WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          d(s(X)) -> s(s(d(X)))
          f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
          p(s(X),s(Y)) -> s(s(p(X,Y)))
          q(s(X)) -> s(p(q(X),d(X)))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            p(X,0()) -> X
            p(0(),X) -> X
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          a#(X) -> c_1()
          a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
          a#(ns(X)) -> c_3(s#(a(X)))
          a#(nt(X)) -> c_4(t#(a(X)))
          d#(0()) -> c_5()
          f#(X1,X2) -> c_6()
          f#(0(),X) -> c_7()
          p#(X,0()) -> c_8()
          p#(0(),X) -> c_9()
          q#(0()) -> c_10()
          s#(X) -> c_11()
          t#(N) -> c_12(q#(N))
          t#(X) -> c_13()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(X) -> c_1()
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            d#(0()) -> c_5()
            f#(X1,X2) -> c_6()
            f#(0(),X) -> c_7()
            p#(X,0()) -> c_8()
            p#(0(),X) -> c_9()
            q#(0()) -> c_10()
            s#(X) -> c_11()
            t#(N) -> c_12(q#(N))
            t#(X) -> c_13()
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            p(X,0()) -> X
            p(0(),X) -> X
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          a(X) -> X
          a(nf(X1,X2)) -> f(a(X1),a(X2))
          a(ns(X)) -> s(a(X))
          a(nt(X)) -> t(a(X))
          f(X1,X2) -> nf(X1,X2)
          f(0(),X) -> nil()
          q(0()) -> 0()
          s(X) -> ns(X)
          t(N) -> cs(r(q(N)),nt(ns(N)))
          t(X) -> nt(X)
          a#(X) -> c_1()
          a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
          a#(ns(X)) -> c_3(s#(a(X)))
          a#(nt(X)) -> c_4(t#(a(X)))
          d#(0()) -> c_5()
          f#(X1,X2) -> c_6()
          f#(0(),X) -> c_7()
          p#(X,0()) -> c_8()
          p#(0(),X) -> c_9()
          q#(0()) -> c_10()
          s#(X) -> c_11()
          t#(N) -> c_12(q#(N))
          t#(X) -> c_13()
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(X) -> c_1()
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            d#(0()) -> c_5()
            f#(X1,X2) -> c_6()
            f#(0(),X) -> c_7()
            p#(X,0()) -> c_8()
            p#(0(),X) -> c_9()
            q#(0()) -> c_10()
            s#(X) -> c_11()
            t#(N) -> c_12(q#(N))
            t#(X) -> c_13()
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + 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(cs) = {1},
            uargs(f) = {1,2},
            uargs(r) = {1},
            uargs(s) = {1},
            uargs(t) = {1},
            uargs(f#) = {1,2},
            uargs(s#) = {1},
            uargs(t#) = {1},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_4) = {1},
            uargs(c_12) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]                   
               p(a) = [4] x1 + [1]          
              p(cs) = [1] x1 + [0]          
               p(d) = [0]                   
               p(f) = [1] x1 + [1] x2 + [14]
              p(nf) = [1] x1 + [1] x2 + [4] 
             p(nil) = [0]                   
              p(ns) = [1] x1 + [4]          
              p(nt) = [1] x1 + [6]          
               p(p) = [1] x2 + [0]          
               p(q) = [1] x1 + [2]          
               p(r) = [1] x1 + [3]          
               p(s) = [1] x1 + [8]          
               p(t) = [1] x1 + [8]          
              p(a#) = [5] x1 + [0]          
              p(d#) = [0]                   
              p(f#) = [1] x1 + [1] x2 + [0] 
              p(p#) = [5]                   
              p(q#) = [1] x1 + [13]         
              p(s#) = [1] x1 + [0]          
              p(t#) = [1] x1 + [0]          
             p(c_1) = [0]                   
             p(c_2) = [1] x1 + [0]          
             p(c_3) = [1] x1 + [0]          
             p(c_4) = [1] x1 + [0]          
             p(c_5) = [1]                   
             p(c_6) = [1]                   
             p(c_7) = [8]                   
             p(c_8) = [2]                   
             p(c_9) = [2]                   
            p(c_10) = [2]                   
            p(c_11) = [1]                   
            p(c_12) = [1] x1 + [5]          
            p(c_13) = [0]                   
          
          Following rules are strictly oriented:
          a#(nf(X1,X2)) = [5] X1 + [5] X2 + [20]
                        > [4] X1 + [4] X2 + [2] 
                        = c_2(f#(a(X1),a(X2)))  
          
              a#(ns(X)) = [5] X + [20]          
                        > [4] X + [1]           
                        = c_3(s#(a(X)))         
          
              a#(nt(X)) = [5] X + [30]          
                        > [4] X + [1]           
                        = c_4(t#(a(X)))         
          
              p#(X,0()) = [5]                   
                        > [2]                   
                        = c_8()                 
          
              p#(0(),X) = [5]                   
                        > [2]                   
                        = c_9()                 
          
                q#(0()) = [13]                  
                        > [2]                   
                        = c_10()                
          
                   a(X) = [4] X + [1]           
                        > [1] X + [0]           
                        = X                     
          
           a(nf(X1,X2)) = [4] X1 + [4] X2 + [17]
                        > [4] X1 + [4] X2 + [16]
                        = f(a(X1),a(X2))        
          
               a(ns(X)) = [4] X + [17]          
                        > [4] X + [9]           
                        = s(a(X))               
          
               a(nt(X)) = [4] X + [25]          
                        > [4] X + [9]           
                        = t(a(X))               
          
               f(X1,X2) = [1] X1 + [1] X2 + [14]
                        > [1] X1 + [1] X2 + [4] 
                        = nf(X1,X2)             
          
               f(0(),X) = [1] X + [14]          
                        > [0]                   
                        = nil()                 
          
                 q(0()) = [2]                   
                        > [0]                   
                        = 0()                   
          
                   s(X) = [1] X + [8]           
                        > [1] X + [4]           
                        = ns(X)                 
          
                   t(N) = [1] N + [8]           
                        > [1] N + [5]           
                        = cs(r(q(N)),nt(ns(N))) 
          
                   t(X) = [1] X + [8]           
                        > [1] X + [6]           
                        = nt(X)                 
          
          
          Following rules are (at-least) weakly oriented:
              a#(X) =  [5] X + [0]          
                    >= [0]                  
                    =  c_1()                
          
            d#(0()) =  [0]                  
                    >= [1]                  
                    =  c_5()                
          
          f#(X1,X2) =  [1] X1 + [1] X2 + [0]
                    >= [1]                  
                    =  c_6()                
          
          f#(0(),X) =  [1] X + [0]          
                    >= [8]                  
                    =  c_7()                
          
              s#(X) =  [1] X + [0]          
                    >= [1]                  
                    =  c_11()               
          
              t#(N) =  [1] N + [0]          
                    >= [1] N + [18]         
                    =  c_12(q#(N))          
          
              t#(X) =  [1] X + [0]          
                    >= [0]                  
                    =  c_13()               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            a#(X) -> c_1()
            d#(0()) -> c_5()
            f#(X1,X2) -> c_6()
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12(q#(N))
            t#(X) -> c_13()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            p#(X,0()) -> c_8()
            p#(0(),X) -> c_9()
            q#(0()) -> c_10()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {}.
        Here rules are labelled as follows:
          1: a#(X) -> c_1()
          2: d#(0()) -> c_5()
          3: f#(X1,X2) -> c_6()
          4: f#(0(),X) -> c_7()
          5: s#(X) -> c_11()
          6: t#(N) -> c_12(q#(N))
          7: t#(X) -> c_13()
          8: a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
          9: a#(ns(X)) -> c_3(s#(a(X)))
          10: a#(nt(X)) -> c_4(t#(a(X)))
          11: p#(X,0()) -> c_8()
          12: p#(0(),X) -> c_9()
          13: q#(0()) -> c_10()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X1,X2) -> c_6()
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12(q#(N))
            t#(X) -> c_13()
        - Weak DPs:
            a#(X) -> c_1()
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            d#(0()) -> c_5()
            p#(X,0()) -> c_8()
            p#(0(),X) -> c_9()
            q#(0()) -> c_10()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(X1,X2) -> c_6()
             
          
          2:S:f#(0(),X) -> c_7()
             
          
          3:S:s#(X) -> c_11()
             
          
          4:S:t#(N) -> c_12(q#(N))
             -->_1 q#(0()) -> c_10():13
          
          5:S:t#(X) -> c_13()
             
          
          6:W:a#(X) -> c_1()
             
          
          7:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(0(),X) -> c_7():2
             -->_1 f#(X1,X2) -> c_6():1
          
          8:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():3
          
          9:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(X) -> c_13():5
             -->_1 t#(N) -> c_12(q#(N)):4
          
          10:W:d#(0()) -> c_5()
             
          
          11:W:p#(X,0()) -> c_8()
             
          
          12:W:p#(0(),X) -> c_9()
             
