We consider the following Problem:

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

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