WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
            D(+(x,y)) -> +(D(x),D(y))
            D(-(x,y)) -> -(D(x),D(y))
            D(constant()) -> 0()
            D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2())))
            D(ln(x)) -> div(D(x),x)
            D(minus(x)) -> minus(D(x))
            D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))
            D(t()) -> 1()
        - Signature:
            {D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          D#(*(x,y)) -> c_1(D#(x),D#(y))
          D#(+(x,y)) -> c_2(D#(x),D#(y))
          D#(-(x,y)) -> c_3(D#(x),D#(y))
          D#(constant()) -> c_4()
          D#(div(x,y)) -> c_5(D#(x),D#(y))
          D#(ln(x)) -> c_6(D#(x))
          D#(minus(x)) -> c_7(D#(x))
          D#(pow(x,y)) -> c_8(D#(x),D#(y))
          D#(t()) -> c_9()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(constant()) -> c_4()
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
            D#(t()) -> c_9()
        - Weak TRS:
            D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
            D(+(x,y)) -> +(D(x),D(y))
            D(-(x,y)) -> -(D(x),D(y))
            D(constant()) -> 0()
            D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2())))
            D(ln(x)) -> div(D(x),x)
            D(minus(x)) -> minus(D(x))
            D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))
            D(t()) -> 1()
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,9}
        by application of
          Pre({4,9}) = {1,2,3,5,6,7,8}.
        Here rules are labelled as follows:
          1: D#(*(x,y)) -> c_1(D#(x),D#(y))
          2: D#(+(x,y)) -> c_2(D#(x),D#(y))
          3: D#(-(x,y)) -> c_3(D#(x),D#(y))
          4: D#(constant()) -> c_4()
          5: D#(div(x,y)) -> c_5(D#(x),D#(y))
          6: D#(ln(x)) -> c_6(D#(x))
          7: D#(minus(x)) -> c_7(D#(x))
          8: D#(pow(x,y)) -> c_8(D#(x),D#(y))
          9: D#(t()) -> c_9()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Weak DPs:
            D#(constant()) -> c_4()
            D#(t()) -> c_9()
        - Weak TRS:
            D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
            D(+(x,y)) -> +(D(x),D(y))
            D(-(x,y)) -> -(D(x),D(y))
            D(constant()) -> 0()
            D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2())))
            D(ln(x)) -> div(D(x),x)
            D(minus(x)) -> minus(D(x))
            D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))
            D(t()) -> 1()
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:D#(*(x,y)) -> c_1(D#(x),D#(y))
             -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_2 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_2 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(t()) -> c_9():9
             -->_1 D#(t()) -> c_9():9
             -->_2 D#(constant()) -> c_4():8
             -->_1 D#(constant()) -> c_4():8
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          2:S:D#(+(x,y)) -> c_2(D#(x),D#(y))
             -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_2 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_2 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(t()) -> c_9():9
             -->_1 D#(t()) -> c_9():9
             -->_2 D#(constant()) -> c_4():8
             -->_1 D#(constant()) -> c_4():8
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          3:S:D#(-(x,y)) -> c_3(D#(x),D#(y))
             -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_2 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_2 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_2 D#(t()) -> c_9():9
             -->_1 D#(t()) -> c_9():9
             -->_2 D#(constant()) -> c_4():8
             -->_1 D#(constant()) -> c_4():8
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          4:S:D#(div(x,y)) -> c_5(D#(x),D#(y))
             -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_2 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_2 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_2 D#(t()) -> c_9():9
             -->_1 D#(t()) -> c_9():9
             -->_2 D#(constant()) -> c_4():8
             -->_1 D#(constant()) -> c_4():8
             -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          5:S:D#(ln(x)) -> c_6(D#(x))
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(t()) -> c_9():9
             -->_1 D#(constant()) -> c_4():8
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          6:S:D#(minus(x)) -> c_7(D#(x))
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(t()) -> c_9():9
             -->_1 D#(constant()) -> c_4():8
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          7:S:D#(pow(x,y)) -> c_8(D#(x),D#(y))
             -->_2 D#(t()) -> c_9():9
             -->_1 D#(t()) -> c_9():9
             -->_2 D#(constant()) -> c_4():8
             -->_1 D#(constant()) -> c_4():8
             -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
             -->_2 D#(minus(x)) -> c_7(D#(x)):6
             -->_1 D#(minus(x)) -> c_7(D#(x)):6
             -->_2 D#(ln(x)) -> c_6(D#(x)):5
             -->_1 D#(ln(x)) -> c_6(D#(x)):5
             -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          8:W:D#(constant()) -> c_4()
             
