*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {dx}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      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.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP 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))
      Strict TRS Rules:
        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)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
    Applied Processor:
      UsableRules
    Proof:
      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))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP 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))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
    Applied Processor:
      Succeeding
    Proof:
      ()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP 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))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      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))
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        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 Rules:
        
      Weak DP Rules:
        dx#(X) -> c_1()
        dx#(a()) -> c_2()
      Weak TRS Rules:
        
      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
        basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      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()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        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 Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {dx#}/{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, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        4: dx#(minus(ALPHA,BETA)) ->  
             c_6(dx#(ALPHA),dx#(BETA))
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          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 Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        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
          basic terms: {dx#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        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) = [1] x1 + [1]          
            p(exp) = [1] x1 + [1] x2 + [0] 
             p(ln) = [1] x1 + [0]          
          p(minus) = [1] x1 + [1] x2 + [2] 
            p(neg) = [1] x1 + [0]          
            p(one) = [0]                   
           p(plus) = [1] x1 + [1] x2 + [0] 
          p(times) = [1] x1 + [1] x2 + [0] 
            p(two) = [0]                   
           p(zero) = [1]                   
            p(dx#) = [8] 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 + [0] 
            p(c_9) = [1] x1 + [1] x2 + [0] 
        
        Following rules are strictly oriented:
        dx#(minus(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                               > [8] ALPHA + [8] BETA + [12]
                               = c_6(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#(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)) 
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          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#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        Strict TRS Rules:
          
        Weak DP Rules:
          dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
        Weak TRS Rules:
          
        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
          basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.2 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          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#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
        Strict TRS Rules:
          
        Weak DP Rules:
          dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
        Weak TRS Rules:
          
        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
          basic terms: {dx#}/{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, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: dx#(div(ALPHA,BETA)) ->    
               c_3(dx#(ALPHA),dx#(BETA))
          2: dx#(exp(ALPHA,BETA)) ->    
               c_4(dx#(ALPHA),dx#(BETA))
          6: dx#(times(ALPHA,BETA)) ->  
               c_9(dx#(ALPHA),dx#(BETA))
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            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#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
          Strict TRS Rules:
            
          Weak DP Rules:
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          Weak TRS Rules:
            
          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
            basic terms: {dx#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
        Proof:
          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 + [3] 
               p(dx) = [1] x1 + [1]          
              p(exp) = [1] x1 + [1] x2 + [1] 
               p(ln) = [1] x1 + [0]          
            p(minus) = [1] x1 + [1] x2 + [0] 
              p(neg) = [1] x1 + [0]          
              p(one) = [8]                   
             p(plus) = [1] x1 + [1] x2 + [0] 
            p(times) = [1] x1 + [1] x2 + [2] 
              p(two) = [1]                   
             p(zero) = [1]                   
              p(dx#) = [8] x1 + [0]          
              p(c_1) = [2]                   
              p(c_2) = [0]                   
              p(c_3) = [1] x1 + [1] x2 + [12]
              p(c_4) = [1] x1 + [1] x2 + [4] 
              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 + [12]
          
          Following rules are strictly oriented:
            dx#(div(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [24]
                                 > [8] ALPHA + [8] BETA + [12]
                                 = c_3(dx#(ALPHA),dx#(BETA))  
          
            dx#(exp(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [8] 
                                 > [8] ALPHA + [8] BETA + [4] 
                                 = c_4(dx#(ALPHA),dx#(BETA))  
          
          dx#(times(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                                 > [8] ALPHA + [8] BETA + [12]
                                 = c_9(dx#(ALPHA),dx#(BETA))  
          
          
          Following rules are (at-least) weakly oriented:
                  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)) 
          
    *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          Strict TRS Rules:
            
          Weak DP Rules:
            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#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
          Weak TRS Rules:
            
          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
            basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.1.2.2 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
          Strict TRS Rules:
            
          Weak DP Rules:
            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#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
          Weak TRS Rules:
            
          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
            basic terms: {dx#}/{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, greedy = NoGreedy}}
        Proof:
          We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
            2: dx#(neg(ALPHA)) ->
                 c_7(dx#(ALPHA)) 
            
          The strictly oriented rules are moved into the weak component.
      *** 1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))]  ***
          Considered Problem:
            Strict DP Rules:
              dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
              dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
              dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            Strict TRS Rules:
              
            Weak DP Rules:
              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#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
            Weak TRS Rules:
              
