We consider the following Problem:

  Strict Trs:
    {  a__f(f(X)) -> a__c(f(g(f(X))))
     , a__c(X) -> d(X)
     , a__h(X) -> a__c(d(X))
     , mark(f(X)) -> a__f(mark(X))
     , mark(c(X)) -> a__c(X)
     , mark(h(X)) -> a__h(mark(X))
     , mark(g(X)) -> g(X)
     , mark(d(X)) -> d(X)
     , a__f(X) -> f(X)
     , a__c(X) -> c(X)
     , a__h(X) -> h(X)}
  StartTerms: basic terms
  Strategy: innermost

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

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