MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib(N) -> sel(N,fib1(s(0()),s(0())))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
            sel(0(),cons(X,XS)) -> X
            sel(s(N),cons(X,XS)) -> sel(N,XS)
        - Signature:
            {add/2,fib/1,fib1/2,sel/2} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,fib,fib1,sel} and constructors {0,cons,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
          sel#(0(),cons(X,XS)) -> c_5()
          sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(0(),cons(X,XS)) -> c_5()
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib(N) -> sel(N,fib1(s(0()),s(0())))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
            sel(0(),cons(X,XS)) -> X
            sel(s(N),cons(X,XS)) -> sel(N,XS)
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
          sel#(0(),cons(X,XS)) -> c_5()
          sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(0(),cons(X,XS)) -> c_5()
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,5}
        by application of
          Pre({1,5}) = {2,3,4,6}.
        Here rules are labelled as follows:
          1: add#(0(),X) -> c_1()
          2: add#(s(X),Y) -> c_2(add#(X,Y))
          3: fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
          4: fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
          5: sel#(0(),cons(X,XS)) -> c_5()
          6: sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak DPs:
            add#(0(),X) -> c_1()
            sel#(0(),cons(X,XS)) -> c_5()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(0(),X) -> c_1():5
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:S:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):4
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):3
             -->_1 sel#(0(),cons(X,XS)) -> c_5():6
          
          3:S:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
             -->_2 add#(0(),X) -> c_1():5
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):3
             -->_2 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          4:S:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(0(),cons(X,XS)) -> c_5():6
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):4
          
          5:W:add#(0(),X) -> c_1()
             
          
          6:W:sel#(0(),cons(X,XS)) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: sel#(0(),cons(X,XS)) -> c_5()
          5: add#(0(),X) -> c_1()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,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:
              add#(s(X),Y) -> c_2(add#(X,Y))
          - Weak DPs:
              fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
              fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
              sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
          - Signature:
              {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
              ,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
        
        Problem (S)
          - Strict DPs:
              fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
              fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
              sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
          - Weak DPs:
              add#(s(X),Y) -> c_2(add#(X,Y))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
          - Signature:
              {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
              ,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:W:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):4
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):3
          
          3:W:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
             -->_2 add#(s(X),Y) -> c_2(add#(X,Y)):1
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):3
          
          4:W:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
** Step 5.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:W:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):3
          
          3:W:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
             -->_2 add#(s(X),Y) -> c_2(add#(X,Y)):1
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fib#(N) -> c_3(fib1#(s(0()),s(0())))
** Step 5.a:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            fib#(N) -> c_3(fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          add#(s(X),Y) -> c_2(add#(X,Y))
          fib#(N) -> c_3(fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
** Step 5.a:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          add#(s(X),Y) -> c_2(add#(X,Y))
      - Weak DPs:
          fib#(N) -> c_3(fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
      - Weak TRS:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
      - Signature:
          {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0
          ,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):2
          
          2:S:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
             -->_2 add#(s(X),Y) -> c_2(add#(X,Y)):4
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):2
          
          3:S:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
          
          4:W:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: add#(s(X),Y) -> c_2(add#(X,Y))
** Step 5.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):2
          
          2:S:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y))
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)),add#(X,Y)):2
          
          3:S:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
** Step 5.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,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:
              fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
              sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
          - Weak DPs:
              fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
          - Signature:
              {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
              ,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
        
        Problem (S)
          - Strict DPs:
              fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
          - Weak DPs:
              fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
              sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
          - Signature:
              {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
              ,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
*** Step 5.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak DPs:
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y))):2
          
          2:W:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y))):2
          
          3:S:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
*** Step 5.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
          
          3:S:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
*** Step 5.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
          2: sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {add#,fib#,fib1#,sel#}
        TcT has computed the following interpretation:
              p(0) = [2]                  
            p(add) = [4] x1 + [2] x2 + [8]
           p(cons) = [4]                  
            p(fib) = [1]                  
           p(fib1) = [2] x2 + [0]         
              p(s) = [1] x1 + [1]         
            p(sel) = [8] x1 + [2] x2 + [0]
           p(add#) = [8] x1 + [0]         
           p(fib#) = [8] x1 + [10]        
          p(fib1#) = [2] x1 + [2]         
           p(sel#) = [2] x1 + [0]         
            p(c_1) = [2]                  
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [4] x1 + [0]         
            p(c_4) = [1] x1 + [1]         
            p(c_5) = [0]                  
            p(c_6) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
                      fib#(N) = [8] N + [10]                    
                              > [8] N + [0]                     
                              = c_3(sel#(N,fib1(s(0()),s(0()))))
        
        sel#(s(N),cons(X,XS)) = [2] N + [2]                     
                              > [2] N + [0]                     
                              = c_6(sel#(N,XS))                 
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 5.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):2
          
          2:W:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))))
          2: sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
**** Step 5.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.b:3.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
        - Weak DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y))):1
          
          2:W:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y))):1
          
          3:W:sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
             -->_1 sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sel#(s(N),cons(X,XS)) -> c_6(sel#(N,XS))
*** Step 5.b:3.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
        - Weak DPs:
            fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/2,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
             -->_1 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y))):1
          
          2:W:fib#(N) -> c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_2 fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fib#(N) -> c_3(fib1#(s(0()),s(0())))
*** Step 5.b:3.b:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
        - Weak DPs:
            fib#(N) -> c_3(fib1#(s(0()),s(0())))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
        - Signature:
            {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          fib#(N) -> c_3(fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
*** Step 5.b:3.b:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          fib1#(X,Y) -> c_4(fib1#(Y,add(X,Y)))
      - Weak DPs:
          fib#(N) -> c_3(fib1#(s(0()),s(0())))
      - Weak TRS:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
      - Signature:
          {add/2,fib/1,fib1/2,sel/2,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
          ,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,fib#,fib1#,sel#} and constructors {0,cons,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE