We consider the following Problem:

  Strict Trs:
    {  from(X) -> cons(X, n__from(n__s(X)))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^2))

Proof:
  Arguments of following rules are not normal-forms:
  {sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  from(X) -> cons(X, n__from(n__s(X)))
       , sel(0(), cons(X, Y)) -> X
       , from(X) -> n__from(X)
       , s(X) -> n__s(X)
       , activate(n__from(X)) -> from(activate(X))
       , activate(n__s(X)) -> s(activate(X))
       , activate(X) -> X}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {activate(n__s(X)) -> s(activate(X))}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
        Uargs(n__s) = {}, Uargs(sel) = {}, Uargs(s) = {1},
        Uargs(activate) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       from(x1) = [1 1] x1 + [0]
                  [0 0]      [1]
       cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                      [0 0]      [0 0]      [1]
       n__from(x1) = [0 0] x1 + [0]
                     [1 1]      [0]
       n__s(x1) = [0 0] x1 + [0]
                  [1 1]      [2]
       sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                     [0 0]      [0 0]      [1]
       0() = [0]
             [0]
       s(x1) = [1 0] x1 + [0]
               [0 0