We consider the following Problem:

  Strict Trs:
    {  a(c(d(x))) -> c(x)
     , u(b(d(d(x)))) -> b(x)
     , v(a(a(x))) -> u(v(x))
     , v(a(c(x))) -> u(b(d(x)))
     , v(c(x)) -> b(x)
     , w(a(a(x))) -> u(w(x))
     , w(a(c(x))) -> u(b(d(x)))
     , w(c(x)) -> b(x)}
  StartTerms: basic terms
  Strategy: innermost

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

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