*** 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:
        Full
        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 dependency pairs:
      
      Strict DPs
        dx#(X) -> c_1()
        dx#(a()) -> c_2()
        dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
      Obligation:
        Full
        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),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
      Obligation:
        Full
        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),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
      Obligation:
        Full
        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)                      
                ,BETA                            
                ,ALPHA                           
                ,dx#(BETA)                       
                ,BETA)                           
        4: dx#(exp(ALPHA,BETA)) -> c_4(BETA      
                                      ,ALPHA     
                                      ,BETA      
                                      ,dx#(ALPHA)
                                      ,ALPHA     
                                      ,BETA      
                                      ,ALPHA     
                                      ,dx#(BETA))
        5: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)      
                                ,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(BETA                            
                ,dx#(ALPHA)                      
                ,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),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
      Obligation:
        Full
        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),BETA,ALPHA,dx#(BETA),BETA)
           -->_5 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_4 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_3 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_5 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_4 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_3 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_5 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_4 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_3 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_5 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_4 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_3 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_5 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_4 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_3 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_5 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_4 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_3 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_5 dx#(a()) -> c_2():9
           -->_4 dx#(a()) -> c_2():9
           -->_3 dx#(a()) -> c_2():9
           -->_2 dx#(a()) -> c_2():9
           -->_1 dx#(a()) -> c_2():9
           -->_5 dx#(X) -> c_1():8
           -->_4 dx#(X) -> c_1():8
           -->_3 dx#(X) -> c_1():8
           -->_2 dx#(X) -> c_1():8
           -->_1 dx#(X) -> c_1():8
           -->_5 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_4 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_3 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
        
        2:S:dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
           -->_8 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_7 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_6 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_5 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_4 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_3 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_8 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_7 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_6 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_5 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_4 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_3 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_8 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_7 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_6 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_5 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_4 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_3 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_8 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_7 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_6 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_5 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_4 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_3 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_8 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_7 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_6 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_5 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_4 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_3 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_8 dx#(a()) -> c_2():9
           -->_7 dx#(a()) -> c_2():9
           -->_6 dx#(a()) -> c_2():9
           -->_5 dx#(a()) -> c_2():9
           -->_4 dx#(a()) -> c_2():9
           -->_3 dx#(a()) -> c_2():9
           -->_2 dx#(a()) -> c_2():9
           -->_1 dx#(a()) -> c_2():9
           -->_8 dx#(X) -> c_1():8
           -->_7 dx#(X) -> c_1():8
           -->_6 dx#(X) -> c_1():8
           -->_5 dx#(X) -> c_1():8
           -->_4 dx#(X) -> c_1():8
           -->_3 dx#(X) -> c_1():8
           -->_2 dx#(X) -> c_1():8
           -->_1 dx#(X) -> c_1():8
           -->_8 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_7 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_6 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_5 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_4 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_3 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_8 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_7 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_6 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_5 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_4 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_3 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
        
        3:S:dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA)
           -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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#(a()) -> c_2():9
           -->_1 dx#(a()) -> c_2():9
           -->_2 dx#(X) -> c_1():8
           -->_1 dx#(X) -> c_1():8
           -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
        
        4:S:dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
           -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
           -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
        
        5:S:dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
        
        6:S:dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
           -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
           -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
        