            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
              basic terms: {dx#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
          Proof:
            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) = [1]                  
                p(exp) = [1] x1 + [1] x2 + [2]
                 p(ln) = [1] x1 + [0]         
              p(minus) = [1] x1 + [1] x2 + [0]
                p(neg) = [1] x1 + [4]         
                p(one) = [0]                  
               p(plus) = [1] x1 + [1] x2 + [0]
              p(times) = [1] x1 + [1] x2 + [0]
                p(two) = [4]                  
               p(zero) = [1]                  
                p(dx#) = [4] x1 + [0]         
                p(c_1) = [0]                  
                p(c_2) = [1]                  
                p(c_3) = [1] x1 + [1] x2 + [0]
                p(c_4) = [1] x1 + [1] x2 + [2]
                p(c_5) = [1] x1 + [0]         
                p(c_6) = [1] x1 + [1] x2 + [0]
                p(c_7) = [1] x1 + [1]         
                p(c_8) = [1] x1 + [1] x2 + [0]
                p(c_9) = [1] x1 + [1] x2 + [0]
            
            Following rules are strictly oriented:
            dx#(neg(ALPHA)) = [4] ALPHA + [16]
                            > [4] ALPHA + [1] 
                            = c_7(dx#(ALPHA)) 
            
            
            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 + [8]
                                   >= [4] ALPHA + [4] BETA + [2]
                                   =  c_4(dx#(ALPHA),dx#(BETA)) 
            
                    dx#(ln(ALPHA)) =  [4] ALPHA + [0]           
                                   >= [4] ALPHA + [0]           
                                   =  c_5(dx#(ALPHA))           
            
            dx#(minus(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [0]
                                   >= [4] ALPHA + [4] BETA + [0]
                                   =  c_6(dx#(ALPHA),dx#(BETA)) 
            
             dx#(plus(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [0]
                                   >= [4] ALPHA + [4] BETA + [0]
                                   =  c_8(dx#(ALPHA),dx#(BETA)) 
            
            dx#(times(ALPHA,BETA)) =  [4] ALPHA + [4] BETA + [0]
                                   >= [4] ALPHA + [4] BETA + [0]
                                   =  c_9(dx#(ALPHA),dx#(BETA)) 
            
      *** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
              dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            Strict TRS Rules:
              
            Weak DP Rules:
              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#(neg(ALPHA)) -> c_7(dx#(ALPHA))
              dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
            Weak TRS Rules:
              
            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
              basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
          Applied Processor:
            Assumption
          Proof:
            ()
      
      *** 1.1.1.1.1.1.2.2.2 Progress [(?,O(n^1))]  ***
          Considered Problem:
            Strict DP Rules:
              dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
              dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
            Strict TRS Rules:
              
            Weak DP Rules:
              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#(neg(ALPHA)) -> c_7(dx#(ALPHA))
              dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
            Weak TRS Rules:
              
            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
              basic terms: {dx#}/{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, greedy = NoGreedy}}
          Proof:
            We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
              1: dx#(ln(ALPHA)) ->
                   c_5(dx#(ALPHA))
              
            The strictly oriented rules are moved into the weak component.
        *** 1.1.1.1.1.1.2.2.2.1 Progress [(?,O(n^1))]  ***
            Considered Problem:
              Strict DP Rules:
                dx#(ln(ALPHA)) -> c_5(dx#(ALPHA))
                dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
              Strict TRS Rules:
                
              Weak DP Rules:
                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#(neg(ALPHA)) -> c_7(dx#(ALPHA))
                dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
              Weak TRS Rules:
                
              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
                basic terms: {dx#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
            Proof:
              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) = [1] x1 + [8]         
                  p(exp) = [1] x1 + [1] x2 + [0]
                   p(ln) = [1] x1 + [1]         
                p(minus) = [1] x1 + [1] x2 + [1]
                  p(neg) = [1] x1 + [0]         
                  p(one) = [1]                  
                 p(plus) = [1] x1 + [1] x2 + [0]
                p(times) = [1] x1 + [1] x2 + [0]
                  p(two) = [0]                  
                 p(zero) = [2]                  
                  p(dx#) = [8] x1 + [0]         
                  p(c_1) = [8]                  
                  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 + [6]         
                  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#(ln(ALPHA)) = [8] ALPHA + [8]
                             > [8] ALPHA + [6]
                             = c_5(dx#(ALPHA))
              
