Problem:
 -(+(x,y)) -> +(-(x),-(y))
 -(-(x)) -> x
 +(+(x,y),z) -> +(x,+(y,z))
 +(x,y) -> +(y,x)

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  -(+(x,y)) -> +(-(x),-(y))
  -(-(x)) -> x
  +(+(x,y),z) -> +(x,+(y,z))
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(+(x,y),z) -> +(x,+(y,z))
         +(x,y) -> +(y,x)
  
   TRS S:-(+(x,y)) -> +(-(x),-(y))
         -(-(x)) -> x
  
  S is linear and P is reversible.
  
   CP(S,S) = 
  -(+(-(x98),-(x99))) = +(x98,x99), -(x100) = -(x100)
  
   CP(S,P union P^-1) = 
  
   CP(P union P^-1,S) = 
  -(+(x159,+(x160,y))) = +(-(+(x159,x160)),-(y)), -(+(y,x)) = +(-(x),-(y)), 
  -(+(+(x,x165),x166)) = +(-(x),-(+(x165,x166)))
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [-](x0) = 2x0 + 1,
    
    [+](x0, x1) = x0 + x1 + 1
   orientation:
    -(+(x,y)) = 2x + 2y + 3 >= 2x + 2y + 3 = +(-(x),-(y))
    
    -(-(x)) = 4x + 3 >= x = x
   problem:
    -(+(x,y)) -> +(-(x),-(y))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [-](x0) = 4x0,
     
     [+](x0, x1) = 2x0 + x1 + 4
    orientation:
     -(+(x,y)) = 8x + 4y + 16 >= 8x + 4y + 4 = +(-(x),-(y))
    problem:
     
    Qed