        7:S:dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA))
           -->_4 dx#(a()) -> c_2():9
           -->_3 dx#(a()) -> c_2():9
           -->_2 dx#(a()) -> c_2():9
           -->_1 dx#(a()) -> c_2():9
           -->_4 dx#(X) -> c_1():8
           -->_3 dx#(X) -> c_1():8
           -->_2 dx#(X) -> c_1():8
           -->_1 dx#(X) -> c_1():8
           -->_4 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_3 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
           -->_4 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_3 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
           -->_4 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_3 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
           -->_4 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_3 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
           -->_4 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_3 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
           -->_4 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_3 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
           -->_4 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_3 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
           -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),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),BETA,ALPHA,dx#(BETA),BETA)
        dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
        dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
      Obligation:
        Full
        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#(exp(ALPHA,BETA)) -> c_4(BETA      
                                      ,ALPHA     
                                      ,BETA      
                                      ,dx#(ALPHA)
                                      ,ALPHA     
                                      ,BETA      
                                      ,ALPHA     
                                      ,dx#(BETA))
        3: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)      
                                ,ALPHA)          
        6: dx#(plus(ALPHA,BETA)) ->              
             c_8(dx#(ALPHA),dx#(BETA))           
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
          dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
          dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
        Obligation:
          Full
          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) = {1,4},
          uargs(c_4) = {4,8},
          uargs(c_5) = {1},
          uargs(c_6) = {1,2},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_9) = {2,4}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
              p(a) = [1]                   
            p(div) = [1] x1 + [1] x2 + [0] 
             p(dx) = [2] x1 + [1]          
            p(exp) = [1] x1 + [1] x2 + [2] 
             p(ln) = [1] x1 + [1]          
          p(minus) = [1] x1 + [1] x2 + [0] 
            p(neg) = [1] x1 + [0]          
            p(one) = [1]                   
           p(plus) = [1] x1 + [1] x2 + [2] 
          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) = [2]                   
            p(c_3) = [1] x1 + [1] x4 + [0] 
            p(c_4) = [1] x4 + [1] x8 + [14]
            p(c_5) = [1] x1 + [6]          
            p(c_6) = [1] x1 + [1] x2 + [0] 
            p(c_7) = [1] x1 + [0]          
            p(c_8) = [1] x1 + [1] x2 + [11]
            p(c_9) = [1] x2 + [1] x4 + [0] 
        
        Following rules are strictly oriented:
         dx#(exp(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                              > [8] ALPHA + [8] BETA + [14]
                              = c_4(BETA                   
                                   ,ALPHA                  
                                   ,BETA                   
                                   ,dx#(ALPHA)             
                                   ,ALPHA                  
                                   ,BETA                   
                                   ,ALPHA                  
                                   ,dx#(BETA))             
        
               dx#(ln(ALPHA)) = [8] ALPHA + [8]            
                              > [8] ALPHA + [6]            
                              = c_5(dx#(ALPHA),ALPHA)      
        
        dx#(plus(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                              > [8] ALPHA + [8] BETA + [11]
                              = 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)            
                                     ,BETA                  
                                     ,ALPHA                 
                                     ,dx#(BETA)             
                                     ,BETA)                 
        
        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#(times(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0]
                               >= [8] ALPHA + [8] BETA + [0]
                               =  c_9(BETA                  
                                     ,dx#(ALPHA)            
                                     ,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),BETA,ALPHA,dx#(BETA),BETA)
          dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA))
        Strict TRS Rules:
          
        Weak DP Rules:
          dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
          dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA)
          dx#(plus(ALPHA,BETA)) -> c_8(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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
        Obligation:
          Full
          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),BETA,ALPHA,dx#(BETA),BETA)
          dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
          dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
          dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA))
        Strict TRS Rules:
          
        Weak DP Rules:
          dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
          dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA)
          dx#(plus(ALPHA,BETA)) -> c_8(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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
        Obligation:
          Full
          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#(minus(ALPHA,BETA)) ->  
               c_6(dx#(ALPHA),dx#(BETA))
          3: dx#(neg(ALPHA)) ->         
               c_7(dx#(ALPHA))          
          4: dx#(times(ALPHA,BETA)) ->  
               c_9(BETA                 
                  ,dx#(ALPHA)           
                  ,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),BETA,ALPHA,dx#(BETA),BETA)
            dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
            dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
            dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA))
          Strict TRS Rules:
            
          Weak DP Rules:
            dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA)
            dx#(plus(ALPHA,BETA)) -> c_8(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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
          Obligation:
            Full
            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) = {1,4},
            uargs(c_4) = {4,8},
            uargs(c_5) = {1},
            uargs(c_6) = {1,2},
            uargs(c_7) = {1},
            uargs(c_8) = {1,2},
            uargs(c_9) = {2,4}
          
