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: NaturalMI 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:
        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_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {add#,fib#,fib1#,sel#}
        TcT has computed the following interpretation:
              p(0) = [1]                  
            p(add) = [2]                  
           p(cons) = [13]                 
            p(fib) = [1] x1 + [1]         
           p(fib1) = [0]                  
              p(s) = [3]                  
            p(sel) = [1] x1 + [8]         
           p(add#) = [0]                  
           p(fib#) = [2] x1 + [11]        
          p(fib1#) = [0]                  
           p(sel#) = [0]                  
            p(c_1) = [0]                  
            p(c_2) = [2] x1 + [0]         
            p(c_3) = [8] x1 + [2] x2 + [7]
            p(c_4) = [2] x1 + [8] x2 + [0]
            p(c_5) = [1]                  
            p(c_6) = [8] x1 + [0]         
        
        Following rules are strictly oriented:
        fib#(N) = [2] N + [11]                                         
                > [7]                                                  
                = c_3(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0())))
        
        
        Following rules are (at-least) weakly oriented:
                 add#(s(X),Y) =  [0]                             
                              >= [0]                             
                              =  c_2(add#(X,Y))                  
        
                   fib1#(X,Y) =  [0]                             
                              >= [0]                             
                              =  c_4(fib1#(Y,add(X,Y)),add#(X,Y))
        
        sel#(s(N),cons(X,XS)) =  [0]                             
                              >= [0]                             
                              =  c_6(sel#(N,XS))                 
        
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          add#(s(X),Y) -> c_2(add#(X,Y))
          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:
          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_4/2,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.
MAYBE