MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            aver(sum,z) -> if(gt(sum,double(z)),sum,z)
            average(x,y) -> aver(plus(x,y),0())
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),sum,z) -> z
            if(true(),sum,z) -> aver(sum,s(z))
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver,average,double,gt,if,plus} and constructors {0,false
            ,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
          average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          gt#(0(),y) -> c_5()
          gt#(s(x),0()) -> c_6()
          gt#(s(x),s(y)) -> c_7(gt#(x,y))
          if#(false(),sum,z) -> c_8()
          if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
          plus#(0(),y) -> c_10()
          plus#(s(x),y) -> c_11(plus#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            gt#(0(),y) -> c_5()
            gt#(s(x),0()) -> c_6()
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(false(),sum,z) -> c_8()
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
            plus#(0(),y) -> c_10()
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak TRS:
            aver(sum,z) -> if(gt(sum,double(z)),sum,z)
            average(x,y) -> aver(plus(x,y),0())
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),sum,z) -> z
            if(true(),sum,z) -> aver(sum,s(z))
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          double(0()) -> 0()
          double(s(x)) -> s(s(double(x)))
          gt(0(),y) -> false()
          gt(s(x),0()) -> true()
          gt(s(x),s(y)) -> gt(x,y)
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
          average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          gt#(0(),y) -> c_5()
          gt#(s(x),0()) -> c_6()
          gt#(s(x),s(y)) -> c_7(gt#(x,y))
          if#(false(),sum,z) -> c_8()
          if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
          plus#(0(),y) -> c_10()
          plus#(s(x),y) -> c_11(plus#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            gt#(0(),y) -> c_5()
            gt#(s(x),0()) -> c_6()
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(false(),sum,z) -> c_8()
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
            plus#(0(),y) -> c_10()
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,5,6,8,10}
        by application of
          Pre({3,5,6,8,10}) = {1,2,4,7,11}.
        Here rules are labelled as follows:
          1: aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
          2: average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
          3: double#(0()) -> c_3()
          4: double#(s(x)) -> c_4(double#(x))
          5: gt#(0(),y) -> c_5()
          6: gt#(s(x),0()) -> c_6()
          7: gt#(s(x),s(y)) -> c_7(gt#(x,y))
          8: if#(false(),sum,z) -> c_8()
          9: if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
          10: plus#(0(),y) -> c_10()
          11: plus#(s(x),y) -> c_11(plus#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            double#(s(x)) -> c_4(double#(x))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak DPs:
            double#(0()) -> c_3()
            gt#(0(),y) -> c_5()
            gt#(s(x),0()) -> c_6()
            if#(false(),sum,z) -> c_8()
            plus#(0(),y) -> c_10()
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
             -->_1 if#(true(),sum,z) -> c_9(aver#(sum,s(z))):5
             -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):4
             -->_3 double#(s(x)) -> c_4(double#(x)):3
             -->_1 if#(false(),sum,z) -> c_8():10
             -->_2 gt#(s(x),0()) -> c_6():9
             -->_2 gt#(0(),y) -> c_5():8
             -->_3 double#(0()) -> c_3():7
          
          2:S:average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
             -->_2 plus#(s(x),y) -> c_11(plus#(x,y)):6
             -->_2 plus#(0(),y) -> c_10():11
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):1
          
          3:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(0()) -> c_3():7
             -->_1 double#(s(x)) -> c_4(double#(x)):3
          
          4:S:gt#(s(x),s(y)) -> c_7(gt#(x,y))
             -->_1 gt#(s(x),0()) -> c_6():9
             -->_1 gt#(0(),y) -> c_5():8
             -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):4
          
          5:S:if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):1
          
          6:S:plus#(s(x),y) -> c_11(plus#(x,y))
             -->_1 plus#(0(),y) -> c_10():11
             -->_1 plus#(s(x),y) -> c_11(plus#(x,y)):6
          
          7:W:double#(0()) -> c_3()
             
          
          8:W:gt#(0(),y) -> c_5()
             
          
          9:W:gt#(s(x),0()) -> c_6()
             
          
          10:W:if#(false(),sum,z) -> c_8()
             
          
          11:W:plus#(0(),y) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          11: plus#(0(),y) -> c_10()
          10: if#(false(),sum,z) -> c_8()
          7: double#(0()) -> c_3()
          8: gt#(0(),y) -> c_5()
          9: gt#(s(x),0()) -> c_6()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            double#(s(x)) -> c_4(double#(x))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
              double#(s(x)) -> c_4(double#(x))
              gt#(s(x),s(y)) -> c_7(gt#(x,y))
              if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
          - Weak DPs:
              average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
              plus#(s(x),y) -> c_11(plus#(x,y))
          - Weak TRS:
              double(0()) -> 0()
              double(s(x)) -> s(s(double(x)))
              gt(0(),y) -> false()
              gt(s(x),0()) -> true()
              gt(s(x),s(y)) -> gt(x,y)
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
          - Signature:
              {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0
              ,false/0,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#
              ,plus#} and constructors {0,false,s,true}
        
        Problem (S)
          - Strict DPs:
              average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
              plus#(s(x),y) -> c_11(plus#(x,y))
          - Weak DPs:
              aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
              double#(s(x)) -> c_4(double#(x))
              gt#(s(x),s(y)) -> c_7(gt#(x,y))
              if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
          - Weak TRS:
              double(0()) -> 0()
              double(s(x)) -> s(s(double(x)))
              gt(0(),y) -> false()
              gt(s(x),0()) -> true()
              gt(s(x),s(y)) -> gt(x,y)
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
          - Signature:
              {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0
              ,false/0,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#
              ,plus#} and constructors {0,false,s,true}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            double#(s(x)) -> c_4(double#(x))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
        - Weak DPs:
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
             -->_1 if#(true(),sum,z) -> c_9(aver#(sum,s(z))):5
             -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):4
             -->_3 double#(s(x)) -> c_4(double#(x)):3
          
          2:W:average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):1
             -->_2 plus#(s(x),y) -> c_11(plus#(x,y)):6
          
          3:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(s(x)) -> c_4(double#(x)):3
          
          4:S:gt#(s(x),s(y)) -> c_7(gt#(x,y))
             -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):4
          
          5:S:if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):1
          
          6:W:plus#(s(x),y) -> c_11(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_11(plus#(x,y)):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: plus#(s(x),y) -> c_11(plus#(x,y))
** Step 5.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            double#(s(x)) -> c_4(double#(x))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
        - Weak DPs:
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
             -->_1 if#(true(),sum,z) -> c_9(aver#(sum,s(z))):5
             -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):4
             -->_3 double#(s(x)) -> c_4(double#(x)):3
          
          2:W:average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):1
          
          3:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(s(x)) -> c_4(double#(x)):3
          
          4:S:gt#(s(x),s(y)) -> c_7(gt#(x,y))
             -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):4
          
          5:S:if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          average#(x,y) -> c_2(aver#(plus(x,y),0()))
** Step 5.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
          double#(s(x)) -> c_4(double#(x))
          gt#(s(x),s(y)) -> c_7(gt#(x,y))
          if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
      - Weak DPs:
          average#(x,y) -> c_2(aver#(plus(x,y),0()))
      - Weak TRS:
          double(0()) -> 0()
          double(s(x)) -> s(s(double(x)))
          gt(0(),y) -> false()
          gt(s(x),0()) -> true()
          gt(s(x),s(y)) -> gt(x,y)
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
      - Signature:
          {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
          ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
          ,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak DPs:
            aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
            double#(s(x)) -> c_4(double#(x))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):3
             -->_2 plus#(s(x),y) -> c_11(plus#(x,y)):2
          
          2:S:plus#(s(x),y) -> c_11(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_11(plus#(x,y)):2
          
          3:W:aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
             -->_1 if#(true(),sum,z) -> c_9(aver#(sum,s(z))):6
             -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):5
             -->_3 double#(s(x)) -> c_4(double#(x)):4
          
          4:W:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(s(x)) -> c_4(double#(x)):4
          
          5:W:gt#(s(x),s(y)) -> c_7(gt#(x,y))
             -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):5
          
          6:W:if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
             -->_1 aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: aver#(sum,z) -> c_1(if#(gt(sum,double(z)),sum,z),gt#(sum,double(z)),double#(z))
          6: if#(true(),sum,z) -> c_9(aver#(sum,s(z)))
          4: double#(s(x)) -> c_4(double#(x))
          5: gt#(s(x),s(y)) -> c_7(gt#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:average#(x,y) -> c_2(aver#(plus(x,y),0()),plus#(x,y))
             -->_2 plus#(s(x),y) -> c_11(plus#(x,y)):2
          
          2:S:plus#(s(x),y) -> c_11(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_11(plus#(x,y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          average#(x,y) -> c_2(plus#(x,y))
** Step 5.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,y) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Weak TRS:
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          average#(x,y) -> c_2(plus#(x,y))
          plus#(s(x),y) -> c_11(plus#(x,y))
** Step 5.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,y) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + 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: average#(x,y) -> c_2(plus#(x,y))
          2: plus#(s(x),y) -> c_11(plus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,y) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + 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_2) = {1},
          uargs(c_11) = {1}
        
        Following symbols are considered usable:
          {aver#,average#,double#,gt#,if#,plus#}
        TcT has computed the following interpretation:
                 p(0) = [1]                           
              p(aver) = [2] x2 + [0]                  
           p(average) = [1] x1 + [0]                  
            p(double) = [1]                           
             p(false) = [0]                           
                p(gt) = [2] x2 + [0]                  
                p(if) = [1] x1 + [1] x3 + [8]         
              p(plus) = [1] x2 + [1]                  
                 p(s) = [1] x1 + [2]                  
              p(true) = [1]                           
             p(aver#) = [0]                           
          p(average#) = [4] x1 + [7]                  
           p(double#) = [1] x1 + [8]                  
               p(gt#) = [2] x1 + [2]                  
               p(if#) = [1] x2 + [2] x3 + [0]         
             p(plus#) = [2] x1 + [3]                  
               p(c_1) = [1] x1 + [1] x2 + [2] x3 + [0]
               p(c_2) = [1] x1 + [0]                  
               p(c_3) = [2]                           
               p(c_4) = [8] x1 + [2]                  
               p(c_5) = [1]                           
               p(c_6) = [1]                           
               p(c_7) = [0]                           
               p(c_8) = [1]                           
               p(c_9) = [2]                           
              p(c_10) = [0]                           
              p(c_11) = [1] x1 + [0]                  
        
        Following rules are strictly oriented:
        average#(x,y) = [4] x + [7]     
                      > [2] x + [3]     
                      = c_2(plus#(x,y)) 
        
        plus#(s(x),y) = [2] x + [7]     
                      > [2] x + [3]     
                      = c_11(plus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 5.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            average#(x,y) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            average#(x,y) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_11(plus#(x,y))
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:average#(x,y) -> c_2(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_11(plus#(x,y)):2
          
          2:W:plus#(s(x),y) -> c_11(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_11(plus#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: average#(x,y) -> c_2(plus#(x,y))
          2: plus#(s(x),y) -> c_11(plus#(x,y))
*** Step 5.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {aver/2,average/2,double/1,gt/2,if/3,plus/2,aver#/2,average#/2,double#/1,gt#/2,if#/3,plus#/2} / {0/0,false/0
            ,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {aver#,average#,double#,gt#,if#,plus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE