We consider the following Problem:

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

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

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