MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> if(true(),0(),s(s(x)),0(),0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            if(false(),c,s(s(x)),a,b) -> sum(fibo(a),fibo(b))
            if(true(),c,s(s(x)),a,b) -> if(lt(s(c),s(s(x))),s(c),s(s(x)),b,c)
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib,fibo,if,lt,sum} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          fib#(0()) -> c_1()
          fib#(s(0())) -> c_2()
          fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
          fibo#(0()) -> c_4(fib#(0()))
          fibo#(s(0())) -> c_5(fib#(s(0())))
          fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          lt#(x,0()) -> c_9()
          lt#(0(),s(x)) -> c_10()
          lt#(s(x),s(y)) -> c_11(lt#(x,y))
          sum#(x,0()) -> c_12()
          sum#(x,s(y)) -> c_13(sum#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(0()) -> c_1()
            fib#(s(0())) -> c_2()
            fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
            fibo#(0()) -> c_4(fib#(0()))
            fibo#(s(0())) -> c_5(fib#(s(0())))
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(x,0()) -> c_9()
            lt#(0(),s(x)) -> c_10()
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
            sum#(x,0()) -> c_12()
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> if(true(),0(),s(s(x)),0(),0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            if(false(),c,s(s(x)),a,b) -> sum(fibo(a),fibo(b))
            if(true(),c,s(s(x)),a,b) -> if(lt(s(c),s(s(x))),s(c),s(s(x)),b,c)
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          fib(0()) -> s(0())
          fib(s(0())) -> s(0())
          fibo(0()) -> fib(0())
          fibo(s(0())) -> fib(s(0()))
          fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          sum(x,0()) -> x
          sum(x,s(y)) -> s(sum(x,y))
          fib#(0()) -> c_1()
          fib#(s(0())) -> c_2()
          fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
          fibo#(0()) -> c_4(fib#(0()))
          fibo#(s(0())) -> c_5(fib#(s(0())))
          fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          lt#(x,0()) -> c_9()
          lt#(0(),s(x)) -> c_10()
          lt#(s(x),s(y)) -> c_11(lt#(x,y))
          sum#(x,0()) -> c_12()
          sum#(x,s(y)) -> c_13(sum#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(0()) -> c_1()
            fib#(s(0())) -> c_2()
            fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
            fibo#(0()) -> c_4(fib#(0()))
            fibo#(s(0())) -> c_5(fib#(s(0())))
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(x,0()) -> c_9()
            lt#(0(),s(x)) -> c_10()
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
            sum#(x,0()) -> c_12()
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,9,10,12}
        by application of
          Pre({1,2,9,10,12}) = {4,5,6,7,11,13}.
        Here rules are labelled as follows:
          1: fib#(0()) -> c_1()
          2: fib#(s(0())) -> c_2()
          3: fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
          4: fibo#(0()) -> c_4(fib#(0()))
          5: fibo#(s(0())) -> c_5(fib#(s(0())))
          6: fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          7: if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          8: if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          9: lt#(x,0()) -> c_9()
          10: lt#(0(),s(x)) -> c_10()
          11: lt#(s(x),s(y)) -> c_11(lt#(x,y))
          12: sum#(x,0()) -> c_12()
          13: sum#(x,s(y)) -> c_13(sum#(x,y))
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
            fibo#(0()) -> c_4(fib#(0()))
            fibo#(s(0())) -> c_5(fib#(s(0())))
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak DPs:
            fib#(0()) -> c_1()
            fib#(s(0())) -> c_2()
            lt#(x,0()) -> c_9()
            lt#(0(),s(x)) -> c_10()
            sum#(x,0()) -> c_12()
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3}
        by application of
          Pre({2,3}) = {4,5}.
        Here rules are labelled as follows:
          1: fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
          2: fibo#(0()) -> c_4(fib#(0()))
          3: fibo#(s(0())) -> c_5(fib#(s(0())))
          4: fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          5: if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          6: if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          7: lt#(s(x),s(y)) -> c_11(lt#(x,y))
          8: sum#(x,s(y)) -> c_13(sum#(x,y))
          9: fib#(0()) -> c_1()
          10: fib#(s(0())) -> c_2()
          11: lt#(x,0()) -> c_9()
          12: lt#(0(),s(x)) -> c_10()
          13: sum#(x,0()) -> c_12()
* Step 5: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak DPs:
            fib#(0()) -> c_1()
            fib#(s(0())) -> c_2()
            fibo#(0()) -> c_4(fib#(0()))
            fibo#(s(0())) -> c_5(fib#(s(0())))
            lt#(x,0()) -> c_9()
            lt#(0(),s(x)) -> c_10()
            sum#(x,0()) -> c_12()
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
             -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):4
          
