We consider the following Problem:

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

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

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

Hurray, we answered YES(?,O(n^1))