MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,fib} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(x,0()) -> c_1()
          +#(x,s(y)) -> c_2(+#(x,y))
          fib#(0()) -> c_3()
          fib#(s(0())) -> c_4()
          fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1()
            +#(x,s(y)) -> c_2(+#(x,y))
            fib#(0()) -> c_3()
            fib#(s(0())) -> c_4()
            fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4}
        by application of
          Pre({1,3,4}) = {2,5}.
        Here rules are labelled as follows:
          1: +#(x,0()) -> c_1()
          2: +#(x,s(y)) -> c_2(+#(x,y))
          3: fib#(0()) -> c_3()
          4: fib#(s(0())) -> c_4()
          5: fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
        - Weak DPs:
            +#(x,0()) -> c_1()
            fib#(0()) -> c_3()
            fib#(s(0())) -> c_4()
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(x,s(y)) -> c_2(+#(x,y))
             -->_1 +#(x,0()) -> c_1():3
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          2:S:fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
             -->_3 fib#(s(0())) -> c_4():5
             -->_2 fib#(s(0())) -> c_4():5
             -->_3 fib#(0()) -> c_3():4
             -->_1 +#(x,0()) -> c_1():3
             -->_3 fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x)):2
             -->_2 fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x)):2
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          3:W:+#(x,0()) -> c_1()
             
          
          4:W:fib#(0()) -> c_3()
             
          
          5:W:fib#(s(0())) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: fib#(0()) -> c_3()
          5: fib#(s(0())) -> c_4()
          3: +#(x,0()) -> c_1()
* Step 4: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              +#(x,s(y)) -> c_2(+#(x,y))
          - Weak DPs:
              fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
          - Weak TRS:
              +(x,0()) -> x
              +(x,s(y)) -> s(+(x,y))
              fib(0()) -> 0()
              fib(s(0())) -> s(0())
              fib(s(s(x))) -> +(fib(s(x)),fib(x))
          - Signature:
              {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
          - Weak DPs:
              +#(x,s(y)) -> c_2(+#(x,y))
          - Weak TRS:
              +(x,0()) -> x
              +(x,s(y)) -> s(+(x,y))
              fib(0()) -> 0()
              fib(s(0())) -> s(0())
              fib(s(s(x))) -> +(fib(s(x)),fib(x))
          - Signature:
              {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
** Step 4.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          +#(x,s(y)) -> c_2(+#(x,y))
      - Weak DPs:
          fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
      - Weak TRS:
          +(x,0()) -> x
          +(x,s(y)) -> s(+(x,y))
          fib(0()) -> 0()
          fib(s(0())) -> s(0())
          fib(s(s(x))) -> +(fib(s(x)),fib(x))
      - Signature:
          {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 4.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
        - Weak DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):2
             -->_3 fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x)):1
             -->_2 fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x)):1
          
          2:W:+#(x,s(y)) -> c_2(+#(x,y))
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: +#(x,s(y)) -> c_2(+#(x,y))
** Step 4.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x))
             -->_3 fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x)):1
             -->_2 fib#(s(s(x))) -> c_5(+#(fib(s(x)),fib(x)),fib#(s(x)),fib#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fib#(s(s(x))) -> c_5(fib#(s(x)),fib#(x))
** Step 4.b:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(s(s(x))) -> c_5(fib#(s(x)),fib#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(x))) -> +(fib(s(x)),fib(x))
        - Signature:
            {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          fib#(s(s(x))) -> c_5(fib#(s(x)),fib#(x))
** Step 4.b:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          fib#(s(s(x))) -> c_5(fib#(s(x)),fib#(x))
      - Signature:
          {+/2,fib/1,+#/2,fib#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {+#,fib#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE