We consider the following Problem:

  Strict Trs:
    {  not(true()) -> false()
     , not(false()) -> true()
     , evenodd(x, 0()) -> not(evenodd(x, s(0())))
     , evenodd(0(), s(0())) -> false()
     , evenodd(s(x), s(0())) -> evenodd(x, 0())}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  not(true()) -> false()
       , not(false()) -> true()
       , evenodd(x, 0()) -> not(evenodd(x, s(0())))
       , evenodd(0(), s(0())) -> false()
       , evenodd(s(x), s(0())) -> evenodd(x, 0())}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  not(true()) -> false()
       , not(false()) -> true()
       , evenodd(0(), s(0())) -> false()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(not) = {1}, Uargs(evenodd) = {}, Uargs(s) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       not(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
       true() = [0]
                [0]
       false() = [0]
                 [0]
       evenodd(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                         [0 0]      [0 0]      [1]
       0() = [0]
             [0]
       s(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:
        {  evenodd(x, 0()) -> not(evenodd(x, s(0())))
         , evenodd(s(x), s(0())) -> evenodd(x, 0())}
      Weak Trs:
        {  not(true()) -> false()
         , not(false()) -> true()
         , evenodd(0(), s(0())) -> false()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {evenodd(s(x), s(0())) -> evenodd(x, 0())}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(not) = {1}, Uargs(evenodd) = {}, Uargs(s) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         not(x1) = [1 2] x1 + [0]
                   [0 0]      [1]
         true() = [0]
                  [0]
         false() = [0]
                   [0]
         evenodd(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                           [0 0]      [0 1]      [3]
         0() = [0]
               [1]
         s(x1) = [0 1] x1 + [1]
                 [0 0]      [2]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
        Weak Trs:
          {  evenodd(s(x), s(0())) -> evenodd(x, 0())
           , not(true()) -> false()
           , not(false()) -> true()
           , evenodd(0(), s(0())) -> false()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(not) = {1}, Uargs(evenodd) = {}, Uargs(s) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           not(x1) = [1 0] x1 + [0]
                     [0 0]      [1]
           true() = [0]
                    [0]
           false() = [0]
                     [0]
           evenodd(x1, x2) = [0 1] x1 + [1 1] x2 + [1]
                             [0 0]      [0 0]      [1]
           0() = [2]
                 [2]
           s(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:
            {  evenodd(x, 0()) -> not(evenodd(x, s(0())))
             , evenodd(s(x), s(0())) -> evenodd(x, 0())
             , not(true()) -> false()
             , not(false()) -> true()
             , evenodd(0(), s(0())) -> false()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          We consider the following Problem:
          
            Weak Trs:
              {  evenodd(x, 0()) -> not(evenodd(x, s(0())))
               , evenodd(s(x), s(0())) -> evenodd(x, 0())
               , not(true()) -> false()
               , not(false()) -> true()
               , evenodd(0(), s(0())) -> false()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            Empty rules are trivially bounded

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