          2:S:fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
             -->_3 fibo#(s(0())) -> c_5(fib#(s(0()))):10
             -->_2 fibo#(s(0())) -> c_5(fib#(s(0()))):10
             -->_3 fibo#(0()) -> c_4(fib#(0())):9
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
             -->_1 sum#(x,0()) -> c_12():13
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
          
          3:S:if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
             -->_3 fibo#(s(0())) -> c_5(fib#(s(0()))):10
             -->_2 fibo#(s(0())) -> c_5(fib#(s(0()))):10
             -->_3 fibo#(0()) -> c_4(fib#(0())):9
             -->_2 fibo#(0()) -> c_4(fib#(0())):9
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
             -->_1 sum#(x,0()) -> c_12():13
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
          
          4:S:if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
             -->_2 lt#(s(x),s(y)) -> c_11(lt#(x,y)):5
             -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):4
             -->_1 if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b)):3
          
          5:S:lt#(s(x),s(y)) -> c_11(lt#(x,y))
             -->_1 lt#(0(),s(x)) -> c_10():12
             -->_1 lt#(x,0()) -> c_9():11
             -->_1 lt#(s(x),s(y)) -> c_11(lt#(x,y)):5
          
          6:S:sum#(x,s(y)) -> c_13(sum#(x,y))
             -->_1 sum#(x,0()) -> c_12():13
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
          
          7:W:fib#(0()) -> c_1()
             
          
          8:W:fib#(s(0())) -> c_2()
             
          
          9:W:fibo#(0()) -> c_4(fib#(0()))
             -->_1 fib#(0()) -> c_1():7
          
          10:W:fibo#(s(0())) -> c_5(fib#(s(0())))
             -->_1 fib#(s(0())) -> c_2():8
          
          11:W:lt#(x,0()) -> c_9()
             
          
          12:W:lt#(0(),s(x)) -> c_10()
             
          
          13:W:sum#(x,0()) -> c_12()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          13: sum#(x,0()) -> c_12()
          9: fibo#(0()) -> c_4(fib#(0()))
          7: fib#(0()) -> c_1()
          10: fibo#(s(0())) -> c_5(fib#(s(0())))
          8: fib#(s(0())) -> c_2()
          11: lt#(x,0()) -> c_9()
          12: lt#(0(),s(x)) -> c_10()
* Step 6: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0()))
           -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):4
        
        2:S:fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
           -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
           -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
           -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
        
        3:S:if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
           -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
           -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
           -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
        
        4:S:if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
           -->_2 lt#(s(x),s(y)) -> c_11(lt#(x,y)):5
           -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):4
           -->_1 if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b)):3
        
        5:S:lt#(s(x),s(y)) -> c_11(lt#(x,y))
           -->_1 lt#(s(x),s(y)) -> c_11(lt#(x,y)):5
        
        6:S:sum#(x,s(y)) -> c_13(sum#(x,y))
           -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
        
        
        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).
        
        [(1,fib#(s(s(x))) -> c_3(if#(true(),0(),s(s(x)),0(),0())))]
* Step 7: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} 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:
              fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
              sum#(x,s(y)) -> c_13(sum#(x,y))
          - Weak DPs:
              if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
              if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
              lt#(s(x),s(y)) -> c_11(lt#(x,y))
          - Weak TRS:
              fib(0()) -> s(0())
              fib(s(0())) -> s(0())
              fibo(0()) -> fib(0())
              fibo(s(0())) -> fib(s(0()))
              fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
              lt(x,0()) -> false()
              lt(0(),s(x)) -> true()
              lt(s(x),s(y)) -> lt(x,y)
              sum(x,0()) -> x
              sum(x,s(y)) -> s(sum(x,y))
          - Signature:
              {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
              ,c_3/1,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
              ,true}
        
