We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  sum(0()) -> 0()
       , sum(s(x)) -> +(sum(x), s(x))
       , +(x, 0()) -> x
       , +(x, s(y)) -> s(+(x, y))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  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) = {1}, Uargs(+) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       sum(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 + [0]
                   [0 0]      [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  sum(s(x)) -> +(sum(x), s(x))
         , +(x, 0()) -> x
         , +(x, s(y)) -> s(+(x, y))}
      Weak Trs: {sum(0()) -> 0()}
      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(sum) = {}, Uargs(s) = {1}, Uargs(+) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Int