          Following symbols are considered usable:
            {}
          TcT has computed the following interpretation:
                p(a) = [2]                   
              p(div) = [1] x1 + [1] x2 + [1] 
               p(dx) = [1]                   
              p(exp) = [1] x1 + [1] x2 + [1] 
               p(ln) = [1] x1 + [0]          
            p(minus) = [1] x1 + [1] x2 + [2] 
              p(neg) = [1] x1 + [1]          
              p(one) = [1]                   
             p(plus) = [1] x1 + [1] x2 + [2] 
            p(times) = [1] x1 + [1] x2 + [3] 
              p(two) = [2]                   
             p(zero) = [1]                   
              p(dx#) = [8] x1 + [0]          
              p(c_1) = [0]                   
              p(c_2) = [2]                   
              p(c_3) = [1] x1 + [1] x4 + [8] 
              p(c_4) = [1] x4 + [1] x8 + [8] 
              p(c_5) = [1] x1 + [0]          
              p(c_6) = [1] x1 + [1] x2 + [12]
              p(c_7) = [1] x1 + [5]          
              p(c_8) = [1] x1 + [1] x2 + [12]
              p(c_9) = [1] x2 + [1] x4 + [12]
          
          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))  
          
                 dx#(neg(ALPHA)) = [8] ALPHA + [8]            
                                 > [8] ALPHA + [5]            
                                 = c_7(dx#(ALPHA))            
          
          dx#(times(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [24]
                                 > [8] ALPHA + [8] BETA + [12]
                                 = c_9(BETA                   
                                      ,dx#(ALPHA)             
                                      ,ALPHA                  
                                      ,dx#(BETA))             
          
          
          Following rules are (at-least) weakly oriented:
           dx#(div(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [8] 
                                >= [8] ALPHA + [8] BETA + [8] 
                                =  c_3(dx#(ALPHA)             
                                      ,BETA                   
                                      ,ALPHA                  
                                      ,dx#(BETA)              
                                      ,BETA)                  
          
           dx#(exp(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [8] 
                                >= [8] ALPHA + [8] BETA + [8] 
                                =  c_4(BETA                   
                                      ,ALPHA                  
                                      ,BETA                   
                                      ,dx#(ALPHA)             
                                      ,ALPHA                  
                                      ,BETA                   
                                      ,ALPHA                  
                                      ,dx#(BETA))             
          
                 dx#(ln(ALPHA)) =  [8] ALPHA + [0]            
                                >= [8] ALPHA + [0]            
                                =  c_5(dx#(ALPHA),ALPHA)      
          
          dx#(plus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [16]
                                >= [8] ALPHA + [8] BETA + [12]
                                =  c_8(dx#(ALPHA),dx#(BETA))  
          
    *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
          Strict TRS Rules:
            
          Weak DP Rules:
            dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
          Obligation:
            Full
            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#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
          Strict TRS Rules:
            
          Weak DP Rules:
            dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
            dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
          Obligation:
            Full
            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)       
                    ,BETA             
                    ,ALPHA            
                    ,dx#(BETA)        
                    ,BETA)            
            
          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#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
            Strict TRS Rules:
              
            Weak DP Rules:
              dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
              dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
            Obligation:
              Full
              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) = {1,4},
              uargs(c_4) = {4,8},
              uargs(c_5) = {1},
              uargs(c_6) = {1,2},
              uargs(c_7) = {1},
              uargs(c_8) = {1,2},
              uargs(c_9) = {2,4}
            
            Following symbols are considered usable:
              {}
            TcT has computed the following interpretation:
                  p(a) = [2]                   
                p(div) = [1] x1 + [1] x2 + [2] 
                 p(dx) = [4] x1 + [1]          
                p(exp) = [1] x1 + [1] x2 + [0] 
                 p(ln) = [1] x1 + [2]          
              p(minus) = [1] x1 + [1] x2 + [2] 
                p(neg) = [1] x1 + [1]          
                p(one) = [1]                   
               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) = [1]                   
                p(c_2) = [1]                   
                p(c_3) = [1] x1 + [1] x4 + [13]
                p(c_4) = [1] x4 + [1] x8 + [0] 
                p(c_5) = [1] x1 + [9]          
                p(c_6) = [1] x1 + [1] x2 + [13]
                p(c_7) = [1] x1 + [0]          
                p(c_8) = [1] x1 + [1] x2 + [0] 
                p(c_9) = [1] x2 + [1] x4 + [0] 
            