        Problem (S)
          - Strict DPs:
              if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
              if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
              lt#(s(x),s(y)) -> c_11(lt#(x,y))
          - Weak DPs:
              fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
              sum#(x,s(y)) -> c_13(sum#(x,y))
          - Weak TRS:
              fib(0()) -> s(0())
              fib(s(0())) -> s(0())
              fibo(0()) -> fib(0())
              fibo(s(0())) -> fib(s(0()))
              fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
              lt(x,0()) -> false()
              lt(0(),s(x)) -> true()
              lt(s(x),s(y)) -> lt(x,y)
              sum(x,0()) -> x
              sum(x,s(y)) -> s(sum(x,y))
          - Signature:
              {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
              ,c_3/1,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
              ,true}
** Step 7.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak DPs:
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          2:S:fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
          
          3:W:if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
          
          4:W:if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
             -->_1 if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b)):3
             -->_2 lt#(s(x),s(y)) -> c_11(lt#(x,y)):5
             -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):4
          
          5:W:lt#(s(x),s(y)) -> c_11(lt#(x,y))
             -->_1 lt#(s(x),s(y)) -> c_11(lt#(x,y)):5
          
          6:S:sum#(x,s(y)) -> c_13(sum#(x,y))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: lt#(s(x),s(y)) -> c_11(lt#(x,y))
** Step 7.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak DPs:
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          2:S:fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
          
          3:W:if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):2
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
          
          4:W:if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
             -->_1 if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b)):3
             -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):4
          
          6:S:sum#(x,s(y)) -> c_13(sum#(x,y))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):6
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c))
** Step 7.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          sum#(x,s(y)) -> c_13(sum#(x,y))
      - Weak DPs:
          if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c))
      - Weak TRS:
          fib(0()) -> s(0())
          fib(s(0())) -> s(0())
          fibo(0()) -> fib(0())
          fibo(s(0())) -> fib(s(0()))
          fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          sum(x,0()) -> x
          sum(x,s(y)) -> s(sum(x,y))
      - Signature:
          {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
          ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 7.b:1: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
        - Weak DPs:
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          2: if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          3: lt#(s(x),s(y)) -> c_11(lt#(x,y))
          4: fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          5: sum#(x,s(y)) -> c_13(sum#(x,y))
** Step 7.b:2: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
        - Weak DPs:
            fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
            if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
            sum#(x,s(y)) -> c_13(sum#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
             -->_1 if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b)):4
             -->_2 lt#(s(x),s(y)) -> c_11(lt#(x,y)):2
             -->_1 if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x)))):1
          
          2:S:lt#(s(x),s(y)) -> c_11(lt#(x,y))
             -->_1 lt#(s(x),s(y)) -> c_11(lt#(x,y)):2
          
          3:W:fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):5
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):3
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):3
          
          4:W:if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):5
             -->_3 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):3
             -->_2 fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x)):3
          
          5:W:sum#(x,s(y)) -> c_13(sum#(x,y))
             -->_1 sum#(x,s(y)) -> c_13(sum#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: if#(false(),c,s(s(x)),a,b) -> c_7(sum#(fibo(a),fibo(b)),fibo#(a),fibo#(b))
          3: fibo#(s(s(x))) -> c_6(sum#(fibo(s(x)),fibo(x)),fibo#(s(x)),fibo#(x))
          5: sum#(x,s(y)) -> c_13(sum#(x,y))
** Step 7.b:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
            lt#(s(x),s(y)) -> c_11(lt#(x,y))
        - Weak TRS:
            fib(0()) -> s(0())
            fib(s(0())) -> s(0())
            fibo(0()) -> fib(0())
            fibo(s(0())) -> fib(s(0()))
            fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            sum(x,0()) -> x
            sum(x,s(y)) -> s(sum(x,y))
        - Signature:
            {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
            ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          lt#(s(x),s(y)) -> c_11(lt#(x,y))
** Step 7.b:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          if#(true(),c,s(s(x)),a,b) -> c_8(if#(lt(s(c),s(s(x))),s(c),s(s(x)),b,c),lt#(s(c),s(s(x))))
          lt#(s(x),s(y)) -> c_11(lt#(x,y))
      - Weak TRS:
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
      - Signature:
          {fib/1,fibo/1,if/5,lt/2,sum/2,fib#/1,fibo#/1,if#/5,lt#/2,sum#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1
          ,c_4/1,c_5/1,c_6/3,c_7/3,c_8/2,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {fib#,fibo#,if#,lt#,sum#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE