WORST_CASE(?,O(n^5))
* 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_ = DT}
    + Details:
        We add the following dependency tuples:
        
        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: 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))
        - Weak 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:
        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 3: 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()
        - Weak 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:
        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 4: 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))
        - Weak 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:
        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 5: UsableRules 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))
        - Weak 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_7/1,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:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          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))
* 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: DecomposeDG 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:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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: 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_8/2,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: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.a:2: WeightGap 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_8/2,c_9/2
            ,c_10/2,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_11) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(a) = [0]                  
               p(b) = [1] x1 + [1] x2 + [0]
             p(f_0) = [0]                  
             p(f_1) = [2] x1 + [0]         
             p(f_2) = [2]                  
             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) = [4] x1 + [2]         
             p(g_5) = [0]                  
               p(s) = [1] x1 + [10]        
            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] x1 + [5]         
             p(c_1) = [0]                  
             p(c_2) = [0]                  
             p(c_3) = [2] x1 + [0]         
             p(c_4) = [0]                  
             p(c_5) = [0]                  
             p(c_6) = [0]                  
             p(c_7) = [0]                  
             p(c_8) = [0]                  
             p(c_9) = [0]                  
            p(c_10) = [0]                  
            p(c_11) = [1] x1 + [3]         
          
          Following rules are strictly oriented:
          g_5#(s(x),y) = [1] x + [15]   
                       > [1] x + [8]    
                       = c_11(g_5#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 7.a:3: EmptyProcessor 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_8/2,c_9/2
            ,c_10/2,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).

** Step 7.b:1: DecomposeDG WORST_CASE(?,O(n^4))
    + 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))
        - Weak DPs:
            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 = Nothing, 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.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + 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))
        - Weak DPs:
            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:
        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: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
          
        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:1.a:2: WeightGap WORST_CASE(?,O(n^1))
    + 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) -> 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_10/1,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_5) = {1},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          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 + [1]         
            p(f_0#) = [0]                  
            p(f_1#) = [0]                  
            p(f_2#) = [0]                  
            p(f_3#) = [1]                  
            p(f_4#) = [9] x1 + [0]         
            p(f_5#) = [0]                  
            p(g_1#) = [0]                  
            p(g_2#) = [0]                  
            p(g_3#) = [0]                  
            p(g_4#) = [5] x1 + [4] x2 + [0]
            p(g_5#) = [9] x2 + [0]         
             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) = [0]                  
             p(c_8) = [1] x2 + [0]         
             p(c_9) = [1] x2 + [8]         
            p(c_10) = [1] x1 + [0]         
            p(c_11) = [0]                  
          
          Following rules are strictly oriented:
          g_4#(s(x),y) = [5] x + [4] y + [5]
                       > [5] x + [4] y + [0]
                       = c_10(g_4#(x,y))    
          
          
          Following rules are (at-least) weakly oriented:
               f_4#(x) =  [9] x + [0]   
                       >= [9] x + [0]   
                       =  c_5(g_4#(x,x))
          
          g_5#(s(x),y) =  [9] y + [0]   
                       >= [9] y + [0]   
                       =  f_4#(y)       
          
          g_5#(s(x),y) =  [9] y + [0]   
                       >= [9] y + [0]   
                       =  g_5#(x,y)     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 7.b:1.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f_4#(x) -> c_5(g_4#(x,x))
        - Weak DPs:
            g_4#(s(x),y) -> c_10(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_10/1,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_5) = {1},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          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) = [1]                  
             p(g_2) = [0]                  
             p(g_3) = [0]                  
             p(g_4) = [0]                  
             p(g_5) = [0]                  
               p(s) = [1] x1 + [0]         
            p(f_0#) = [1] x1 + [0]         
            p(f_1#) = [0]                  
            p(f_2#) = [1]                  
            p(f_3#) = [0]                  
            p(f_4#) = [1]                  
            p(f_5#) = [0]                  
            p(g_1#) = [0]                  
            p(g_2#) = [0]                  
            p(g_3#) = [0]                  
            p(g_4#) = [0]                  
            p(g_5#) = [1]                  
             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) = [0]                  
             p(c_8) = [0]                  
             p(c_9) = [0]                  
            p(c_10) = [1] x1 + [0]         
            p(c_11) = [0]                  
          
          Following rules are strictly oriented:
          f_4#(x) = [1]           
                  > [0]           
                  = c_5(g_4#(x,x))
          
          
          Following rules are (at-least) weakly oriented:
          g_4#(s(x),y) =  [0]            
                       >= [0]            
                       =  c_10(g_4#(x,y))
          
          g_5#(s(x),y) =  [1]            
                       >= [1]            
                       =  f_4#(y)        
          
          g_5#(s(x),y) =  [1]            
                       >= [1]            
                       =  g_5#(x,y)      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 7.b:1.a:4: EmptyProcessor 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) -> 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_10/1,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:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
    + 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))
            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))
        - Weak DPs:
            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:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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.b:1.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f_3#(x) -> c_4(g_3#(x,x))
            g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
        - Weak DPs:
            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:
        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)):2
          
          2: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)):2
          
          3: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
          
          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):3
          
          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
          
        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:1.b:1.a:2: WeightGap WORST_CASE(?,O(n^1))
    + 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) -> 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_9/1
            ,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_4) = {1},
            uargs(c_9) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(a) = [0]                  
               p(b) = [1] x1 + [1] x2 + [0]
             p(f_0) = [0]                  
             p(f_1) = [1]                  
             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 + [1]         
            p(f_0#) = [0]                  
            p(f_1#) = [0]                  
            p(f_2#) = [0]                  
            p(f_3#) = [1] x1 + [11]        
            p(f_4#) = [1] x1 + [11]        
            p(f_5#) = [0]                  
            p(g_1#) = [0]                  
            p(g_2#) = [0]                  
            p(g_3#) = [1] x1 + [0]         
            p(g_4#) = [1] x2 + [11]        
            p(g_5#) = [4] x1 + [1] x2 + [7]
             p(c_1) = [0]                  
             p(c_2) = [0]                  
             p(c_3) = [0]                  
             p(c_4) = [1] x1 + [0]         
             p(c_5) = [0]                  
             p(c_6) = [2]                  
             p(c_7) = [1] x1 + [1]         
             p(c_8) = [2] x2 + [0]         
             p(c_9) = [1] x1 + [0]         
            p(c_10) = [2] x1 + [1] x2 + [0]
            p(c_11) = [1] x1 + [8] x2 + [1]
          
          Following rules are strictly oriented:
               f_3#(x) = [1] x + [11]  
                       > [1] x + [0]   
                       = c_4(g_3#(x,x))
          
          g_3#(s(x),y) = [1] x + [1]   
                       > [1] x + [0]   
                       = c_9(g_3#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
               f_4#(x) =  [1] x + [11]        
                       >= [1] x + [11]        
                       =  g_4#(x,x)           
          
          g_4#(s(x),y) =  [1] y + [11]        
                       >= [1] y + [11]        
                       =  f_3#(y)             
          
          g_4#(s(x),y) =  [1] y + [11]        
                       >= [1] y + [11]        
                       =  g_4#(x,y)           
          
          g_5#(s(x),y) =  [4] x + [1] y + [11]
                       >= [1] y + [11]        
                       =  f_4#(y)             
          
          g_5#(s(x),y) =  [4] x + [1] y + [11]
                       >= [4] x + [1] y + [7] 
                       =  g_5#(x,y)           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
**** Step 7.b:1.b:1.a:3: EmptyProcessor 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(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_9/1
            ,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:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak DPs:
            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)
        - 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 = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          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)
        and a lower component
          f_1#(x) -> c_2(g_1#(x,x))
          g_1#(s(x),y) -> c_7(g_1#(x,y))
        Further, following extension rules are added to the lower component.
          f_2#(x) -> g_2#(x,x)
          f_3#(x) -> g_3#(x,x)
          f_4#(x) -> g_4#(x,x)
          g_2#(s(x),y) -> f_1#(y)
          g_2#(s(x),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)
***** Step 7.b:1.b:1.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f_2#(x) -> c_3(g_2#(x,x))
            g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
        - Weak DPs:
            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)
        - 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)):2
          
          2: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)):2
          
          3:W:f_3#(x) -> g_3#(x,x)
             -->_1 g_3#(s(x),y) -> g_3#(x,y):6
             -->_1 g_3#(s(x),y) -> f_2#(y):5
          
          4:W:f_4#(x) -> g_4#(x,x)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):8
             -->_1 g_4#(s(x),y) -> f_3#(y):7
          
          5:W:g_3#(s(x),y) -> f_2#(y)
             -->_1 f_2#(x) -> c_3(g_2#(x,x)):1
          
          6:W:g_3#(s(x),y) -> g_3#(x,y)
             -->_1 g_3#(s(x),y) -> g_3#(x,y):6
             -->_1 g_3#(s(x),y) -> f_2#(y):5
          
          7:W:g_4#(s(x),y) -> f_3#(y)
             -->_1 f_3#(x) -> g_3#(x,x):3
          
          8:W:g_4#(s(x),y) -> g_4#(x,y)
             -->_1 g_4#(s(x),y) -> g_4#(x,y):8
             -->_1 g_4#(s(x),y) -> f_3#(y):7
          
          9:W:g_5#(s(x),y) -> f_4#(y)
             -->_1 f_4#(x) -> g_4#(x,x):4
          
          10:W:g_5#(s(x),y) -> g_5#(x,y)
             -->_1 g_5#(s(x),y) -> g_5#(x,y):10
             -->_1 g_5#(s(x),y) -> f_4#(y):9
          
        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:1.b:1.b:1.a:2: NaturalMI WORST_CASE(?,O(n^1))
    + 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) -> 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)
        - 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 any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_8) = {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) = [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) = [0]                  
          p(f_0#) = [0]                  
          p(f_1#) = [0]                  
          p(f_2#) = [9]                  
          p(f_3#) = [9]                  
          p(f_4#) = [9]                  
          p(f_5#) = [0]                  
          p(g_1#) = [0]                  
          p(g_2#) = [1]                  
          p(g_3#) = [9]                  
          p(g_4#) = [9]                  
          p(g_5#) = [9]                  
           p(c_1) = [0]                  
           p(c_2) = [0]                  
           p(c_3) = [1] x1 + [0]         
           p(c_4) = [0]                  
           p(c_5) = [0]                  
           p(c_6) = [0]                  
           p(c_7) = [0]                  
           p(c_8) = [1] x1 + [0]         
           p(c_9) = [0]                  
          p(c_10) = [0]                  
          p(c_11) = [0]                  
        
        Following rules are strictly oriented:
        f_2#(x) = [9]           
                > [1]           
                = c_3(g_2#(x,x))
        
        
        Following rules are (at-least) weakly oriented:
             f_3#(x) =  [9]           
                     >= [9]           
                     =  g_3#(x,x)     
        
             f_4#(x) =  [9]           
                     >= [9]           
                     =  g_4#(x,x)     
        
        g_2#(s(x),y) =  [1]           
                     >= [1]           
                     =  c_8(g_2#(x,y))
        
        g_3#(s(x),y) =  [9]           
                     >= [9]           
                     =  f_2#(y)       
        
        g_3#(s(x),y) =  [9]           
                     >= [9]           
                     =  g_3#(x,y)     
        
        g_4#(s(x),y) =  [9]           
                     >= [9]           
                     =  f_3#(y)       
        
        g_4#(s(x),y) =  [9]           
                     >= [9]           
                     =  g_4#(x,y)     
        
        g_5#(s(x),y) =  [9]           
                     >= [9]           
                     =  f_4#(y)       
        
        g_5#(s(x),y) =  [9]           
                     >= [9]           
                     =  g_5#(x,y)     
        
***** Step 7.b:1.b:1.b:1.a:3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g_2#(s(x),y) -> c_8(g_2#(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_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_8/1,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 any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_8) = {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) = [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] x1 + [1] x2 + [4]  
             p(s) = [1] x1 + [2]           
          p(f_0#) = [1]                    
          p(f_1#) = [1] x1 + [1]           
          p(f_2#) = [4] x1 + [8]           
          p(f_3#) = [12] x1 + [1]          
          p(f_4#) = [13] x1 + [1]          
          p(f_5#) = [8]                    
          p(g_1#) = [8]                    
          p(g_2#) = [1] x1 + [2]           
          p(g_3#) = [8] x1 + [4] x2 + [1]  
          p(g_4#) = [13] x2 + [1]          
          p(g_5#) = [12] x1 + [13] x2 + [3]
           p(c_1) = [0]                    
           p(c_2) = [0]                    
           p(c_3) = [1] x1 + [6]           
           p(c_4) = [1] x1 + [8]           
           p(c_5) = [0]                    
           p(c_6) = [1] x1 + [0]           
           p(c_7) = [1] x1 + [0]           
           p(c_8) = [1] x1 + [0]           
           p(c_9) = [4]                    
          p(c_10) = [1] x1 + [1]           
          p(c_11) = [1] x1 + [1]           
        
        Following rules are strictly oriented:
        g_2#(s(x),y) = [1] x + [4]   
                     > [1] x + [2]   
                     = c_8(g_2#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             f_2#(x) =  [4] x + [8]           
                     >= [1] x + [8]           
                     =  c_3(g_2#(x,x))        
        
             f_3#(x) =  [12] x + [1]          
                     >= [12] x + [1]          
                     =  g_3#(x,x)             
        
             f_4#(x) =  [13] x + [1]          
                     >= [13] x + [1]          
                     =  g_4#(x,x)             
        
        g_3#(s(x),y) =  [8] x + [4] y + [17]  
                     >= [4] y + [8]           
                     =  f_2#(y)               
        
        g_3#(s(x),y) =  [8] x + [4] y + [17]  
                     >= [8] x + [4] y + [1]   
                     =  g_3#(x,y)             
        
        g_4#(s(x),y) =  [13] y + [1]          
                     >= [12] y + [1]          
                     =  f_3#(y)               
        
        g_4#(s(x),y) =  [13] y + [1]          
                     >= [13] y + [1]          
                     =  g_4#(x,y)             
        
        g_5#(s(x),y) =  [12] x + [13] y + [27]
                     >= [13] y + [1]          
                     =  f_4#(y)               
        
        g_5#(s(x),y) =  [12] x + [13] y + [27]
                     >= [12] x + [13] y + [3] 
                     =  g_5#(x,y)             
        