          
          9:W:D#(t()) -> c_9()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: D#(constant()) -> c_4()
          9: D#(t()) -> c_9()
* Step 4: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Weak TRS:
            D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
            D(+(x,y)) -> +(D(x),D(y))
            D(-(x,y)) -> -(D(x),D(y))
            D(constant()) -> 0()
            D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2())))
            D(ln(x)) -> div(D(x),x)
            D(minus(x)) -> minus(D(x))
            D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))
            D(t()) -> 1()
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          D#(*(x,y)) -> c_1(D#(x),D#(y))
          D#(+(x,y)) -> c_2(D#(x),D#(y))
          D#(-(x,y)) -> c_3(D#(x),D#(y))
          D#(div(x,y)) -> c_5(D#(x),D#(y))
          D#(ln(x)) -> c_6(D#(x))
          D#(minus(x)) -> c_7(D#(x))
          D#(pow(x,y)) -> c_8(D#(x),D#(y))
* Step 5: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1},
            uargs(c_7) = {1},
            uargs(c_8) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(*) = [1] x1 + [1] x2 + [1]
                   p(+) = [1] x1 + [1] x2 + [0]
                   p(-) = [1] x1 + [1] x2 + [0]
                   p(0) = [0]                  
                   p(1) = [0]                  
                   p(2) = [0]                  
                   p(D) = [1] x1 + [8]         
            p(constant) = [0]                  
                 p(div) = [1] x1 + [1] x2 + [0]
                  p(ln) = [1] x1 + [0]         
               p(minus) = [1] x1 + [0]         
                 p(pow) = [1] x1 + [1] x2 + [0]
                   p(t) = [1]                  
                  p(D#) = [8] x1 + [0]         
                 p(c_1) = [1] x1 + [1] x2 + [5]
                 p(c_2) = [1] x1 + [1] x2 + [0]
                 p(c_3) = [1] x1 + [1] x2 + [0]
                 p(c_4) = [1]                  
                 p(c_5) = [1] x1 + [1] x2 + [3]
                 p(c_6) = [1] x1 + [0]         
                 p(c_7) = [1] x1 + [0]         
                 p(c_8) = [1] x1 + [1] x2 + [2]
                 p(c_9) = [0]                  
          
          Following rules are strictly oriented:
          D#(*(x,y)) = [8] x + [8] y + [8]
                     > [8] x + [8] y + [5]
                     = c_1(D#(x),D#(y))   
          
          
          Following rules are (at-least) weakly oriented:
            D#(+(x,y)) =  [8] x + [8] y + [0]
                       >= [8] x + [8] y + [0]
                       =  c_2(D#(x),D#(y))   
          
            D#(-(x,y)) =  [8] x + [8] y + [0]
                       >= [8] x + [8] y + [0]
                       =  c_3(D#(x),D#(y))   
          
          D#(div(x,y)) =  [8] x + [8] y + [0]
                       >= [8] x + [8] y + [3]
                       =  c_5(D#(x),D#(y))   
          
             D#(ln(x)) =  [8] x + [0]        
                       >= [8] x + [0]        
                       =  c_6(D#(x))         
          
          D#(minus(x)) =  [8] x + [0]        
                       >= [8] x + [0]        
                       =  c_7(D#(x))         
          
          D#(pow(x,y)) =  [8] x + [8] y + [0]
                       >= [8] x + [8] y + [2]
                       =  c_8(D#(x),D#(y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1},
            uargs(c_7) = {1},
            uargs(c_8) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(*) = [1] x1 + [1] x2 + [4]
                   p(+) = [1] x1 + [1] x2 + [3]
                   p(-) = [1] x1 + [1] x2 + [0]
                   p(0) = [0]                  
                   p(1) = [0]                  
                   p(2) = [0]                  
                   p(D) = [0]                  
            p(constant) = [0]                  
                 p(div) = [1] x1 + [1] x2 + [0]
                  p(ln) = [1] x1 + [0]         
               p(minus) = [1] x1 + [0]         
                 p(pow) = [1] x1 + [1] x2 + [0]
                   p(t) = [1]                  
                  p(D#) = [2] x1 + [1]         
                 p(c_1) = [1] x1 + [1] x2 + [0]
                 p(c_2) = [1] x1 + [1] x2 + [0]
                 p(c_3) = [1] x1 + [1] x2 + [0]
                 p(c_4) = [0]                  
                 p(c_5) = [1] x1 + [1] x2 + [0]
                 p(c_6) = [1] x1 + [0]         
                 p(c_7) = [1] x1 + [0]         
                 p(c_8) = [1] x1 + [1] x2 + [1]
                 p(c_9) = [0]                  
          
