WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            fold(a,xs) -> Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
            foldl(a,Nil()) -> a
            foldl(x,Cons(S(0()),xs)) -> foldl(S(x),xs)
            foldl(S(0()),Cons(x,xs)) -> foldl(S(x),xs)
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold,foldl,foldr,notEmpty,op} and constructors {0,Cons
            ,False,Nil,S,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
          foldl#(a,Nil()) -> c_2()
          foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
          foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
          foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
          foldr#(a,Nil()) -> c_6()
          notEmpty#(Cons(x,xs)) -> c_7()
          notEmpty#(Nil()) -> c_8()
          op#(x,S(0())) -> c_9()
          op#(S(0()),y) -> c_10()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
            foldl#(a,Nil()) -> c_2()
            foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
            foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
            foldr#(a,Nil()) -> c_6()
            notEmpty#(Cons(x,xs)) -> c_7()
            notEmpty#(Nil()) -> c_8()
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Strict TRS:
            fold(a,xs) -> Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
            foldl(a,Nil()) -> a
            foldl(x,Cons(S(0()),xs)) -> foldl(S(x),xs)
            foldl(S(0()),Cons(x,xs)) -> foldl(S(x),xs)
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/2,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
          foldr(a,Nil()) -> a
          op(x,S(0())) -> S(x)
          op(S(0()),y) -> S(y)
          fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
          foldl#(a,Nil()) -> c_2()
          foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
          foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
          foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
          foldr#(a,Nil()) -> c_6()
          notEmpty#(Cons(x,xs)) -> c_7()
          notEmpty#(Nil()) -> c_8()
          op#(x,S(0())) -> c_9()
          op#(S(0()),y) -> c_10()
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
            foldl#(a,Nil()) -> c_2()
            foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
            foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
            foldr#(a,Nil()) -> c_6()
            notEmpty#(Cons(x,xs)) -> c_7()
            notEmpty#(Nil()) -> c_8()
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Strict TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/2,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + 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(op) = {2},
            uargs(op#) = {2},
            uargs(c_1) = {1,2},
            uargs(c_3) = {1},
            uargs(c_4) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [1]                  
                 p(Cons) = [1] x2 + [4]         
                p(False) = [0]                  
                  p(Nil) = [2]                  
                    p(S) = [0]                  
                 p(True) = [0]                  
                 p(fold) = [0]                  
                p(foldl) = [0]                  
                p(foldr) = [2] x1 + [4] x2 + [1]
             p(notEmpty) = [0]                  
                   p(op) = [1] x2 + [6]         
                p(fold#) = [5] x1 + [5] x2 + [0]
               p(foldl#) = [1] x2 + [0]         
               p(foldr#) = [2] x1 + [4] x2 + [0]
            p(notEmpty#) = [6] x1 + [0]         
                  p(op#) = [1] x2 + [0]         
                  p(c_1) = [1] x1 + [1] x2 + [0]
                  p(c_2) = [0]                  
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [1] x1 + [0]         
                  p(c_5) = [1] x1 + [0]         
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [0]                  
          
          Following rules are strictly oriented:
                    foldl#(a,Nil()) = [2]                    
                                    > [0]                    
                                    = c_2()                  
          
          foldl#(x,Cons(S(0()),xs)) = [1] xs + [4]           
                                    > [1] xs + [0]           
                                    = c_3(foldl#(S(x),xs))   
          
          foldl#(S(0()),Cons(x,xs)) = [1] xs + [4]           
                                    > [1] xs + [0]           
                                    = c_4(foldl#(S(x),xs))   
          
               foldr#(a,Cons(x,xs)) = [2] a + [4] xs + [16]  
                                    > [2] a + [4] xs + [1]   
                                    = c_5(op#(x,foldr(a,xs)))
          
                    foldr#(a,Nil()) = [2] a + [8]            
                                    > [0]                    
                                    = c_6()                  
          
              notEmpty#(Cons(x,xs)) = [6] xs + [24]          
                                    > [0]                    
                                    = c_7()                  
          
                   notEmpty#(Nil()) = [12]                   
                                    > [0]                    
                                    = c_8()                  
          
                foldr(a,Cons(x,xs)) = [2] a + [4] xs + [17]  
                                    > [2] a + [4] xs + [7]   
                                    = op(x,foldr(a,xs))      
          
                     foldr(a,Nil()) = [2] a + [9]            
                                    > [1] a + [0]            
                                    = a                      
          
                       op(x,S(0())) = [6]                    
                                    > [0]                    
                                    = S(x)                   
          
                       op(S(0()),y) = [1] y + [6]            
                                    > [0]                    
                                    = S(y)                   
          
          
          Following rules are (at-least) weakly oriented:
            fold#(a,xs) =  [5] a + [5] xs + [0]          
                        >= [2] a + [5] xs + [0]          
                        =  c_1(foldl#(a,xs),foldr#(a,xs))
          
          op#(x,S(0())) =  [0]                           
                        >= [0]                           
                        =  c_9()                         
          
          op#(S(0()),y) =  [1] y + [0]                   
                        >= [0]                           
                        =  c_10()                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            foldl#(a,Nil()) -> c_2()
            foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
            foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
            foldr#(a,Nil()) -> c_6()
            notEmpty#(Cons(x,xs)) -> c_7()
            notEmpty#(Nil()) -> c_8()
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/2,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
          2: op#(x,S(0())) -> c_9()
          3: op#(S(0()),y) -> c_10()
          4: foldl#(a,Nil()) -> c_2()
          5: foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
          6: foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
          7: foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
          8: foldr#(a,Nil()) -> c_6()
          9: notEmpty#(Cons(x,xs)) -> c_7()
          10: notEmpty#(Nil()) -> c_8()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
            foldl#(a,Nil()) -> c_2()
            foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
            foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
            foldr#(a,Nil()) -> c_6()
            notEmpty#(Cons(x,xs)) -> c_7()
            notEmpty#(Nil()) -> c_8()
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/2,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:op#(x,S(0())) -> c_9()
             
