We consider the following Problem:

  Strict Trs:
    {  g(c(x, s(y))) -> g(c(s(x), y))
     , f(c(s(x), y)) -> f(c(x, s(y)))
     , f(f(x)) -> f(d(f(x)))
     , f(x) -> x}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  Arguments of following rules are not normal-forms:
  {f(f(x)) -> f(d(f(x)))}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  g(c(x, s(y))) -> g(c(s(x), y))
       , f(c(s(x), y)) -> f(c(x, s(y)))
       , f(x) -> 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(x) -> x}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(g) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(f) = {},
        Uargs(d) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       g(x1) = [0 0] x1 + [0]
               [0 0]      [1]
       c(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                   [0 0]      [0 0]      [0]
       s(x1) = [1 0] x1 + [0]
               [0 1]      [0]
       f(x1) = [1 0] x1 + [1]
               [0 1]      [1]
       d(x1) = [0 0] x1 + [0]
               [0 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  g(c(x, s(y))) -> g(c(s(x), y))
         , f(c(s(x), y)) -> f(c(x, s(y)))}
      Weak Trs: {f(x) -> x}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {g(c(x, s(y))) -> g(c(s(x), y))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(g) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(f) = {},
          Uargs(d) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         g(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
         c(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
                     [0 0]      [0 1]      [0]
         s(x1) = [1 0] x1 + [2]
                 [0 0]      [0]
         f(x1) = [1 0] x1 + [0]
                 [0 1]      [1]
         d(x1) = [0 0] x1 + [0]
                 [0 0]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {f(c(s(x), y)) -> f(c(x, s(y)))}
        Weak Trs:
          {  g(c(x, s(y))) -> g(c(s(x), y))
           , f(x) -> 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(c(s(x), y)) -> f(c(x, s(y)))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(g) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(f) = {},
            Uargs(d) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           g(x1) = [0 0] x1 + [1]
                   [0 0]      [0]
           c(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                       [0 0]      [0 0]      [0]
           s(x1) = [1 0] x1 + [3]
                   [0 1]      [0]
           f(x1) = [1 0] x1 + [0]
                   [0 1]      [1]
           d(x1) = [0 0] x1 + [0]
                   [0 0]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Weak Trs:
            {  f(c(s(x), y)) -> f(c(x, s(y)))
             , g(c(x, s(y))) -> g(c(s(x), y))
             , f(x) -> x}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          We consider the following Problem:
          
            Weak Trs:
              {  f(c(s(x), y)) -> f(c(x, s(y)))
               , g(c(x, s(y))) -> g(c(s(x), y))
               , f(x) -> x}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            Empty rules are trivially bounded

Hurray, we answered YES(?,O(n^1))