          Following rules are strictly oriented:
          D#(+(x,y)) = [2] x + [2] y + [7]
                     > [2] x + [2] y + [2]
                     = c_2(D#(x),D#(y))   
          
          
          Following rules are (at-least) weakly oriented:
            D#(*(x,y)) =  [2] x + [2] y + [9]
                       >= [2] x + [2] y + [2]
                       =  c_1(D#(x),D#(y))   
          
            D#(-(x,y)) =  [2] x + [2] y + [1]
                       >= [2] x + [2] y + [2]
                       =  c_3(D#(x),D#(y))   
          
          D#(div(x,y)) =  [2] x + [2] y + [1]
                       >= [2] x + [2] y + [2]
                       =  c_5(D#(x),D#(y))   
          
             D#(ln(x)) =  [2] x + [1]        
                       >= [2] x + [1]        
                       =  c_6(D#(x))         
          
          D#(minus(x)) =  [2] x + [1]        
                       >= [2] x + [1]        
                       =  c_7(D#(x))         
          
          D#(pow(x,y)) =  [2] x + [2] y + [1]
                       >= [2] x + [2] y + [3]
                       =  c_8(D#(x),D#(y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1},
            uargs(c_7) = {1},
            uargs(c_8) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(*) = [1] x1 + [1] x2 + [5] 
                   p(+) = [1] x1 + [1] x2 + [3] 
                   p(-) = [1] x1 + [1] x2 + [1] 
                   p(0) = [1]                   
                   p(1) = [0]                   
                   p(2) = [1]                   
                   p(D) = [1] x1 + [2]          
            p(constant) = [1]                   
                 p(div) = [1] x1 + [1] x2 + [0] 
                  p(ln) = [1] x1 + [13]         
               p(minus) = [1] x1 + [1]          
                 p(pow) = [1] x1 + [1] x2 + [12]
                   p(t) = [0]                   
                  p(D#) = [1] x1 + [3]          
                 p(c_1) = [1] x1 + [1] x2 + [2] 
                 p(c_2) = [1] x1 + [1] x2 + [0] 
                 p(c_3) = [1] x1 + [1] x2 + [0] 
                 p(c_4) = [0]                   
                 p(c_5) = [1] x1 + [1] x2 + [0] 
                 p(c_6) = [1] x1 + [0]          
                 p(c_7) = [1] x1 + [0]          
                 p(c_8) = [1] x1 + [1] x2 + [0] 
                 p(c_9) = [0]                   
          
          Following rules are strictly oriented:
             D#(ln(x)) = [1] x + [16]        
                       > [1] x + [3]         
                       = c_6(D#(x))          
          
          D#(minus(x)) = [1] x + [4]         
                       > [1] x + [3]         
                       = c_7(D#(x))          
          
          D#(pow(x,y)) = [1] x + [1] y + [15]
                       > [1] x + [1] y + [6] 
                       = c_8(D#(x),D#(y))    
          
          
          Following rules are (at-least) weakly oriented:
            D#(*(x,y)) =  [1] x + [1] y + [8]
                       >= [1] x + [1] y + [8]
                       =  c_1(D#(x),D#(y))   
          
            D#(+(x,y)) =  [1] x + [1] y + [6]
                       >= [1] x + [1] y + [6]
                       =  c_2(D#(x),D#(y))   
          
            D#(-(x,y)) =  [1] x + [1] y + [4]
                       >= [1] x + [1] y + [6]
                       =  c_3(D#(x),D#(y))   
          
          D#(div(x,y)) =  [1] x + [1] y + [3]
                       >= [1] x + [1] y + [6]
                       =  c_5(D#(x),D#(y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1},
            uargs(c_7) = {1},
            uargs(c_8) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(*) = [1] x1 + [1] x2 + [2] 
                   p(+) = [1] x1 + [1] x2 + [1] 
                   p(-) = [1] x1 + [1] x2 + [1] 
                   p(0) = [0]                   
                   p(1) = [0]                   
                   p(2) = [0]                   
                   p(D) = [8]                   
            p(constant) = [8]                   
                 p(div) = [1] x1 + [1] x2 + [10]
                  p(ln) = [1] x1 + [0]          
               p(minus) = [1] x1 + [0]          
                 p(pow) = [1] x1 + [1] x2 + [1] 
                   p(t) = [0]                   
                  p(D#) = [1] x1 + [1]          
                 p(c_1) = [1] x1 + [1] x2 + [1] 
                 p(c_2) = [1] x1 + [1] x2 + [0] 
                 p(c_3) = [1] x1 + [1] x2 + [0] 
                 p(c_4) = [0]                   
                 p(c_5) = [1] x1 + [1] x2 + [0] 
                 p(c_6) = [1] x1 + [0]          
                 p(c_7) = [1] x1 + [0]          
                 p(c_8) = [1] x1 + [1] x2 + [0] 
                 p(c_9) = [0]                   
          