              
              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#(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)) 
              
        *** 1.1.1.1.1.1.2.2.2.1.1 Progress [(?,O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
              Strict TRS Rules:
                
              Weak DP Rules:
                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#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
              Weak TRS Rules:
                
              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
                basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
            Applied Processor:
              Assumption
            Proof:
              ()
        
        *** 1.1.1.1.1.1.2.2.2.2 Progress [(?,O(n^1))]  ***
            Considered Problem:
              Strict DP Rules:
                dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
              Strict TRS Rules:
                
              Weak DP Rules:
                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#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
              Weak TRS Rules:
                
              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
                basic terms: {dx#}/{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, greedy = NoGreedy}}
            Proof:
              We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
                1: dx#(plus(ALPHA,BETA)) ->   
                     c_8(dx#(ALPHA),dx#(BETA))
                
              The strictly oriented rules are moved into the weak component.
          *** 1.1.1.1.1.1.2.2.2.2.1 Progress [(?,O(n^1))]  ***
              Considered Problem:
                Strict DP Rules:
                  dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
                Strict TRS Rules:
                  
                Weak DP Rules:
                  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#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA))
                Weak TRS Rules:
                  
                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
                  basic terms: {dx#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
              Proof:
                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 + [0]
                     p(dx) = [4] x1 + [1]         
                    p(exp) = [1] x1 + [1] x2 + [1]
                     p(ln) = [1] x1 + [2]         
                  p(minus) = [1] x1 + [1] x2 + [2]
                    p(neg) = [1] x1 + [0]         
                    p(one) = [0]                  
                   p(plus) = [1] x1 + [1] x2 + [1]
                  p(times) = [1] x1 + [1] x2 + [0]
                    p(two) = [8]                  
                   p(zero) = [0]                  
                    p(dx#) = [8] x1 + [0]         
                    p(c_1) = [8]                  
                    p(c_2) = [1]                  
                    p(c_3) = [1] x1 + [1] x2 + [0]
                    p(c_4) = [1] x1 + [1] x2 + [8]
                    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 + [6]
                    p(c_9) = [1] x1 + [1] x2 + [0]
                
                Following rules are strictly oriented:
                dx#(plus(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [8]
                                      > [8] ALPHA + [8] BETA + [6]
                                      = c_8(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 + [8] 
                                       >= [8] ALPHA + [8] BETA + [8] 
                                       =  c_4(dx#(ALPHA),dx#(BETA))  
                
                        dx#(ln(ALPHA)) =  [8] ALPHA + [16]           
                                       >= [8] ALPHA + [0]            
                                       =  c_5(dx#(ALPHA))            
                
                dx#(minus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [16]
                                       >= [8] ALPHA + [8] BETA + [4] 
                                       =  c_6(dx#(ALPHA),dx#(BETA))  
                
                       dx#(neg(ALPHA)) =  [8] ALPHA + [0]            
                                       >= [8] ALPHA + [0]            
                                       =  c_7(dx#(ALPHA))            
                
                dx#(times(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0] 
                                       >= [8] ALPHA + [8] BETA + [0] 
                                       =  c_9(dx#(ALPHA),dx#(BETA))  
                
          *** 1.1.1.1.1.1.2.2.2.2.1.1 Progress [(?,O(1))]  ***
              Considered Problem:
                Strict DP Rules:
                  
                Strict TRS Rules:
                  
                Weak DP Rules:
                  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 TRS Rules:
                  
                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
                  basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
              Applied Processor:
                Assumption
              Proof:
                ()
          
          *** 1.1.1.1.1.1.2.2.2.2.2 Progress [(O(1),O(1))]  ***
              Considered Problem:
                Strict DP Rules:
                  
                Strict TRS Rules:
                  
                Weak DP Rules:
                  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 TRS Rules:
                  
                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
                  basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
              Applied Processor:
                RemoveWeakSuffixes
              Proof:
                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))
          *** 1.1.1.1.1.1.2.2.2.2.2.1 Progress [(O(1),O(1))]  ***
              Considered Problem:
                Strict DP Rules:
                  
                Strict TRS Rules:
                  
                Weak DP Rules:
                  
                Weak TRS Rules:
                  
                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
                  basic terms: {dx#}/{a,div,exp,ln,minus,neg,one,plus,times,two,zero}
              Applied Processor:
                EmptyProcessor
              Proof:
                The problem is already closed. The intended complexity is O(1).