          
          13:W:q#(0()) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: p#(0(),X) -> c_9()
          11: p#(X,0()) -> c_8()
          10: d#(0()) -> c_5()
          6: a#(X) -> c_1()
          13: q#(0()) -> c_10()
* Step 7: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X1,X2) -> c_6()
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12(q#(N))
            t#(X) -> c_13()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(X1,X2) -> c_6()
             
          
          2:S:f#(0(),X) -> c_7()
             
          
          3:S:s#(X) -> c_11()
             
          
          4:S:t#(N) -> c_12(q#(N))
             
          
          5:S:t#(X) -> c_13()
             
          
          7:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(0(),X) -> c_7():2
             -->_1 f#(X1,X2) -> c_6():1
          
          8:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():3
          
          9:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(X) -> c_13():5
             -->_1 t#(N) -> c_12(q#(N)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          t#(N) -> c_12()
* Step 8: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X1,X2) -> c_6()
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              f#(X1,X2) -> c_6()
          - Weak DPs:
              a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
              a#(ns(X)) -> c_3(s#(a(X)))
              a#(nt(X)) -> c_4(t#(a(X)))
              f#(0(),X) -> c_7()
              s#(X) -> c_11()
              t#(N) -> c_12()
              t#(X) -> c_13()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
        
        Problem (S)
          - Strict DPs:
              f#(0(),X) -> c_7()
              s#(X) -> c_11()
              t#(N) -> c_12()
              t#(X) -> c_13()
          - Weak DPs:
              a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
              a#(ns(X)) -> c_3(s#(a(X)))
              a#(nt(X)) -> c_4(t#(a(X)))
              f#(X1,X2) -> c_6()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X1,X2) -> c_6()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(X1,X2) -> c_6()
             
          
          2:W:f#(0(),X) -> c_7()
             
          
          3:W:s#(X) -> c_11()
             
          
          4:W:t#(N) -> c_12()
             
          
          5:W:t#(X) -> c_13()
             
          
          6:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(X1,X2) -> c_6():1
             -->_1 f#(0(),X) -> c_7():2
          
          7:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():3
          
          8:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(N) -> c_12():4
             -->_1 t#(X) -> c_13():5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: a#(nt(X)) -> c_4(t#(a(X)))
          7: a#(ns(X)) -> c_3(s#(a(X)))
          5: t#(X) -> c_13()
          4: t#(N) -> c_12()
          3: s#(X) -> c_11()
          2: f#(0(),X) -> c_7()
** Step 8.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X1,X2) -> c_6()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:f#(X1,X2) -> c_6()
             
          
          6:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(X1,X2) -> c_6():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
** Step 8.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            f#(X1,X2) -> c_6()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(0(),X) -> c_7()
             
          
          2:S:s#(X) -> c_11()
             
          
          3:S:t#(N) -> c_12()
             
          
          4:S:t#(X) -> c_13()
             
          
          5:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(X1,X2) -> c_6():8
             -->_1 f#(0(),X) -> c_7():1
          
          6:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():2
          
          7:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(X) -> c_13():4
             -->_1 t#(N) -> c_12():3
          
          8:W:f#(X1,X2) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: f#(X1,X2) -> c_6()
** Step 8.b:2: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),X) -> c_7()
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              f#(0(),X) -> c_7()
          - Weak DPs:
              a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
              a#(ns(X)) -> c_3(s#(a(X)))
              a#(nt(X)) -> c_4(t#(a(X)))
              s#(X) -> c_11()
              t#(N) -> c_12()
              t#(X) -> c_13()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
        
        Problem (S)
          - Strict DPs:
              s#(X) -> c_11()
              t#(N) -> c_12()
              t#(X) -> c_13()
          - Weak DPs:
              a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
              a#(ns(X)) -> c_3(s#(a(X)))
              a#(nt(X)) -> c_4(t#(a(X)))
              f#(0(),X) -> c_7()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
*** Step 8.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),X) -> c_7()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(0(),X) -> c_7()
             
          
          2:W:s#(X) -> c_11()
             
          
          3:W:t#(N) -> c_12()
             
          
          4:W:t#(X) -> c_13()
             
          
          5:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(0(),X) -> c_7():1
          
          6:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():2
          
          7:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(N) -> c_12():3
             -->_1 t#(X) -> c_13():4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: a#(nt(X)) -> c_4(t#(a(X)))
          6: a#(ns(X)) -> c_3(s#(a(X)))
          4: t#(X) -> c_13()
          3: t#(N) -> c_12()
          2: s#(X) -> c_11()
*** Step 8.b:2.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),X) -> c_7()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:f#(0(),X) -> c_7()
             
          
          5:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(0(),X) -> c_7():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
*** Step 8.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 8.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            f#(0(),X) -> c_7()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_11()
             
