We consider the following Problem:

  Strict Trs:
    {  active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
     , mark(f(X1, X2)) -> active(f(mark(X1), X2))
     , mark(g(X)) -> active(g(mark(X)))
     , f(mark(X1), X2) -> f(X1, X2)
     , f(X1, mark(X2)) -> f(X1, X2)
     , f(active(X1), X2) -> f(X1, X2)
     , f(X1, active(X2)) -> f(X1, X2)
     , g(mark(X)) -> g(X)
     , g(active(X)) -> g(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
       , mark(f(X1, X2)) -> active(f(mark(X1), X2))
       , mark(g(X)) -> active(g(mark(X)))
       , f(mark(X1), X2) -> f(X1, X2)
       , f(X1, mark(X2)) -> f(X1, X2)
       , f(active(X1), X2) -> f(X1, X2)
       , f(X1, active(X2)) -> f(X1, X2)
       , g(mark(X)) -> g(X)
       , g(active(X)) -> g(X)}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  f(mark(X1), X2) -> f(X1, X2)
       , f(active(X1), X2) -> f(X1, X2)
       , g(mark(X)) -> g(X)
       , g(active(X)) -> g(X)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(active) = {1}, Uargs(f) = {1}, Uargs(g) = {1},
        Uargs(mark) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       active(x1) = [1 0] x1 + [1]
                    [0 0]      [1]
       f(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
                   [0 0]      [0 0]      [1]
       g(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       mark(x1) = [1 0] x1 + [3]
                  [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  active(f(g(X), Y))