We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  not(true()) -> false()
       , not(false()) -> true()
       , odd(0()) -> false()
       , odd(s(x)) -> not(odd(x))
       , +(x, 0()) -> x
       , +(x, s(y)) -> s(+(x, y))
       , +(s(x), y) -> s(+(x, y))}
    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()
       , odd(0()) -> false()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(not) = {1}, Uargs(odd) = {}, Uargs(s) = {1}, Uargs(+) = {}
      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]
       odd(x1) = [0 0] x1 + [1]
                 [1 0]      [1]
       0() = [0]
             [0]
       s(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       +(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                   [0 0]      [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  odd(s(x)) -> not(odd(x))
         , +(x, 0()) -> x
         , +(x, s(y)) -> s(+(x, y))
         , +(s(x), y) -> s(+(x, y))}
      Weak Trs:
        {  not(true()) -> false()
         , not(false()) -> true()
         , odd(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: {+(x, 0()) -> x}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(not) = {1}, Uargs(odd) = {}, Uargs(s) = {1}, Uargs(+) = {}
        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]
         odd(x1) = [0 0] x1 + [1]
                   [1 0]      [1]
         0() = [0]
               [0]
         s(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
         +(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                     [0 1]      [0 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  odd(s(x)) -> not(odd(x))
           , +(x, s(y)) -> s(+(x, y))
           , +(s(x), y) -> s(+(x, y))}
        Weak Trs:
          {  +(x, 0()) -> x
           , not(true()) -> false()
           , not(false()) -> true()
           , odd(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: {+(x, s(y)) -> s(+(x, y))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(not) = {1}, Uargs(odd) = {}, Uargs(s) = {1}, Uargs(+) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           not(x1) = [1 2] x1 + [1]
                     [0 0]      [1]
           true() = [0]
                    [0]
           false() = [0]
                     [0]
           odd(x1) = [0 0] x1 + [0]
                     [0 0]      [2]
           0() = [0]
                 [0]
           s(x1) = [1 0] x1 + [0]
                   [0 1]      [1]
           +(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
                       [0 1]      [0 1]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  odd(s(x)) -> not(odd(x))
             , +(s(x), y) -> s(+(x, y))}
          Weak Trs:
            {  +(x, s(y)) -> s(+(x, y))
             , +(x, 0()) -> x
             , not(true()) -> false()
             , not(false()) -> true()
             , odd(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: {odd(s(x)) -> not(odd(x))}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(not) = {1}, Uargs(odd) = {}, Uargs(s) = {1}, Uargs(+) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             not(x1) = [1 0] x1 + [0]
                       [0 0]      [0]
             true() = [0]
                      [0]
             false() = [0]
                       [0]
             odd(x1) = [0 3] x1 + [0]
                       [0 0]      [1]
             0() = [0]
                   [0]
             s(x1) = [1 0] x1 + [0]
                     [0 1]      [3]
             +(x1, x2) = [1 0] x1 + [0 2] x2 + [0]
                         [0 1]      [0 1]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs: {+(s(x), y) -> s(+(x, y))}
            Weak Trs:
              {  odd(s(x)) -> not(odd(x))
               , +(x, s(y)) -> s(+(x, y))
               , +(x, 0()) -> x
               , not(true()) -> false()
               , not(false()) -> true()
               , odd(0()) -> false()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            We consider the following Problem:
            
              Strict Trs: {+(s(x), y) -> s(+(x, y))}
              Weak Trs:
                {  odd(s(x)) -> not(odd(x))
                 , +(x, s(y)) -> s(+(x, y))
                 , +(x, 0()) -> x
                 , not(true()) -> false()
                 , not(false()) -> true()
                 , odd(0()) -> false()}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^1))
            
            Proof:
              The problem is match-bounded by 1.
              The enriched problem is compatible with the following automaton:
              {  not_0(1) -> 1
               , not_0(2) -> 1
               , true_0() -> 1
               , true_0() -> 2
               , true_0() -> 3
               , false_0() -> 1
               , false_0() -> 2
               , false_0() -> 3
               , odd_0(2) -> 1
               , 0_0() -> 1
               , 0_0() -> 2
               , 0_0() -> 3
               , s_0(1) -> 1
               , s_0(2) -> 1
               , s_0(2) -> 2
               , s_0(2) -> 3
               , s_1(3) -> 1
               , s_1(3) -> 3
               , +_0(2, 2) -> 1
               , +_1(2, 2) -> 3}

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