We consider the following Problem:

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

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

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