We consider the following Problem:

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

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

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