We consider the following Problem:

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

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

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

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