MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__fib1(X1,X2)) -> fib1(X1,X2)
            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,n__fib1(Y,add(X,Y)))
            fib1(X1,X2) -> n__fib1(X1,X2)
            sel(0(),cons(X,XS)) -> X
            sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
        - Signature:
            {activate/1,add/2,fib/1,fib1/2,sel/2} / {0/0,cons/2,n__fib1/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,fib,fib1,sel} and constructors {0,cons
            ,n__fib1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
          add#(0(),X) -> c_3()
          add#(s(X),Y) -> c_4(add#(X,Y))
          fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_6(add#(X,Y))
          fib1#(X1,X2) -> c_7()
          sel#(0(),cons(X,XS)) -> c_8()
          sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
            add#(0(),X) -> c_3()
            add#(s(X),Y) -> c_4(add#(X,Y))
            fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_6(add#(X,Y))
            fib1#(X1,X2) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__fib1(X1,X2)) -> fib1(X1,X2)
            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,n__fib1(Y,add(X,Y)))
            fib1(X1,X2) -> n__fib1(X1,X2)
            sel(0(),cons(X,XS)) -> X
            sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
        - Signature:
            {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons
            ,n__fib1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__fib1(X1,X2)) -> fib1(X1,X2)
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y)))
          fib1(X1,X2) -> n__fib1(X1,X2)
          activate#(X) -> c_1()
          activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
          add#(0(),X) -> c_3()
          add#(s(X),Y) -> c_4(add#(X,Y))
          fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
          fib1#(X,Y) -> c_6(add#(X,Y))
          fib1#(X1,X2) -> c_7()
          sel#(0(),cons(X,XS)) -> c_8()
          sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
            add#(0(),X) -> c_3()
            add#(s(X),Y) -> c_4(add#(X,Y))
            fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_6(add#(X,Y))
            fib1#(X1,X2) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__fib1(X1,X2)) -> fib1(X1,X2)
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y)))
            fib1(X1,X2) -> n__fib1(X1,X2)
        - Signature:
            {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons
            ,n__fib1,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,7,8}
        by application of
          Pre({1,3,7,8}) = {2,4,5,6,9}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
          3: add#(0(),X) -> c_3()
          4: add#(s(X),Y) -> c_4(add#(X,Y))
          5: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
          6: fib1#(X,Y) -> c_6(add#(X,Y))
          7: fib1#(X1,X2) -> c_7()
          8: sel#(0(),cons(X,XS)) -> c_8()
          9: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
            add#(s(X),Y) -> c_4(add#(X,Y))
            fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_6(add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
        - Weak DPs:
            activate#(X) -> c_1()
            add#(0(),X) -> c_3()
            fib1#(X1,X2) -> c_7()
            sel#(0(),cons(X,XS)) -> c_8()
        - Weak TRS:
            activate(X) -> X
            activate(n__fib1(X1,X2)) -> fib1(X1,X2)
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y)))
            fib1(X1,X2) -> n__fib1(X1,X2)
        - Signature:
            {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons
            ,n__fib1,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
             -->_1 fib1#(X,Y) -> c_6(add#(X,Y)):4
             -->_1 fib1#(X1,X2) -> c_7():8
          
          2:S:add#(s(X),Y) -> c_4(add#(X,Y))
             -->_1 add#(0(),X) -> c_3():7
             -->_1 add#(s(X),Y) -> c_4(add#(X,Y)):2
          
          3:S:fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
             -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):5
             -->_2 fib1#(X,Y) -> c_6(add#(X,Y)):4
             -->_1 sel#(0(),cons(X,XS)) -> c_8():9
             -->_2 fib1#(X1,X2) -> c_7():8
          
          4:S:fib1#(X,Y) -> c_6(add#(X,Y))
             -->_1 add#(0(),X) -> c_3():7
             -->_1 add#(s(X),Y) -> c_4(add#(X,Y)):2
          
          5:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
             -->_1 sel#(0(),cons(X,XS)) -> c_8():9
             -->_2 activate#(X) -> c_1():6
             -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):5
             -->_2 activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)):1
          
          6:W:activate#(X) -> c_1()
             
          
          7:W:add#(0(),X) -> c_3()
             
          
          8:W:fib1#(X1,X2) -> c_7()
             
          
          9:W:sel#(0(),cons(X,XS)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: activate#(X) -> c_1()
          9: sel#(0(),cons(X,XS)) -> c_8()
          8: fib1#(X1,X2) -> c_7()
          7: add#(0(),X) -> c_3()
* Step 5: NaturalMI MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
            add#(s(X),Y) -> c_4(add#(X,Y))
            fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
            fib1#(X,Y) -> c_6(add#(X,Y))
            sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__fib1(X1,X2)) -> fib1(X1,X2)
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y)))
            fib1(X1,X2) -> n__fib1(X1,X2)
        - Signature:
            {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons
            ,n__fib1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_4) = {1},
          uargs(c_5) = {1,2},
          uargs(c_6) = {1},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {activate#,add#,fib#,fib1#,sel#}
        TcT has computed the following interpretation:
                  p(0) = [4]                  
           p(activate) = [2]                  
                p(add) = [1] x2 + [0]         
               p(cons) = [0]                  
                p(fib) = [1] x1 + [0]         
               p(fib1) = [1]                  
            p(n__fib1) = [3]                  
                  p(s) = [0]                  
                p(sel) = [1] x2 + [1]         
          p(activate#) = [0]                  
               p(add#) = [0]                  
               p(fib#) = [8] x1 + [4]         
              p(fib1#) = [0]                  
               p(sel#) = [0]                  
                p(c_1) = [4]                  
                p(c_2) = [8] x1 + [0]         
                p(c_3) = [0]                  
                p(c_4) = [2] x1 + [0]         
                p(c_5) = [1] x1 + [8] x2 + [2]
                p(c_6) = [1] x1 + [0]         
                p(c_7) = [0]                  
                p(c_8) = [1]                  
                p(c_9) = [4] x1 + [8] x2 + [0]
        
        Following rules are strictly oriented:
        fib#(N) = [8] N + [4]                                          
                > [2]                                                  
                = c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__fib1(X1,X2)) =  [0]                                    
                                  >= [0]                                    
                                  =  c_2(fib1#(X1,X2))                      
        
                     add#(s(X),Y) =  [0]                                    
                                  >= [0]                                    
                                  =  c_4(add#(X,Y))                         
        
                       fib1#(X,Y) =  [0]                                    
                                  >= [0]                                    
                                  =  c_6(add#(X,Y))                         
        
            sel#(s(N),cons(X,XS)) =  [0]                                    
                                  >= [0]                                    
                                  =  c_9(sel#(N,activate(XS)),activate#(XS))
        
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2))
          add#(s(X),Y) -> c_4(add#(X,Y))
          fib1#(X,Y) -> c_6(add#(X,Y))
          sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS))
      - Weak DPs:
          fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
      - Weak TRS:
          activate(X) -> X
          activate(n__fib1(X1,X2)) -> fib1(X1,X2)
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y)))
          fib1(X1,X2) -> n__fib1(X1,X2)
      - Signature:
          {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1
          ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons
          ,n__fib1,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE