* Step 1: DependencyPairs WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict TRS:
            f_0(x) -> a()
            f_1(x) -> g_1(x,x)
            f_2(x) -> g_2(x,x)
            f_3(x) -> g_3(x,x)
            f_4(x) -> g_4(x,x)
            f_5(x) -> g_5(x,x)
            g_1(s(x),y) -> b(f_0(y),g_1(x,y))
            g_2(s(x),y) -> b(f_1(y),g_2(x,y))
            g_3(s(x),y) -> b(f_2(y),g_3(x,y))
            g_4(s(x),y) -> b(f_3(y),g_4(x,y))
            g_5(s(x),y) -> b(f_4(y),g_5(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2} / {a/0,b/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0,f_1,f_2,f_3,f_4,f_5,g_1,g_2,g_3,g_4
            ,g_5} and constructors {a,b,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          f_0#(x) -> c_1()
          f_1#(x) -> c_2(g_1#(x,x))
          f_2#(x) -> c_3(g_2#(x,x))
          f_3#(x) -> c_4(g_3#(x,x))
          f_4#(x) -> c_5(g_4#(x,x))
          f_5#(x) -> c_6(g_5#(x,x))
          g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
          g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_0#(x) -> c_1()
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            f_5#(x) -> c_6(g_5#(x,x))
            g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Strict TRS:
            f_0(x) -> a()
            f_1(x) -> g_1(x,x)
            f_2(x) -> g_2(x,x)
            f_3(x) -> g_3(x,x)
            f_4(x) -> g_4(x,x)
            f_5(x) -> g_5(x,x)
            g_1(s(x),y) -> b(f_0(y),g_1(x,y))
            g_2(s(x),y) -> b(f_1(y),g_2(x,y))
            g_3(s(x),y) -> b(f_2(y),g_3(x,y))
            g_4(s(x),y) -> b(f_3(y),g_4(x,y))
            g_5(s(x),y) -> b(f_4(y),g_5(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f_0#(x) -> c_1()
          f_1#(x) -> c_2(g_1#(x,x))
          f_2#(x) -> c_3(g_2#(x,x))
          f_3#(x) -> c_4(g_3#(x,x))
          f_4#(x) -> c_5(g_4#(x,x))
          f_5#(x) -> c_6(g_5#(x,x))
          g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
          g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_0#(x) -> c_1()
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            f_5#(x) -> c_6(g_5#(x,x))
            g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {7}.
        Here rules are labelled as follows:
          1: f_0#(x) -> c_1()
          2: f_1#(x) -> c_2(g_1#(x,x))
          3: f_2#(x) -> c_3(g_2#(x,x))
          4: f_3#(x) -> c_4(g_3#(x,x))
          5: f_4#(x) -> c_5(g_4#(x,x))
          6: f_5#(x) -> c_6(g_5#(x,x))
          7: g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
          8: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
          9: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          10: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          11: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            f_5#(x) -> c_6(g_5#(x,x))
            g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Weak DPs:
            f_0#(x) -> c_1()
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f_1#(x) -> c_2(g_1#(x,x))
             -->_1 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6
          
          2:S:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
          
          3:S:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
          
          4:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
          
          5:S:f_5#(x) -> c_6(g_5#(x,x))
             -->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
          
          6:S:g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
             -->_1 f_0#(x) -> c_1():11
             -->_2 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6
          
          7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
             -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
             -->_1 f_1#(x) -> c_2(g_1#(x,x)):1
          
          8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):2
          
          9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):3
          
          10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):4
          
          11:W:f_0#(x) -> c_1()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          11: f_0#(x) -> c_1()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            f_5#(x) -> c_6(g_5#(x,x))
            g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f_1#(x) -> c_2(g_1#(x,x))
             -->_1 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6
          
          2:S:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
          
          3:S:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
          
          4:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
          
          5:S:f_5#(x) -> c_6(g_5#(x,x))
             -->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
          
          6:S:g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
             -->_2 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6
          
          7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
             -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
             -->_1 f_1#(x) -> c_2(g_1#(x,x)):1
          
          8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):2
          
          9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):3
          
          10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g_1#(s(x),y) -> c_7(g_1#(x,y))
* Step 6: RemoveHeads WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            f_5#(x) -> c_6(g_5#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:f_1#(x) -> c_2(g_1#(x,x))
           -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6
        
        2:S:f_2#(x) -> c_3(g_2#(x,x))
           -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
        
        3:S:f_3#(x) -> c_4(g_3#(x,x))
           -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
        
        4:S:f_4#(x) -> c_5(g_4#(x,x))
           -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
        
        5:S:f_5#(x) -> c_6(g_5#(x,x))
           -->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
        
        6:S:g_1#(s(x),y) -> c_7(g_1#(x,y))
           -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6
        
        7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
           -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
           -->_1 f_1#(x) -> c_2(g_1#(x,x)):1
        
        8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
           -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
           -->_1 f_2#(x) -> c_3(g_2#(x,x)):2
        
        9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
           -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
           -->_1 f_3#(x) -> c_4(g_3#(x,x)):3
        
        10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
           -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
           -->_1 f_4#(x) -> c_5(g_4#(x,x)):4
        
        
        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).
        
        [(5,f_5#(x) -> c_6(g_5#(x,x)))]
* Step 7: Decompose WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + 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_1#(x) -> c_2(g_1#(x,x))
              g_1#(s(x),y) -> c_7(g_1#(x,y))
          - Weak DPs:
              f_2#(x) -> c_3(g_2#(x,x))
              f_3#(x) -> c_4(g_3#(x,x))
              f_4#(x) -> c_5(g_4#(x,x))
              g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
              g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
              g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2
              ,c_9/2,c_10/2,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
        
        Problem (S)
          - Strict DPs:
              f_2#(x) -> c_3(g_2#(x,x))
              f_3#(x) -> c_4(g_3#(x,x))
              f_4#(x) -> c_5(g_4#(x,x))
              g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
              g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
              g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Weak DPs:
              f_1#(x) -> c_2(g_1#(x,x))
              g_1#(s(x),y) -> c_7(g_1#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2
              ,c_9/2,c_10/2,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
** Step 7.a:1: DecomposeDG WORST_CASE(?,O(n^5))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        and a lower component
          f_1#(x) -> c_2(g_1#(x,x))
          f_2#(x) -> c_3(g_2#(x,x))
          f_3#(x) -> c_4(g_3#(x,x))
          f_4#(x) -> c_5(g_4#(x,x))
          g_1#(s(x),y) -> c_7(g_1#(x,y))
          g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        Further, following extension rules are added to the lower component.
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
*** Step 7.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 7.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_11) = {2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [0]                   
             p(b) = [1] x1 + [1] x2 + [0] 
           p(f_0) = [0]                   
           p(f_1) = [0]                   
           p(f_2) = [0]                   
           p(f_3) = [0]                   
           p(f_4) = [0]                   
           p(f_5) = [0]                   
           p(g_1) = [0]                   
           p(g_2) = [0]                   
           p(g_3) = [0]                   
           p(g_4) = [0]                   
           p(g_5) = [1] x2 + [0]          
             p(s) = [1] x1 + [1]          
          p(f_0#) = [0]                   
          p(f_1#) = [1]                   
          p(f_2#) = [1] x1 + [1]          
          p(f_3#) = [1] x1 + [0]          
          p(f_4#) = [0]                   
          p(f_5#) = [0]                   
          p(g_1#) = [1] x1 + [4]          
          p(g_2#) = [4] x2 + [1]          
          p(g_3#) = [0]                   
          p(g_4#) = [2] x2 + [1]          
          p(g_5#) = [1] x1 + [11] x2 + [0]
           p(c_1) = [1]                   
           p(c_2) = [0]                   
           p(c_3) = [2]                   
           p(c_4) = [0]                   
           p(c_5) = [1] x1 + [1]          
           p(c_6) = [1] x1 + [0]          
           p(c_7) = [1]                   
           p(c_8) = [2] x1 + [1] x2 + [2] 
           p(c_9) = [2] x1 + [1]          
          p(c_10) = [1] x2 + [1]          
          p(c_11) = [1] x2 + [0]          
        
        Following rules are strictly oriented:
        g_5#(s(x),y) = [1] x + [11] y + [1]   
                     > [1] x + [11] y + [0]   
                     = c_11(f_4#(y),g_5#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 7.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 7.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
**** Step 7.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 7.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          f_4#(x) -> c_5(g_4#(x,x))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
        and a lower component
          f_1#(x) -> c_2(g_1#(x,x))
          f_2#(x) -> c_3(g_2#(x,x))
          f_3#(x) -> c_4(g_3#(x,x))
          g_1#(s(x),y) -> c_7(g_1#(x,y))
          g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
        Further, following extension rules are added to the lower component.
          f_4#(x) -> g_4#(x,x)
          g_4#(s(x),y) -> f_3#(y)
          g_4#(s(x),y) -> g_4#(x,y)
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
**** Step 7.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 7.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_10) = {2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [2]                  
             p(b) = [1] x1 + [2]         
           p(f_0) = [1]                  
           p(f_1) = [0]                  
           p(f_2) = [1] x1 + [0]         
           p(f_3) = [0]                  
           p(f_4) = [2] x1 + [1]         
           p(f_5) = [1] x1 + [1]         
           p(g_1) = [8] x1 + [1] x2 + [1]
           p(g_2) = [1] x1 + [1]         
           p(g_3) = [1] x1 + [1] x2 + [0]
           p(g_4) = [1] x1 + [1] x2 + [0]
           p(g_5) = [1] x1 + [1] x2 + [0]
             p(s) = [1] x1 + [8]         
          p(f_0#) = [0]                  
          p(f_1#) = [1] x1 + [1]         
          p(f_2#) = [1] x1 + [2]         
          p(f_3#) = [0]                  
          p(f_4#) = [2] x1 + [2]         
          p(f_5#) = [8] x1 + [2]         
          p(g_1#) = [8]                  
          p(g_2#) = [2]                  
          p(g_3#) = [1] x2 + [1]         
          p(g_4#) = [1] x1 + [0]         
          p(g_5#) = [8] x2 + [11]        
           p(c_1) = [4]                  
           p(c_2) = [1] x1 + [1]         
           p(c_3) = [1]                  
           p(c_4) = [4]                  
           p(c_5) = [2] x1 + [2]         
           p(c_6) = [1] x1 + [0]         
           p(c_7) = [1]                  
           p(c_8) = [1] x1 + [1]         
           p(c_9) = [1] x1 + [0]         
          p(c_10) = [1] x2 + [2]         
          p(c_11) = [1] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        g_4#(s(x),y) = [1] x + [8]            
                     > [1] x + [2]            
                     = c_10(f_3#(y),g_4#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_4#(x) =  [2] x + [2]   
                     >= [2] x + [2]   
                     =  c_5(g_4#(x,x))
        
        g_5#(s(x),y) =  [8] y + [11]  
                     >= [2] y + [2]   
                     =  f_4#(y)       
        
        g_5#(s(x),y) =  [8] y + [11]  
                     >= [8] y + [11]  
                     =  g_5#(x,y)     
        
***** Step 7.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 7.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          2:W:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          3:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):1
          
          4:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):4
             -->_1 g_5#(s(x),y) -> f_4#(y):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: g_5#(s(x),y) -> g_5#(x,y)
          3: g_5#(s(x),y) -> f_4#(y)
          1: f_4#(x) -> c_5(g_4#(x,x))
          2: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
***** Step 7.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 7.a:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          f_3#(x) -> c_4(g_3#(x,x))
          f_4#(x) -> g_4#(x,x)
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          g_4#(s(x),y) -> f_3#(y)
          g_4#(s(x),y) -> g_4#(x,y)
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
        and a lower component
          f_1#(x) -> c_2(g_1#(x,x))
          f_2#(x) -> c_3(g_2#(x,x))
          g_1#(s(x),y) -> c_7(g_1#(x,y))
          g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
        Further, following extension rules are added to the lower component.
          f_3#(x) -> g_3#(x,x)
          f_4#(x) -> g_4#(x,x)
          g_3#(s(x),y) -> f_2#(y)
          g_3#(s(x),y) -> g_3#(x,y)
          g_4#(s(x),y) -> f_3#(y)
          g_4#(s(x),y) -> g_4#(x,y)
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
***** Step 7.a:1.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.a:1.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_4) = {1},
          uargs(c_9) = {2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [0]                  
             p(b) = [1] x1 + [1] x2 + [0]
           p(f_0) = [0]                  
           p(f_1) = [0]                  
           p(f_2) = [0]                  
           p(f_3) = [0]                  
           p(f_4) = [0]                  
           p(f_5) = [0]                  
           p(g_1) = [0]                  
           p(g_2) = [0]                  
           p(g_3) = [0]                  
           p(g_4) = [0]                  
           p(g_5) = [0]                  
             p(s) = [1] x1 + [5]         
          p(f_0#) = [0]                  
          p(f_1#) = [0]                  
          p(f_2#) = [0]                  
          p(f_3#) = [5] x1 + [10]        
          p(f_4#) = [5] x1 + [10]        
          p(f_5#) = [0]                  
          p(g_1#) = [0]                  
          p(g_2#) = [0]                  
          p(g_3#) = [1] x1 + [0]         
          p(g_4#) = [5] x2 + [10]        
          p(g_5#) = [2] x1 + [5] x2 + [0]
           p(c_1) = [0]                  
           p(c_2) = [1] x1 + [0]         
           p(c_3) = [1]                  
           p(c_4) = [4] x1 + [0]         
           p(c_5) = [0]                  
           p(c_6) = [0]                  
           p(c_7) = [0]                  
           p(c_8) = [0]                  
           p(c_9) = [1] x2 + [0]         
          p(c_10) = [0]                  
          p(c_11) = [0]                  
        
        Following rules are strictly oriented:
        g_3#(s(x),y) = [1] x + [5]           
                     > [1] x + [0]           
                     = c_9(f_2#(y),g_3#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_3#(x) =  [5] x + [10]        
                     >= [4] x + [0]         
                     =  c_4(g_3#(x,x))      
        
             f_4#(x) =  [5] x + [10]        
                     >= [5] x + [10]        
                     =  g_4#(x,x)           
        
        g_4#(s(x),y) =  [5] y + [10]        
                     >= [5] y + [10]        
                     =  f_3#(y)             
        
        g_4#(s(x),y) =  [5] y + [10]        
                     >= [5] y + [10]        
                     =  g_4#(x,y)           
        
        g_5#(s(x),y) =  [2] x + [5] y + [10]
                     >= [5] y + [10]        
                     =  f_4#(y)             
        
        g_5#(s(x),y) =  [2] x + [5] y + [10]
                     >= [2] x + [5] y + [0] 
                     =  g_5#(x,y)           
        
****** Step 7.a:1.b:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.a:1.b:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3
          
          2:W:f_4#(x) -> g_4#(x,x)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):5
             -->_1 g_4#(s(x),y) -> f_3#(y):4
          
          3:W:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3
          
          4:W:g_4#(s(x),y) -> f_3#(y)
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):1
          
          5:W:g_4#(s(x),y) -> g_4#(x,y)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):5
             -->_1 g_4#(s(x),y) -> f_3#(y):4
          
          6:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> g_4#(x,x):2
          
          7:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):7
             -->_1 g_5#(s(x),y) -> f_4#(y):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: g_5#(s(x),y) -> g_5#(x,y)
          6: g_5#(s(x),y) -> f_4#(y)
          2: f_4#(x) -> g_4#(x,x)
          5: g_4#(s(x),y) -> g_4#(x,y)
          4: g_4#(s(x),y) -> f_3#(y)
          1: f_3#(x) -> c_4(g_3#(x,x))
          3: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
****** Step 7.a:1.b:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 7.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> g_3#(x,x)
            f_4#(x) -> g_4#(x,x)
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> f_2#(y)
            g_3#(s(x),y) -> g_3#(x,y)
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f_1#(x) -> c_2(g_1#(x,x))
          2: g_1#(s(x),y) -> c_7(g_1#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.a:1.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> g_3#(x,x)
            f_4#(x) -> g_4#(x,x)
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> f_2#(y)
            g_3#(s(x),y) -> g_3#(x,y)
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_3) = {1},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = 1                                    
             p(b) = x1                                   
           p(f_0) = x1                                   
           p(f_1) = 1                                    
           p(f_2) = 1                                    
           p(f_3) = 1 + x1^2                             
           p(f_4) = 1 + x1 + x1^2                        
           p(f_5) = 2 + 2*x1                             
           p(g_1) = 2 + x1 + x1*x2 + x1^2 + 4*x2^2       
           p(g_2) = 4*x1 + 2*x1*x2                       
           p(g_3) = 1 + x1^2                             
           p(g_4) = 1 + 2*x1*x2 + 4*x1^2 + x2 + x2^2     
           p(g_5) = 0                                    
             p(s) = 1 + x1                               
          p(f_0#) = 4 + x1 + x1^2                        
          p(f_1#) = 2 + x1                               
          p(f_2#) = 5*x1 + 2*x1^2                        
          p(f_3#) = 6*x1 + 4*x1^2                        
          p(f_4#) = 4 + 6*x1 + 7*x1^2                    
          p(f_5#) = 0                                    
          p(g_1#) = x1                                   
          p(g_2#) = 3*x1 + 2*x1*x2 + 2*x2                
          p(g_3#) = 5*x2 + 3*x2^2                        
          p(g_4#) = 1 + x1^2 + 6*x2 + 5*x2^2             
          p(g_5#) = 4*x1 + 4*x1*x2 + x1^2 + 5*x2 + 7*x2^2
           p(c_1) = 0                                    
           p(c_2) = 1 + x1                               
           p(c_3) = x1                                   
           p(c_4) = 0                                    
           p(c_5) = 1                                    
           p(c_6) = 0                                    
           p(c_7) = x1                                   
           p(c_8) = x1 + x2                              
           p(c_9) = x2                                   
          p(c_10) = x1                                   
          p(c_11) = 1 + x2                               
        
        Following rules are strictly oriented:
             f_1#(x) = 2 + x         
                     > 1 + x         
                     = c_2(g_1#(x,x))
        
        g_1#(s(x),y) = 1 + x         
                     > x             
                     = c_7(g_1#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_2#(x) =  5*x + 2*x^2                        
                     >= 5*x + 2*x^2                        
                     =  c_3(g_2#(x,x))                     
        
             f_3#(x) =  6*x + 4*x^2                        
                     >= 5*x + 3*x^2                        
                     =  g_3#(x,x)                          
        
             f_4#(x) =  4 + 6*x + 7*x^2                    
                     >= 1 + 6*x + 6*x^2                    
                     =  g_4#(x,x)                          
        
        g_2#(s(x),y) =  3 + 3*x + 2*x*y + 4*y              
                     >= 2 + 3*x + 2*x*y + 3*y              
                     =  c_8(f_1#(y),g_2#(x,y))             
        
        g_3#(s(x),y) =  5*y + 3*y^2                        
                     >= 5*y + 2*y^2                        
                     =  f_2#(y)                            
        
        g_3#(s(x),y) =  5*y + 3*y^2                        
                     >= 5*y + 3*y^2                        
                     =  g_3#(x,y)                          
        
        g_4#(s(x),y) =  2 + 2*x + x^2 + 6*y + 5*y^2        
                     >= 6*y + 4*y^2                        
                     =  f_3#(y)                            
        
        g_4#(s(x),y) =  2 + 2*x + x^2 + 6*y + 5*y^2        
                     >= 1 + x^2 + 6*y + 5*y^2              
                     =  g_4#(x,y)                          
        
        g_5#(s(x),y) =  5 + 6*x + 4*x*y + x^2 + 9*y + 7*y^2
                     >= 4 + 6*y + 7*y^2                    
                     =  f_4#(y)                            
        
        g_5#(s(x),y) =  5 + 6*x + 4*x*y + x^2 + 9*y + 7*y^2
                     >= 4*x + 4*x*y + x^2 + 5*y + 7*y^2    
                     =  g_5#(x,y)                          
        
****** Step 7.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> g_3#(x,x)
            f_4#(x) -> g_4#(x,x)
            g_1#(s(x),y) -> c_7(g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> f_2#(y)
            g_3#(s(x),y) -> g_3#(x,y)
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.a:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> g_3#(x,x)
            f_4#(x) -> g_4#(x,x)
            g_1#(s(x),y) -> c_7(g_1#(x,y))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> f_2#(y)
            g_3#(s(x),y) -> g_3#(x,y)
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_1#(x) -> c_2(g_1#(x,x))
             -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):5
          
          2:W:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):6
          
          3:W:f_3#(x) -> g_3#(x,x)
             -->_1 g_3#(s(x),y) -> g_3#(x,y):8
             -->_1 g_3#(s(x),y) -> f_2#(y):7
          
          4:W:f_4#(x) -> g_4#(x,x)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):10
             -->_1 g_4#(s(x),y) -> f_3#(y):9
          
          5:W:g_1#(s(x),y) -> c_7(g_1#(x,y))
             -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):5
          
          6:W:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
             -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):6
             -->_1 f_1#(x) -> c_2(g_1#(x,x)):1
          
          7:W:g_3#(s(x),y) -> f_2#(y)
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):2
          
          8:W:g_3#(s(x),y) -> g_3#(x,y)
             -->_1 g_3#(s(x),y) -> g_3#(x,y):8
             -->_1 g_3#(s(x),y) -> f_2#(y):7
          
          9:W:g_4#(s(x),y) -> f_3#(y)
             -->_1 f_3#(x) -> g_3#(x,x):3
          
          10:W:g_4#(s(x),y) -> g_4#(x,y)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):10
             -->_1 g_4#(s(x),y) -> f_3#(y):9
          
          11:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> g_4#(x,x):4
          
          12:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):12
             -->_1 g_5#(s(x),y) -> f_4#(y):11
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: g_5#(s(x),y) -> g_5#(x,y)
          11: g_5#(s(x),y) -> f_4#(y)
          4: f_4#(x) -> g_4#(x,x)
          10: g_4#(s(x),y) -> g_4#(x,y)
          9: g_4#(s(x),y) -> f_3#(y)
          3: f_3#(x) -> g_3#(x,x)
          8: g_3#(s(x),y) -> g_3#(x,y)
          7: g_3#(s(x),y) -> f_2#(y)
          2: f_2#(x) -> c_3(g_2#(x,x))
          6: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
          1: f_1#(x) -> c_2(g_1#(x,x))
          5: g_1#(s(x),y) -> c_7(g_1#(x,y))
****** Step 7.a:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Weak DPs:
            f_1#(x) -> c_2(g_1#(x,x))
            g_1#(s(x),y) -> c_7(g_1#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):4
          
          2:S:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5
          
          3:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):6
          
          4:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
             -->_1 f_1#(x) -> c_2(g_1#(x,x)):8
             -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):4
          
          5:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):1
          
          6:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):6
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):2
          
          7:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):7
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):3
          
          8:W:f_1#(x) -> c_2(g_1#(x,x))
             -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):9
          
          9:W:g_1#(s(x),y) -> c_7(g_1#(x,y))
             -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):9
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: f_1#(x) -> c_2(g_1#(x,x))
          9: g_1#(s(x),y) -> c_7(g_1#(x,y))
** Step 7.b:2: SimplifyRHS WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):4
          
          2:S:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5
          
          3:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):6
          
          4:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
             -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):4
          
          5:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):1
          
          6:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):6
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):2
          
          7:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):7
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g_2#(s(x),y) -> c_8(g_2#(x,y))
** Step 7.b:3: Decompose WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_2#(s(x),y) -> c_8(g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + 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_2#(x) -> c_3(g_2#(x,x))
              g_2#(s(x),y) -> c_8(g_2#(x,y))
          - Weak DPs:
              f_3#(x) -> c_4(g_3#(x,x))
              f_4#(x) -> c_5(g_4#(x,x))
              g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
              g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1
              ,c_9/2,c_10/2,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
        
        Problem (S)
          - Strict DPs:
              f_3#(x) -> c_4(g_3#(x,x))
              f_4#(x) -> c_5(g_4#(x,x))
              g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
              g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Weak DPs:
              f_2#(x) -> c_3(g_2#(x,x))
              g_2#(s(x),y) -> c_8(g_2#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1
              ,c_9/2,c_10/2,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
*** Step 7.b:3.a:1: DecomposeDG WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            g_2#(s(x),y) -> c_8(g_2#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        and a lower component
          f_2#(x) -> c_3(g_2#(x,x))
          f_3#(x) -> c_4(g_3#(x,x))
          f_4#(x) -> c_5(g_4#(x,x))
          g_2#(s(x),y) -> c_8(g_2#(x,y))
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        Further, following extension rules are added to the lower component.
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
**** Step 7.b:3.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 7.b:3.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_11) = {2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [0]                  
             p(b) = [1] x1 + [1] x2 + [0]
           p(f_0) = [0]                  
           p(f_1) = [0]                  
           p(f_2) = [0]                  
           p(f_3) = [0]                  
           p(f_4) = [0]                  
           p(f_5) = [1] x1 + [0]         
           p(g_1) = [1]                  
           p(g_2) = [2] x2 + [1]         
           p(g_3) = [1] x2 + [2]         
           p(g_4) = [1] x1 + [1] x2 + [1]
           p(g_5) = [1] x1 + [1] x2 + [0]
             p(s) = [1] x1 + [5]         
          p(f_0#) = [2] x1 + [2]         
          p(f_1#) = [1] x1 + [4]         
          p(f_2#) = [1] x1 + [4]         
          p(f_3#) = [2] x1 + [2]         
          p(f_4#) = [6]                  
          p(f_5#) = [1]                  
          p(g_1#) = [4] x2 + [0]         
          p(g_2#) = [1] x2 + [0]         
          p(g_3#) = [1] x1 + [1] x2 + [1]
          p(g_4#) = [1] x2 + [2]         
          p(g_5#) = [4] x1 + [9] x2 + [8]
           p(c_1) = [8]                  
           p(c_2) = [2] x1 + [0]         
           p(c_3) = [1] x1 + [0]         
           p(c_4) = [2] x1 + [1]         
           p(c_5) = [0]                  
           p(c_6) = [4] x1 + [1]         
           p(c_7) = [1]                  
           p(c_8) = [2] x1 + [1]         
           p(c_9) = [1] x1 + [1] x2 + [1]
          p(c_10) = [2] x1 + [1] x2 + [1]
          p(c_11) = [2] x1 + [1] x2 + [4]
        
        Following rules are strictly oriented:
        g_5#(s(x),y) = [4] x + [9] y + [28]   
                     > [4] x + [9] y + [24]   
                     = c_11(f_4#(y),g_5#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
***** Step 7.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 7.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
***** Step 7.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 7.b:3.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            g_2#(s(x),y) -> c_8(g_2#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          f_4#(x) -> c_5(g_4#(x,x))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
        and a lower component
          f_2#(x) -> c_3(g_2#(x,x))
          f_3#(x) -> c_4(g_3#(x,x))
          g_2#(s(x),y) -> c_8(g_2#(x,y))
          g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
        Further, following extension rules are added to the lower component.
          f_4#(x) -> g_4#(x,x)
          g_4#(s(x),y) -> f_3#(y)
          g_4#(s(x),y) -> g_4#(x,y)
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
***** Step 7.b:3.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.b:3.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_10) = {2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [0]                   
             p(b) = [1] x1 + [1] x2 + [0] 
           p(f_0) = [0]                   
           p(f_1) = [0]                   
           p(f_2) = [0]                   
           p(f_3) = [0]                   
           p(f_4) = [0]                   
           p(f_5) = [0]                   
           p(g_1) = [0]                   
           p(g_2) = [0]                   
           p(g_3) = [0]                   
           p(g_4) = [0]                   
           p(g_5) = [0]                   
             p(s) = [1] x1 + [8]          
          p(f_0#) = [0]                   
          p(f_1#) = [0]                   
          p(f_2#) = [1] x1 + [0]          
          p(f_3#) = [1]                   
          p(f_4#) = [2] x1 + [9]          
          p(f_5#) = [8] x1 + [0]          
          p(g_1#) = [0]                   
          p(g_2#) = [2] x2 + [0]          
          p(g_3#) = [8] x2 + [0]          
          p(g_4#) = [1] x1 + [1] x2 + [8] 
          p(g_5#) = [1] x1 + [2] x2 + [14]
           p(c_1) = [1]                   
           p(c_2) = [1] x1 + [1]          
           p(c_3) = [1]                   
           p(c_4) = [1]                   
           p(c_5) = [1] x1 + [1]          
           p(c_6) = [1] x1 + [8]          
           p(c_7) = [4]                   
           p(c_8) = [2]                   
           p(c_9) = [1] x1 + [1] x2 + [1] 
          p(c_10) = [3] x1 + [1] x2 + [3] 
          p(c_11) = [1] x2 + [0]          
        
        Following rules are strictly oriented:
        g_4#(s(x),y) = [1] x + [1] y + [16]   
                     > [1] x + [1] y + [14]   
                     = c_10(f_3#(y),g_4#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_4#(x) =  [2] x + [9]         
                     >= [2] x + [9]         
                     =  c_5(g_4#(x,x))      
        
        g_5#(s(x),y) =  [1] x + [2] y + [22]
                     >= [2] y + [9]         
                     =  f_4#(y)             
        
        g_5#(s(x),y) =  [1] x + [2] y + [22]
                     >= [1] x + [2] y + [14]
                     =  g_5#(x,y)           
        
****** Step 7.b:3.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.b:3.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          2:W:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          3:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):1
          
          4:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):4
             -->_1 g_5#(s(x),y) -> f_4#(y):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: g_5#(s(x),y) -> g_5#(x,y)
          3: g_5#(s(x),y) -> f_4#(y)
          1: f_4#(x) -> c_5(g_4#(x,x))
          2: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
****** Step 7.b:3.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 7.b:3.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            g_2#(s(x),y) -> c_8(g_2#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f_2#(x) -> c_3(g_2#(x,x))
          2: g_2#(s(x),y) -> c_8(g_2#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.b:3.a:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            g_2#(s(x),y) -> c_8(g_2#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1},
          uargs(c_8) = {1},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = 0                              
             p(b) = 1 + x1                         
           p(f_0) = 1 + 4*x1 + x1^2                
           p(f_1) = 1                              
           p(f_2) = 1 + x1 + 4*x1^2                
           p(f_3) = 1 + x1^2                       
           p(f_4) = 1                              
           p(f_5) = 2                              
           p(g_1) = 4                              
           p(g_2) = x1 + x1*x2 + x1^2 + x2^2       
           p(g_3) = x1 + x1*x2                     
           p(g_4) = x1 + 4*x1^2                    
           p(g_5) = 1 + x1 + 2*x1*x2 + x2^2        
             p(s) = 1 + x1                         
          p(f_0#) = 4 + x1 + x1^2                  
          p(f_1#) = 0                              
          p(f_2#) = 1 + 4*x1                       
          p(f_3#) = 1 + 6*x1 + 6*x1^2              
          p(f_4#) = 6 + 6*x1 + 7*x1^2              
          p(f_5#) = x1 + 4*x1^2                    
          p(g_1#) = 2 + x2 + 2*x2^2                
          p(g_2#) = x1 + x2                        
          p(g_3#) = 4*x1 + 5*x1*x2 + x2            
          p(g_4#) = 4 + x1*x2 + 5*x2 + 6*x2^2      
          p(g_5#) = 3 + 4*x1 + x1^2 + 7*x2 + 7*x2^2
           p(c_1) = 0                              
           p(c_2) = x1                             
           p(c_3) = x1                             
           p(c_4) = x1                             
           p(c_5) = 1                              
           p(c_6) = 0                              
           p(c_7) = 1                              
           p(c_8) = x1                             
           p(c_9) = x1 + x2                        
          p(c_10) = 0                              
          p(c_11) = x1                             
        
        Following rules are strictly oriented:
             f_2#(x) = 1 + 4*x       
                     > 2*x           
                     = c_3(g_2#(x,x))
        
        g_2#(s(x),y) = 1 + x + y     
                     > x + y         
                     = c_8(g_2#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_3#(x) =  1 + 6*x + 6*x^2            
                     >= 5*x + 5*x^2                
                     =  c_4(g_3#(x,x))             
        
             f_4#(x) =  6 + 6*x + 7*x^2            
                     >= 4 + 5*x + 7*x^2            
                     =  g_4#(x,x)                  
        
        g_3#(s(x),y) =  4 + 4*x + 5*x*y + 6*y      
                     >= 1 + 4*x + 5*x*y + 5*y      
                     =  c_9(f_2#(y),g_3#(x,y))     
        
        g_4#(s(x),y) =  4 + x*y + 6*y + 6*y^2      
                     >= 1 + 6*y + 6*y^2            
                     =  f_3#(y)                    
        
        g_4#(s(x),y) =  4 + x*y + 6*y + 6*y^2      
                     >= 4 + x*y + 5*y + 6*y^2      
                     =  g_4#(x,y)                  
        
        g_5#(s(x),y) =  8 + 6*x + x^2 + 7*y + 7*y^2
                     >= 6 + 6*y + 7*y^2            
                     =  f_4#(y)                    
        
        g_5#(s(x),y) =  8 + 6*x + x^2 + 7*y + 7*y^2
                     >= 3 + 4*x + x^2 + 7*y + 7*y^2
                     =  g_5#(x,y)                  
        
****** Step 7.b:3.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_2#(s(x),y) -> c_8(g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.b:3.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> g_4#(x,x)
            g_2#(s(x),y) -> c_8(g_2#(x,y))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> f_3#(y)
            g_4#(s(x),y) -> g_4#(x,y)
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):4
          
          2:W:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5
          
          3:W:f_4#(x) -> g_4#(x,x)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):7
             -->_1 g_4#(s(x),y) -> f_3#(y):6
          
          4:W:g_2#(s(x),y) -> c_8(g_2#(x,y))
             -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):4
          
          5:W:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):1
          
          6:W:g_4#(s(x),y) -> f_3#(y)
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):2
          
          7:W:g_4#(s(x),y) -> g_4#(x,y)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):7
             -->_1 g_4#(s(x),y) -> f_3#(y):6
          
          8:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> g_4#(x,x):3
          
          9:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):9
             -->_1 g_5#(s(x),y) -> f_4#(y):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: g_5#(s(x),y) -> g_5#(x,y)
          8: g_5#(s(x),y) -> f_4#(y)
          3: f_4#(x) -> g_4#(x,x)
          7: g_4#(s(x),y) -> g_4#(x,y)
          6: g_4#(s(x),y) -> f_3#(y)
          2: f_3#(x) -> c_4(g_3#(x,x))
          5: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
          1: f_2#(x) -> c_3(g_2#(x,x))
          4: g_2#(s(x),y) -> c_8(g_2#(x,y))
****** Step 7.b:3.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 7.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Weak DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            g_2#(s(x),y) -> c_8(g_2#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3
          
          2:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4
          
          3:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):6
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3
          
          4:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):1
          
          5:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):5
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):2
          
          6:W:f_2#(x) -> c_3(g_2#(x,x))
             -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):7
          
          7:W:g_2#(s(x),y) -> c_8(g_2#(x,y))
             -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: f_2#(x) -> c_3(g_2#(x,x))
          7: g_2#(s(x),y) -> c_8(g_2#(x,y))
*** Step 7.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3
          
          2:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4
          
          3:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
             -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3
          
          4:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):1
          
          5:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):5
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g_3#(s(x),y) -> c_9(g_3#(x,y))
*** Step 7.b:3.b:3: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + 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_3#(x) -> c_4(g_3#(x,x))
              g_3#(s(x),y) -> c_9(g_3#(x,y))
          - Weak DPs:
              f_4#(x) -> c_5(g_4#(x,x))
              g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1
              ,c_9/1,c_10/2,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
        
        Problem (S)
          - Strict DPs:
              f_4#(x) -> c_5(g_4#(x,x))
              g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Weak DPs:
              f_3#(x) -> c_4(g_3#(x,x))
              g_3#(s(x),y) -> c_9(g_3#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1
              ,c_9/1,c_10/2,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
**** Step 7.b:3.b:3.a:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        and a lower component
          f_3#(x) -> c_4(g_3#(x,x))
          f_4#(x) -> c_5(g_4#(x,x))
          g_3#(s(x),y) -> c_9(g_3#(x,y))
          g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
        Further, following extension rules are added to the lower component.
          g_5#(s(x),y) -> f_4#(y)
          g_5#(s(x),y) -> g_5#(x,y)
***** Step 7.b:3.b:3.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.b:3.b:3.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_11) = {2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [0]                   
             p(b) = [1] x1 + [1] x2 + [0] 
           p(f_0) = [0]                   
           p(f_1) = [0]                   
           p(f_2) = [0]                   
           p(f_3) = [0]                   
           p(f_4) = [0]                   
           p(f_5) = [0]                   
           p(g_1) = [0]                   
           p(g_2) = [0]                   
           p(g_3) = [0]                   
           p(g_4) = [0]                   
           p(g_5) = [0]                   
             p(s) = [1] x1 + [4]          
          p(f_0#) = [0]                   
          p(f_1#) = [0]                   
          p(f_2#) = [0]                   
          p(f_3#) = [1] x1 + [0]          
          p(f_4#) = [11]                  
          p(f_5#) = [0]                   
          p(g_1#) = [2]                   
          p(g_2#) = [1] x2 + [1]          
          p(g_3#) = [1] x1 + [2] x2 + [0] 
          p(g_4#) = [8] x1 + [2]          
          p(g_5#) = [4] x1 + [4] x2 + [13]
           p(c_1) = [0]                   
           p(c_2) = [1]                   
           p(c_3) = [1] x1 + [1]          
           p(c_4) = [1] x1 + [1]          
           p(c_5) = [1]                   
           p(c_6) = [2] x1 + [0]          
           p(c_7) = [8]                   
           p(c_8) = [2] x1 + [4]          
           p(c_9) = [0]                   
          p(c_10) = [8] x2 + [2]          
          p(c_11) = [1] x1 + [1] x2 + [4] 
        
        Following rules are strictly oriented:
        g_5#(s(x),y) = [4] x + [4] y + [29]   
                     > [4] x + [4] y + [28]   
                     = c_11(f_4#(y),g_5#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
****** Step 7.b:3.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.b:3.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
****** Step 7.b:3.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 7.b:3.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f_3#(x) -> c_4(g_3#(x,x))
          2: g_3#(s(x),y) -> c_9(g_3#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.b:3.b:3.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_4) = {1},
          uargs(c_5) = {1},
          uargs(c_9) = {1},
          uargs(c_10) = {1,2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = 1                           
             p(b) = 0                           
           p(f_0) = x1 + x1^2                   
           p(f_1) = 4 + x1^2                    
           p(f_2) = 1 + 2*x1^2                  
           p(f_3) = 1                           
           p(f_4) = 4 + x1                      
           p(f_5) = x1^2                        
           p(g_1) = 1 + 2*x1^2 + x2^2           
           p(g_2) = 4 + x1                      
           p(g_3) = 4*x1 + x2                   
           p(g_4) = x1 + 2*x1*x2 + 4*x1^2 + x2  
           p(g_5) = x1 + x1^2 + x2^2            
             p(s) = 1 + x1                      
          p(f_0#) = x1 + 2*x1^2                 
          p(f_1#) = 1                           
          p(f_2#) = 1 + 2*x1                    
          p(f_3#) = 2 + x1                      
          p(f_4#) = 1 + 6*x1 + 4*x1^2           
          p(f_5#) = 2 + 2*x1 + 4*x1^2           
          p(g_1#) = 1 + x1 + x1^2               
          p(g_2#) = x1 + 2*x1*x2 + 2*x2 + 4*x2^2
          p(g_3#) = x1                          
          p(g_4#) = 4*x1 + 2*x1*x2 + 2*x2 + x2^2
          p(g_5#) = x1 + 4*x1*x2 + 6*x2 + 4*x2^2
           p(c_1) = 0                           
           p(c_2) = x1                          
           p(c_3) = 1 + x1                      
           p(c_4) = x1                          
           p(c_5) = 1 + x1                      
           p(c_6) = 0                           
           p(c_7) = 1                           
           p(c_8) = 1 + x1                      
           p(c_9) = x1                          
          p(c_10) = x1 + x2                     
          p(c_11) = x1 + x2                     
        
        Following rules are strictly oriented:
             f_3#(x) = 2 + x         
                     > x             
                     = c_4(g_3#(x,x))
        
        g_3#(s(x),y) = 1 + x         
                     > x             
                     = c_9(g_3#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_4#(x) =  1 + 6*x + 4*x^2             
                     >= 1 + 6*x + 3*x^2             
                     =  c_5(g_4#(x,x))              
        
        g_4#(s(x),y) =  4 + 4*x + 2*x*y + 4*y + y^2 
                     >= 2 + 4*x + 2*x*y + 3*y + y^2 
                     =  c_10(f_3#(y),g_4#(x,y))     
        
        g_5#(s(x),y) =  1 + x + 4*x*y + 10*y + 4*y^2
                     >= 1 + 6*y + 4*y^2             
                     =  f_4#(y)                     
        
        g_5#(s(x),y) =  1 + x + 4*x*y + 10*y + 4*y^2
                     >= x + 4*x*y + 6*y + 4*y^2     
                     =  g_5#(x,y)                   
        
****** Step 7.b:3.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.b:3.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            f_4#(x) -> c_5(g_4#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> f_4#(y)
            g_5#(s(x),y) -> g_5#(x,y)
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):3
          
          2:W:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4
          
          3:W:g_3#(s(x),y) -> c_9(g_3#(x,y))
             -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):3
          
          4:W:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):1
          
          5:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):2
          
          6:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):6
             -->_1 g_5#(s(x),y) -> f_4#(y):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: g_5#(s(x),y) -> g_5#(x,y)
          5: g_5#(s(x),y) -> f_4#(y)
          2: f_4#(x) -> c_5(g_4#(x,x))
          4: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
          1: f_3#(x) -> c_4(g_3#(x,x))
          3: g_3#(s(x),y) -> c_9(g_3#(x,y))
****** Step 7.b:3.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 7.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Weak DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            g_3#(s(x),y) -> c_9(g_3#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          2:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_1 f_3#(x) -> c_4(g_3#(x,x)):4
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          3:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):3
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):1
          
          4:W:f_3#(x) -> c_4(g_3#(x,x))
             -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):5
          
          5:W:g_3#(s(x),y) -> c_9(g_3#(x,y))
             -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: f_3#(x) -> c_4(g_3#(x,x))
          5: g_3#(s(x),y) -> c_9(g_3#(x,y))
**** Step 7.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/2,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          2:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
             -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2
          
          3:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):3
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g_4#(s(x),y) -> c_10(g_4#(x,y))
**** Step 7.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + 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_4#(x) -> c_5(g_4#(x,x))
              g_4#(s(x),y) -> c_10(g_4#(x,y))
          - Weak DPs:
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1
              ,c_9/1,c_10/1,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
        