            Following rules are strictly oriented:
            dx#(div(ALPHA,BETA)) = [8] ALPHA + [8] BETA + [16]
                                 > [8] ALPHA + [8] BETA + [13]
                                 = c_3(dx#(ALPHA)             
                                      ,BETA                   
                                      ,ALPHA                  
                                      ,dx#(BETA)              
                                      ,BETA)                  
            
            
            Following rules are (at-least) weakly oriented:
              dx#(exp(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [0] 
                                   >= [8] ALPHA + [8] BETA + [0] 
                                   =  c_4(BETA                   
                                         ,ALPHA                  
                                         ,BETA                   
                                         ,dx#(ALPHA)             
                                         ,ALPHA                  
                                         ,BETA                   
                                         ,ALPHA                  
                                         ,dx#(BETA))             
            
                    dx#(ln(ALPHA)) =  [8] ALPHA + [16]           
                                   >= [8] ALPHA + [9]            
                                   =  c_5(dx#(ALPHA),ALPHA)      
            
            dx#(minus(ALPHA,BETA)) =  [8] ALPHA + [8] BETA + [16]
                                   >= [8] ALPHA + [8] BETA + [13]
                                   =  c_6(dx#(ALPHA),dx#(BETA))  
            
                   dx#(neg(ALPHA)) =  [8] ALPHA + [8]            
                                   >= [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(BETA                   
                                         ,dx#(ALPHA)             
                                         ,ALPHA                  
                                         ,dx#(BETA))             
            
      *** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
              dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
              dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
            Obligation:
              Full
              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(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA)
              dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
              dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),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(BETA,dx#(ALPHA),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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
            Obligation:
              Full
              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),BETA,ALPHA,dx#(BETA),BETA)
                 -->_5 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_4 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_3 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_5 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_4 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_3 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_5 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_4 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_3 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_5 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_4 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_3 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_5 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_4 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_3 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_5 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_4 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_3 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_5 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_4 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_3 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
              
              2:W:dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA))
                 -->_8 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_7 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_6 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_5 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_4 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_3 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_8 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_7 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_6 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_5 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_4 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_3 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_8 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_7 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_6 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_5 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_4 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_3 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_8 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_7 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_6 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_5 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_4 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_3 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_8 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_7 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_6 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_5 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_4 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_3 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_8 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_7 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_6 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_5 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_4 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_3 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_8 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_7 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_6 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_5 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_4 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_3 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
              
              3:W:dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA)
                 -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
                 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
              
              4:W:dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA))
                 -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
                 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
              
              5:W:dx#(neg(ALPHA)) -> c_7(dx#(ALPHA))
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
              
              6:W:dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA))
                 -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),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),ALPHA):3
                 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
              
              7:W:dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA))
                 -->_4 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_3 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_2 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_1 dx#(times(ALPHA,BETA)) -> c_9(BETA,dx#(ALPHA),ALPHA,dx#(BETA)):7
                 -->_4 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_3 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6
                 -->_4 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_3 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5
                 -->_4 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_3 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4
                 -->_4 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_3 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA),ALPHA):3
                 -->_4 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_3 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(BETA,ALPHA,BETA,dx#(ALPHA),ALPHA,BETA,ALPHA,dx#(BETA)):2
                 -->_4 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_3 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),BETA):1
                 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),BETA,ALPHA,dx#(BETA),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)                      
                      ,BETA                            
                      ,ALPHA                           
                      ,dx#(BETA)                       
                      ,BETA)                           
              7: dx#(times(ALPHA,BETA)) ->             
                   c_9(BETA                            
                      ,dx#(ALPHA)                      
                      ,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)      
                                      ,ALPHA)          
              2: dx#(exp(ALPHA,BETA)) -> c_4(BETA      
                                            ,ALPHA     
                                            ,BETA      
                                            ,dx#(ALPHA)
                                            ,ALPHA     
                                            ,BETA      
                                            ,ALPHA     
                                            ,dx#(BETA))
      *** 1.1.1.1.1.1.2.2.2.1 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_3/5,c_4/8,c_5/2,c_6/2,c_7/1,c_8/2,c_9/4}
            Obligation:
              Full
              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).