          
          2:S:op#(S(0()),y) -> c_10()
             
          
          3:W:fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
             -->_2 foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs))):7
             -->_1 foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs)):6
             -->_1 foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs)):5
             -->_2 foldr#(a,Nil()) -> c_6():8
             -->_1 foldl#(a,Nil()) -> c_2():4
          
          4:W:foldl#(a,Nil()) -> c_2()
             
          
          5:W:foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
             -->_1 foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs)):6
             -->_1 foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs)):5
             -->_1 foldl#(a,Nil()) -> c_2():4
          
          6:W:foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
             -->_1 foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs)):6
             -->_1 foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs)):5
             -->_1 foldl#(a,Nil()) -> c_2():4
          
          7:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
             -->_1 op#(S(0()),y) -> c_10():2
             -->_1 op#(x,S(0())) -> c_9():1
          
          8:W:foldr#(a,Nil()) -> c_6()
             
          
          9:W:notEmpty#(Cons(x,xs)) -> c_7()
             
          
          10:W:notEmpty#(Nil()) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: notEmpty#(Nil()) -> c_8()
          9: notEmpty#(Cons(x,xs)) -> c_7()
          8: foldr#(a,Nil()) -> c_6()
          6: foldl#(S(0()),Cons(x,xs)) -> c_4(foldl#(S(x),xs))
          5: foldl#(x,Cons(S(0()),xs)) -> c_3(foldl#(S(x),xs))
          4: foldl#(a,Nil()) -> c_2()
* Step 6: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/2,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:op#(x,S(0())) -> c_9()
             
          
          2:S:op#(S(0()),y) -> c_10()
             
          
          3:W:fold#(a,xs) -> c_1(foldl#(a,xs),foldr#(a,xs))
             -->_2 foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs))):7
          
          7:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
             -->_1 op#(S(0()),y) -> c_10():2
             -->_1 op#(x,S(0())) -> c_9():1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fold#(a,xs) -> c_1(foldr#(a,xs))
* Step 7: RemoveHeads WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            fold#(a,xs) -> c_1(foldr#(a,xs))
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:op#(x,S(0())) -> c_9()
           
        
        2:S:op#(S(0()),y) -> c_10()
           
        
        3:W:fold#(a,xs) -> c_1(foldr#(a,xs))
           -->_1 foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs))):4
        
        4:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
           -->_1 op#(S(0()),y) -> c_10():2
           -->_1 op#(x,S(0())) -> c_9():1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(3,fold#(a,xs) -> c_1(foldr#(a,xs)))]
* Step 8: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(x,S(0())) -> c_9()
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + 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:
              op#(x,S(0())) -> c_9()
          - Weak DPs:
              foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
              op#(S(0()),y) -> c_10()
          - Weak TRS:
              foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
              foldr(a,Nil()) -> a
              op(x,S(0())) -> S(x)
              op(S(0()),y) -> S(y)
          - Signature:
              {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
              ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
              ,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              op#(S(0()),y) -> c_10()
          - Weak DPs:
              foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
              op#(x,S(0())) -> c_9()
          - Weak TRS:
              foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
              foldr(a,Nil()) -> a
              op(x,S(0())) -> S(x)
              op(S(0()),y) -> S(y)
          - Signature:
              {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
              ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
              ,Cons,False,Nil,S,True}
** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(x,S(0())) -> c_9()
        - Weak DPs:
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
            op#(S(0()),y) -> c_10()
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:op#(x,S(0())) -> c_9()
             
          
          2:W:op#(S(0()),y) -> c_10()
             
          
          4:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
             -->_1 op#(x,S(0())) -> c_9():1
             -->_1 op#(S(0()),y) -> c_10():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: op#(S(0()),y) -> c_10()
** Step 8.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(x,S(0())) -> c_9()
        - Weak DPs:
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:op#(x,S(0())) -> c_9()
             
          
          4:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
             -->_1 op#(x,S(0())) -> c_9():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:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + 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:
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
            op#(x,S(0())) -> c_9()
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:op#(S(0()),y) -> c_10()
             
          
          2:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
             -->_1 op#(x,S(0())) -> c_9():3
             -->_1 op#(S(0()),y) -> c_10():1
          
          3:W:op#(x,S(0())) -> c_9()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: op#(x,S(0())) -> c_9()
** Step 8.b:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            op#(S(0()),y) -> c_10()
        - Weak DPs:
            foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:op#(S(0()),y) -> c_10()
             
          
          2:W:foldr#(a,Cons(x,xs)) -> c_5(op#(x,foldr(a,xs)))
             -->_1 op#(S(0()),y) -> c_10():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
** Step 8.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2,fold#/2,foldl#/2,foldr#/2,notEmpty#/1,op#/2} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold#,foldl#,foldr#,notEmpty#,op#} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))