          
          2:S:t#(N) -> c_12()
             
          
          3:S:t#(X) -> c_13()
             
          
          4:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
             -->_1 f#(0(),X) -> c_7():7
          
          5:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():1
          
          6:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(X) -> c_13():3
             -->_1 t#(N) -> c_12():2
          
          7:W:f#(0(),X) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
          7: f#(0(),X) -> c_7()
*** Step 8.b:2.b:2: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_11()
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              s#(X) -> c_11()
          - Weak DPs:
              a#(ns(X)) -> c_3(s#(a(X)))
              a#(nt(X)) -> c_4(t#(a(X)))
              t#(N) -> c_12()
              t#(X) -> c_13()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
        
        Problem (S)
          - Strict DPs:
              t#(N) -> c_12()
              t#(X) -> c_13()
          - Weak DPs:
              a#(ns(X)) -> c_3(s#(a(X)))
              a#(nt(X)) -> c_4(t#(a(X)))
              s#(X) -> c_11()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
**** Step 8.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_11()
        - Weak DPs:
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_11()
             
          
          2:W:t#(N) -> c_12()
             
          
          3:W:t#(X) -> c_13()
             
          
          5:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():1
          
          6:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(N) -> c_12():2
             -->_1 t#(X) -> c_13():3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: a#(nt(X)) -> c_4(t#(a(X)))
          3: t#(X) -> c_13()
          2: t#(N) -> c_12()
**** Step 8.b:2.b:2.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_11()
        - Weak DPs:
            a#(ns(X)) -> c_3(s#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_11()
             
          
          5:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
**** Step 8.b:2.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 8.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(ns(X)) -> c_3(s#(a(X)))
            a#(nt(X)) -> c_4(t#(a(X)))
            s#(X) -> c_11()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:t#(N) -> c_12()
             
          
          2:S:t#(X) -> c_13()
             
          
          3:W:a#(ns(X)) -> c_3(s#(a(X)))
             -->_1 s#(X) -> c_11():5
          
          4:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(X) -> c_13():2
             -->_1 t#(N) -> c_12():1
          
          5:W:s#(X) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: a#(ns(X)) -> c_3(s#(a(X)))
          5: s#(X) -> c_11()
**** Step 8.b:2.b:2.b:2: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(N) -> c_12()
            t#(X) -> c_13()
        - Weak DPs:
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              t#(N) -> c_12()
          - Weak DPs:
              a#(nt(X)) -> c_4(t#(a(X)))
              t#(X) -> c_13()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
        
        Problem (S)
          - Strict DPs:
              t#(X) -> c_13()
          - Weak DPs:
              a#(nt(X)) -> c_4(t#(a(X)))
              t#(N) -> c_12()
          - Weak TRS:
              a(X) -> X
              a(nf(X1,X2)) -> f(a(X1),a(X2))
              a(ns(X)) -> s(a(X))
              a(nt(X)) -> t(a(X))
              f(X1,X2) -> nf(X1,X2)
              f(0(),X) -> nil()
              q(0()) -> 0()
              s(X) -> ns(X)
              t(N) -> cs(r(q(N)),nt(ns(N)))
              t(X) -> nt(X)
          - Signature:
              {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
              ,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
              ,nt,r}
***** Step 8.b:2.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(N) -> c_12()
        - Weak DPs:
            a#(nt(X)) -> c_4(t#(a(X)))
            t#(X) -> c_13()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:t#(N) -> c_12()
             
          
          2:W:t#(X) -> c_13()
             
          
          4:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(N) -> c_12():1
             -->_1 t#(X) -> c_13():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: t#(X) -> c_13()
***** Step 8.b:2.b:2.b:2.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(N) -> c_12()
        - Weak DPs:
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:t#(N) -> c_12()
             
          
          4:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(N) -> c_12():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
***** Step 8.b:2.b:2.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 8.b:2.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(X) -> c_13()
        - Weak DPs:
            a#(nt(X)) -> c_4(t#(a(X)))
            t#(N) -> c_12()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:t#(X) -> c_13()
             
          
          2:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(N) -> c_12():3
             -->_1 t#(X) -> c_13():1
          
          3:W:t#(N) -> c_12()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: t#(N) -> c_12()
***** Step 8.b:2.b:2.b:2.b:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(X) -> c_13()
        - Weak DPs:
            a#(nt(X)) -> c_4(t#(a(X)))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:t#(X) -> c_13()
             
          
          2:W:a#(nt(X)) -> c_4(t#(a(X)))
             -->_1 t#(X) -> c_13():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
***** Step 8.b:2.b:2.b:2.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            q(0()) -> 0()
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))