We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sum(x), s(x))
     , sum1(0()) -> 0()
     , sum1(s(x)) -> s(+(sum1(x), +(x, 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)) -> +(sum(x), s(x))
       , sum1(0()) -> 0()
       , sum1(s(x)) -> s(+(sum1(x), +(x, 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()
       , sum1(0()) -> 0()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(sum) = {}, Uargs(s) = {1}, Uargs(+) = {1}, Uargs(sum1) = {}
      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]
       sum1(x1) = [0 0] x1 + [1]
                  [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))
         , sum1(s(x)) -> s(+(sum1(x), +(x, x)))}
      Weak Trs:
        {  sum(0()) -> 0()
         , sum1(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: {sum1(s(x)) -> s(+(sum1(x), +(x, x)))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(sum) = {}, Uargs(s) = {1}, Uargs(+) = {1}, Uargs(sum1) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         sum(x1) = [0 0] x1 + [0]
                   [0 0]      [1]
         0() = [0]
               [0]
         s(x1) = [1 0] x1 + [0]
                 [0 1]      [2]
         +(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                     [0 0]      [0 0]      [0]
         sum1(x1) = [0 2] x1 + [0]
                    [0 0]      [2]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {sum(s(x)) -> +(sum(x), s(x))}
        Weak Trs:
          {  sum1(s(x)) -> s(+(sum1(x), +(x, x)))
           , sum(0()) -> 0()
           , sum1(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)) -> +(sum(x), s(x))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(sum) = {}, Uargs(s) = {1}, Uargs(+) = {1}, Uargs(sum1) = {}
          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) = [1 0] x1 + [0]
                   [0 1]      [3]
           +(x1, x2) = [1 2] x1 + [0 0] x2 + [3]
                       [0 0]      [0 0]      [0]
           sum1(x1) = [0 3] x1 + [0]
                      [0 0]      [3]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Weak Trs:
            {  sum(s(x)) -> +(sum(x), s(x))
             , sum1(s(x)) -> s(+(sum1(x), +(x, x)))
             , sum(0()) -> 0()
             , sum1(0()) -> 0()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          We consider the following Problem:
          
            Weak Trs:
              {  sum(s(x)) -> +(sum(x), s(x))
               , sum1(s(x)) -> s(+(sum1(x), +(x, x)))
               , sum(0()) -> 0()
               , sum1(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))