Problem:
 +(0(),y) -> y
 +(s(x),y) -> s(+(y,x))
 +(x,y) -> +(y,x)

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(0(),y) -> y
  +(s(x),y) -> s(+(y,x))
  +(x,0()) -> x
  +(x,s(x20)) -> s(+(x20,x))
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(x,y) -> +(y,x)
  
   TRS S:+(0(),y) -> y
         +(s(x),y) -> s(+(y,x))
         +(x,0()) -> x
         +(x,s(x20)) -> s(+(x20,x))
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x20) = s(+(x20,0())), s(+(0(),x172)) = s(x172), s(+(s(x20),x174)) = 
  s(+(x20,s(x174))), s(x) = s(+(0(),x)), s(+(x179,0())) = s(x179), s(+(x181,s(x))) = 
  s(+(s(x181),x))
  
   CP(S,P union P^-1) = 
  y = +(y,0()), x = +(x,0()), s(+(y,x196)) = +(y,s(x196)), s(+(x,x198)) = 
  +(x,s(x198)), x = +(0(),x), y = +(0(),y), s(+(x203,x)) = +(s(x203),x), 
  s(+(x205,y)) = +(s(x205),y)
  
  
   PCP_in(P union P^-1,S) = 
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [s](x0) = x0 + 2,
    
    [+](x0, x1) = 4x0 + 4x1,
    
    [0] = 0
   orientation:
    +(0(),y) = 4y >= y = y
    
    +(s(x),y) = 4x + 4y + 8 >= 4x + 4y + 2 = s(+(y,x))
    
    +(x,0()) = 4x >= x = x
    
    +(x,s(x20)) = 4x + 4x20 + 8 >= 4x + 4x20 + 2 = s(+(x20,x))
   problem:
    +(0(),y) -> y
    +(x,0()) -> x
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [+](x0, x1) = x0 + x1 + 5,
     
     [0] = 0
    orientation:
     +(0(),y) = y + 5 >= y = y
     
     +(x,0()) = x + 5 >= x = x
    problem:
     
    Qed