We consider the following Problem:

  Strict Trs:
    {  f(0()) -> 1()
     , f(s(x)) -> g(x, s(x))
     , g(0(), y) -> y
     , g(s(x), y) -> g(x, +(y, s(x)))
     , +(x, 0()) -> x
     , +(x, s(y)) -> s(+(x, y))
     , g(s(x), y) -> g(x, s(+(y, x)))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^2))

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

Hurray, we answered YES(?,O(n^2))