We consider the following Problem:

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

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

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

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