        Problem (S)
          - Strict DPs:
              g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          - Weak DPs:
              f_4#(x) -> c_5(g_4#(x,x))
              g_4#(s(x),y) -> c_10(g_4#(x,y))
          - Signature:
              {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1
              ,f_5#/1,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1
              ,c_9/1,c_10/1,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
              ,g_5#} and constructors {a,b,s}
***** Step 7.b:3.b:3.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(g_4#(x,y))
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f_4#(x) -> c_5(g_4#(x,x))
          2: g_4#(s(x),y) -> c_10(g_4#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.b:3.b:3.b:3.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(g_4#(x,y))
        - Weak DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_10) = {1},
          uargs(c_11) = {1,2}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = 1                                           
             p(b) = 0                                           
           p(f_0) = 4*x1 + x1^2                                 
           p(f_1) = x1                                          
           p(f_2) = x1 + x1^2                                   
           p(f_3) = 2*x1                                        
           p(f_4) = 1                                           
           p(f_5) = 1 + 2*x1                                    
           p(g_1) = 2 + x1 + 2*x1^2 + 2*x2 + 2*x2^2             
           p(g_2) = 1 + x1 + x1*x2 + x1^2 + x2                  
           p(g_3) = 1 + x1 + x1^2                               
           p(g_4) = 8 + 2*x1^2 + x2^2                           
           p(g_5) = 1 + 8*x1*x2 + x1^2 + x2 + x2^2              
             p(s) = 1 + x1                                      
          p(f_0#) = 1 + x1                                      
          p(f_1#) = 1 + x1^2                                    
          p(f_2#) = 4*x1 + x1^2                                 
          p(f_3#) = 1 + 2*x1 + x1^2                             
          p(f_4#) = 9 + 8*x1                                    
          p(f_5#) = x1 + x1^2                                   
          p(g_1#) = 1                                           
          p(g_2#) = x1*x2 + x1^2 + x2 + x2^2                    
          p(g_3#) = 2 + 2*x1*x2 + 4*x1^2                        
          p(g_4#) = 5 + 2*x1                                    
          p(g_5#) = 8 + 2*x1 + 8*x1*x2 + 7*x1^2 + 4*x2 + 10*x2^2
           p(c_1) = 1                                           
           p(c_2) = 1 + x1                                      
           p(c_3) = 0                                           
           p(c_4) = x1                                          
           p(c_5) = x1                                          
           p(c_6) = 0                                           
           p(c_7) = 1                                           
           p(c_8) = x1                                          
           p(c_9) = 1 + x1                                      
          p(c_10) = x1                                          
          p(c_11) = x1 + x2                                     
        
        Following rules are strictly oriented:
             f_4#(x) = 9 + 8*x        
                     > 5 + 2*x        
                     = c_5(g_4#(x,x)) 
        
        g_4#(s(x),y) = 7 + 2*x        
                     > 5 + 2*x        
                     = c_10(g_4#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        g_5#(s(x),y) =  17 + 16*x + 8*x*y + 7*x^2 + 12*y + 10*y^2
                     >= 17 + 2*x + 8*x*y + 7*x^2 + 12*y + 10*y^2 
                     =  c_11(f_4#(y),g_5#(x,y))                  
        
****** Step 7.b:3.b:3.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.b:3.b:3.b:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(g_4#(x,y))
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):2
          
          2:W:g_4#(s(x),y) -> c_10(g_4#(x,y))
             -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):2
          
          3:W:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):3
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
          1: f_4#(x) -> c_5(g_4#(x,x))
          2: g_4#(s(x),y) -> c_10(g_4#(x,y))
****** Step 7.b:3.b:3.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 7.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Weak DPs:
            f_4#(x) -> c_5(g_4#(x,x))
            g_4#(s(x),y) -> c_10(g_4#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_1 f_4#(x) -> c_5(g_4#(x,x)):2
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1
          
          2:W:f_4#(x) -> c_5(g_4#(x,x))
             -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):3
          
          3:W:g_4#(s(x),y) -> c_10(g_4#(x,y))
             -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f_4#(x) -> c_5(g_4#(x,x))
          3: g_4#(s(x),y) -> c_10(g_4#(x,y))
***** Step 7.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
             -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g_5#(s(x),y) -> c_11(g_5#(x,y))
***** Step 7.b:3.b:3.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g_5#(s(x),y) -> c_11(g_5#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 7.b:3.b:3.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_5#(s(x),y) -> c_11(g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_11) = {1}
        
        Following symbols are considered usable:
          {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#}
        TcT has computed the following interpretation:
             p(a) = [1]                  
             p(b) = [0]                  
           p(f_0) = [1]                  
           p(f_1) = [2]                  
           p(f_2) = [1]                  
           p(f_3) = [1] x1 + [1]         
           p(f_4) = [8] x1 + [0]         
           p(f_5) = [0]                  
           p(g_1) = [2] x1 + [0]         
           p(g_2) = [1] x1 + [1] x2 + [1]
           p(g_3) = [1] x2 + [4]         
           p(g_4) = [4] x1 + [1] x2 + [4]
           p(g_5) = [2] x1 + [2] x2 + [0]
             p(s) = [1] x1 + [4]         
          p(f_0#) = [2] x1 + [0]         
          p(f_1#) = [8] x1 + [0]         
          p(f_2#) = [0]                  
          p(f_3#) = [2] x1 + [0]         
          p(f_4#) = [1] x1 + [0]         
          p(f_5#) = [1] x1 + [0]         
          p(g_1#) = [4] x1 + [1] x2 + [0]
          p(g_2#) = [0]                  
          p(g_3#) = [1] x1 + [0]         
          p(g_4#) = [1] x1 + [0]         
          p(g_5#) = [1] x1 + [12]        
           p(c_1) = [0]                  
           p(c_2) = [0]                  
           p(c_3) = [0]                  
           p(c_4) = [0]                  
           p(c_5) = [1] x1 + [0]         
           p(c_6) = [0]                  
           p(c_7) = [1] x1 + [0]         
           p(c_8) = [1] x1 + [2]         
           p(c_9) = [0]                  
          p(c_10) = [1] x1 + [4]         
          p(c_11) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        g_5#(s(x),y) = [1] x + [16]   
                     > [1] x + [12]   
                     = c_11(g_5#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
****** Step 7.b:3.b:3.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 7.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g_5#(s(x),y) -> c_11(g_5#(x,y))
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g_5#(s(x),y) -> c_11(g_5#(x,y))
             -->_1 g_5#(s(x),y) -> c_11(g_5#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g_5#(s(x),y) -> c_11(g_5#(x,y))
****** Step 7.b:3.b:3.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
            ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1
            ,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
            ,g_5#} and constructors {a,b,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^5))