***** Step 7.b:1.b:1.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - 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(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_8/1,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:1.b:1.b:1.b:1: WeightGap WORST_CASE(?,O(n^1))
    + 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) -> g_2#(x,x)
            f_3#(x) -> g_3#(x,x)
            f_4#(x) -> g_4#(x,x)
            g_2#(s(x),y) -> f_1#(y)
            g_2#(s(x),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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_2) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(a) = [0]                  
               p(b) = [1] x1 + [1] x2 + [4]
             p(f_0) = [0]                  
             p(f_1) = [2] x1 + [1]         
             p(f_2) = [4]                  
             p(f_3) = [4] x1 + [2]         
             p(f_4) = [2]                  
             p(f_5) = [2] x1 + [0]         
             p(g_1) = [1] x1 + [2] x2 + [0]
             p(g_2) = [0]                  
             p(g_3) = [0]                  
             p(g_4) = [0]                  
             p(g_5) = [0]                  
               p(s) = [1] x1 + [1]         
            p(f_0#) = [0]                  
            p(f_1#) = [1] x1 + [7]         
            p(f_2#) = [1] x1 + [7]         
            p(f_3#) = [1] x1 + [7]         
            p(f_4#) = [1] x1 + [7]         
            p(f_5#) = [0]                  
            p(g_1#) = [1] x1 + [4]         
            p(g_2#) = [1] x2 + [7]         
            p(g_3#) = [1] x2 + [7]         
            p(g_4#) = [1] x2 + [7]         
            p(g_5#) = [5] x1 + [1] x2 + [2]
             p(c_1) = [0]                  
             p(c_2) = [1] x1 + [0]         
             p(c_3) = [0]                  
             p(c_4) = [0]                  
             p(c_5) = [0]                  
             p(c_6) = [0]                  
             p(c_7) = [1] x1 + [0]         
             p(c_8) = [1] x2 + [0]         
             p(c_9) = [4] x1 + [1] x2 + [1]
            p(c_10) = [2] x1 + [0]         
            p(c_11) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
               f_1#(x) = [1] x + [7]   
                       > [1] x + [4]   
                       = c_2(g_1#(x,x))
          
          g_1#(s(x),y) = [1] x + [5]   
                       > [1] x + [4]   
                       = c_7(g_1#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
               f_2#(x) =  [1] x + [7]        
                       >= [1] x + [7]        
                       =  g_2#(x,x)          
          
               f_3#(x) =  [1] x + [7]        
                       >= [1] x + [7]        
                       =  g_3#(x,x)          
          
               f_4#(x) =  [1] x + [7]        
                       >= [1] x + [7]        
                       =  g_4#(x,x)          
          
          g_2#(s(x),y) =  [1] y + [7]        
                       >= [1] y + [7]        
                       =  f_1#(y)            
          
          g_2#(s(x),y) =  [1] y + [7]        
                       >= [1] y + [7]        
                       =  g_2#(x,y)          
          
          g_3#(s(x),y) =  [1] y + [7]        
                       >= [1] y + [7]        
                       =  f_2#(y)            
          
          g_3#(s(x),y) =  [1] y + [7]        
                       >= [1] y + [7]        
                       =  g_3#(x,y)          
          
          g_4#(s(x),y) =  [1] y + [7]        
                       >= [1] y + [7]        
                       =  f_3#(y)            
          
          g_4#(s(x),y) =  [1] y + [7]        
                       >= [1] y + [7]        
                       =  g_4#(x,y)          
          
          g_5#(s(x),y) =  [5] x + [1] y + [7]
                       >= [1] y + [7]        
                       =  f_4#(y)            
          
          g_5#(s(x),y) =  [5] x + [1] y + [7]
                       >= [5] x + [1] y + [2]
                       =  g_5#(x,y)          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
***** Step 7.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f_1#(x) -> c_2(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)
            g_1#(s(x),y) -> c_7(g_1#(x,y))
            g_2#(s(x),y) -> f_1#(y)
            g_2#(s(x),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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^5))