          Following rules are strictly oriented:
          D#(div(x,y)) = [1] x + [1] y + [11]
                       > [1] x + [1] y + [2] 
                       = c_5(D#(x),D#(y))    
          
          
          Following rules are (at-least) weakly oriented:
            D#(*(x,y)) =  [1] x + [1] y + [3]
                       >= [1] x + [1] y + [3]
                       =  c_1(D#(x),D#(y))   
          
            D#(+(x,y)) =  [1] x + [1] y + [2]
                       >= [1] x + [1] y + [2]
                       =  c_2(D#(x),D#(y))   
          
            D#(-(x,y)) =  [1] x + [1] y + [2]
                       >= [1] x + [1] y + [2]
                       =  c_3(D#(x),D#(y))   
          
             D#(ln(x)) =  [1] x + [1]        
                       >= [1] x + [1]        
                       =  c_6(D#(x))         
          
          D#(minus(x)) =  [1] x + [1]        
                       >= [1] x + [1]        
                       =  c_7(D#(x))         
          
          D#(pow(x,y)) =  [1] x + [1] y + [2]
                       >= [1] x + [1] y + [2]
                       =  c_8(D#(x),D#(y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 9: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1},
            uargs(c_7) = {1},
            uargs(c_8) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(*) = [1] x1 + [1] x2 + [1]
                   p(+) = [1] x1 + [1] x2 + [0]
                   p(-) = [1] x1 + [1] x2 + [1]
                   p(0) = [0]                  
                   p(1) = [0]                  
                   p(2) = [1]                  
                   p(D) = [1]                  
            p(constant) = [0]                  
                 p(div) = [1] x1 + [1] x2 + [1]
                  p(ln) = [1] x1 + [1]         
               p(minus) = [1] x1 + [1]         
                 p(pow) = [1] x1 + [1] x2 + [1]
                   p(t) = [0]                  
                  p(D#) = [4] x1 + [0]         
                 p(c_1) = [1] x1 + [1] x2 + [2]
                 p(c_2) = [1] x1 + [1] x2 + [0]
                 p(c_3) = [1] x1 + [1] x2 + [3]
                 p(c_4) = [0]                  
                 p(c_5) = [1] x1 + [1] x2 + [0]
                 p(c_6) = [1] x1 + [4]         
                 p(c_7) = [1] x1 + [2]         
                 p(c_8) = [1] x1 + [1] x2 + [1]
                 p(c_9) = [2]                  
          
          Following rules are strictly oriented:
          D#(-(x,y)) = [4] x + [4] y + [4]
                     > [4] x + [4] y + [3]
                     = c_3(D#(x),D#(y))   
          
          
          Following rules are (at-least) weakly oriented:
            D#(*(x,y)) =  [4] x + [4] y + [4]
                       >= [4] x + [4] y + [2]
                       =  c_1(D#(x),D#(y))   
          
            D#(+(x,y)) =  [4] x + [4] y + [0]
                       >= [4] x + [4] y + [0]
                       =  c_2(D#(x),D#(y))   
          
          D#(div(x,y)) =  [4] x + [4] y + [4]
                       >= [4] x + [4] y + [0]
                       =  c_5(D#(x),D#(y))   
          
             D#(ln(x)) =  [4] x + [4]        
                       >= [4] x + [4]        
                       =  c_6(D#(x))         
          
          D#(minus(x)) =  [4] x + [4]        
                       >= [4] x + [2]        
                       =  c_7(D#(x))         
          
          D#(pow(x,y)) =  [4] x + [4] y + [4]
                       >= [4] x + [4] y + [1]
                       =  c_8(D#(x),D#(y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 10: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(div(x,y)) -> c_5(D#(x),D#(y))
            D#(ln(x)) -> c_6(D#(x))
            D#(minus(x)) -> c_7(D#(x))
            D#(pow(x,y)) -> c_8(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2
            ,c_6/1,c_7/1,c_8/2,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus
            ,pow,t}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))