We consider the following Problem:

  Strict Trs:
    {  rev(ls) -> r1(ls, empty())
     , r1(empty(), a) -> a
     , r1(cons(x, k), a) -> r1(k, cons(x, a))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  rev(ls) -> r1(ls, empty())
       , r1(empty(), a) -> a
       , r1(cons(x, k), a) -> r1(k, cons(x, a))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {rev(ls) -> r1(ls, empty())}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(rev) = {}, Uargs(r1) = {}, Uargs(cons) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       rev(x1) = [0 0] x1 + [2]
                 [0 0]      [2]
       r1(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                    [0 0]      [0 0]      [1]
       empty() = [0]
                 [0]
       cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                      [0 0]      [0 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  r1(empty(), a) -> a
         , r1(cons(x, k), a) -> r1(k, cons(x, a))}
      Weak Trs: {rev(ls) -> r1(ls, empty())}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {r1(empty(), a) -> a}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(rev) = {}, Uargs(r1) = {}, Uargs(cons) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         rev(x1) = [1 0] x1 + [2]
                   [0 1]      [2]
         r1(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                      [0 0]      [0 1]      [1]
         empty() = [0]
                   [0]
         cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                        [0 0]      [0 0]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {r1(cons(x, k), a) -> r1(k, cons(x, a))}
        Weak Trs:
          {  r1(empty(), a) -> a
           , rev(ls) -> r1(ls, empty())}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {r1(cons(x, k), a) -> r1(k, cons(x, a))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(rev) = {}, Uargs(r1) = {}, Uargs(cons) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           rev(x1) = [0 2] x1 + [0]
                     [0 1]      [3]
           r1(x1, x2) = [0 2] x1 + [1 0] x2 + [0]
                        [0 1]      [0 1]      [3]
           empty() = [0]
                     [0]
           cons(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                          [0 1]      [0 1]      [2]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Weak Trs:
            {  r1(cons(x, k), a) -> r1(k, cons(x, a))
             , r1(empty(), a) -> a
             , rev(ls) -> r1(ls, empty())}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          We consider the following Problem:
          
            Weak Trs:
              {  r1(cons(x, k), a) -> r1(k, cons(x, a))
               , r1(empty(), a) -> a
               , rev(ls) -> r1(ls, empty())}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            Empty rules are trivially bounded

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