1] x2 + [7] 
            p(c_9) = [1] x1 + [1] x2 + [1] 
        
        Following rules are strictly oriented:
        dx#(ln(ALPHA)) = [2] ALPHA + [4]
                       > [2] ALPHA + [2]
                       = c_5(dx#(ALPHA))
        
        
        Following rules are (at-least) weakly oriented:
          dx#(div(ALPHA,BETA)) =  [2] ALPHA + [2] BETA + [6] 
                               >= [2] ALPHA + [2] BETA + [6] 
                               =  c_3(dx#(ALPHA),dx#(BETA))  
        
          dx#(exp(ALPHA,BETA)) =  [2] ALPHA + [2] BETA + [18]
                               >= [2] ALPHA + [2] BETA + [13]
                               =  c_4(dx#(ALPHA),dx#(BETA))  
        
        dx#(minus(ALPHA,BETA)) =  [2] ALPHA + [2] BETA + [22]
                               >= [2] ALPHA + [2] BETA + [19]
                               =  c_6(dx#(ALPHA),dx#(BETA))  
        
               dx#(neg(ALPHA)) =  [2] ALPHA + [2]            
                               >= [2] ALPHA + [2]            
                               =  c_7(dx#(ALPHA))            
        
         dx#(plus(ALPHA,BETA)) =  [2] ALPHA + [2] BETA + [14]
                               >= [2] ALPHA + [2] BETA + [11]
                               =  c_8(dx#(ALPHA),dx#(BETA))  
        
        dx#(times(ALPHA,BETA)) =  [2] ALPHA + [2] BETA + [8] 
                               >= [2] ALPHA + [2] BETA + [5] 
                               =  c_9(dx#(ALPHA),dx#(BETA))  
        
******* Step 1.b:5.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******* Step 1.b:5.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          
        The strictly oriented rules are moved into the weak component.
******** Step 1.b:5.b:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {dx#}
        TcT has computed the following interpretation:
              p(a) = [1]                   
            p(div) = [1] x1 + [1] x2 + [4] 
             p(dx) = [2] x1 + [1]          
            p(exp) = [1] x1 + [1] x2 + [3] 
             p(ln) = [1] x1 + [4]          
          p(minus) = [1] x1 + [1] x2 + [4] 
            p(neg) = [1] x1 + [2]          
            p(one) = [4]                   
           p(plus) = [1] x1 + [1] x2 + [1] 
          p(times) = [1] x1 + [1] x2 + [3] 
            p(two) = [1]                   
           p(zero) = [0]                   
            p(dx#) = [4] x1 + [2]          
            p(c_1) = [1]                   
            p(c_2) = [1]                   
            p(c_3) = [1] x1 + [1] x2 + [9] 
            p(c_4) = [1] x1 + [1] x2 + [7] 
            p(c_5) = [1] x1 + [1]          
            p(c_6) = [1] x1 + [1] x2 + [14]
            p(c_7) = [1] x1 + [5]          
            p(c_8) = [1] x1 + [1] x2 + [2] 
            p(c_9) = [1] x1 + [1] x2 + [10]
        
        Following rules are strictly oriented:
        dx#(neg(ALPHA)) = [4] ALPHA + [10]
                        > [4] ALPHA + [7] 
                        = c_7(dx#(ALPHA)) 
        
        
        Following rules are (at-least) weakly oriented:
          dx#(div(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [18]
                               >= [4] ALPHA + [4] BETA + [13]
                               =  c_3(dx#(ALPHA),dx#(BETA))  
        
          dx#(exp(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [14]
                               >= [4] ALPHA + [4] BETA + [11]
                               =  c_4(dx#(ALPHA),dx#(BETA))  
        
                dx#(ln(ALPHA)) =  [4] ALPHA + [18]           
                               >= [4] ALPHA + [3]            
                               =  c_5(dx#(ALPHA))            
        
        dx#(minus(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [18]
                               >= [4] ALPHA + [4] BETA + [18]
                               =  c_6(dx#(ALPHA),dx#(BETA))  
        
         dx#(plus(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [6] 
                               >= [4] ALPHA + [4] BETA + [6] 
                               =  c_8(dx#(ALPHA),dx#(BETA))  
        
        dx#(times(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [14]
                               >= [4] ALPHA + [4] BETA + [14]
                               =  c_9(dx#(ALPHA),dx#(BETA))  
        
******** Step 1.b:5.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******** Step 1.b:5.b:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
            dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          2:W:dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          3:W:dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          4:W:dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          5:W:dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          6:W:dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
          7:W:dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
             -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7
             -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
             -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
             -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
             -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3
             -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2
             -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
             -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA))
          7: dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
          6: dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          5: dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          4: dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          3: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
          2: dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA))
******** Step 1.b:5.b:1.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2
            ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus
            ,times,two,zero}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))