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

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(0(),y) -> y
  +(x,s(y)) -> s(+(x,y))
  +(x,0()) -> x
  +(s(x21),y) -> s(+(x21,y))
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(x,y) -> +(y,x)
  
   TRS S:+(0(),y) -> y
         +(x,s(y)) -> s(+(x,y))
         +(x,0()) -> x
         +(s(x21),y) -> s(+(x21,y))
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  s(y) = s(+(0(),y)), 0() = 0(), s(+(0(),x173)) = s(x173), s(+(s(x21),x175)) = 
  s(+(x21,s(x175))), s(x21) = s(+(x21,0())), s(+(x178,s(y))) = s(+(s(x178),y)), 
  s(+(x180,0())) = s(x180)
  
   CP(S,P union P^-1) = 
  y = +(y,0()), x = +(x,0()), s(+(x,x197)) = +(s(x197),x), s(+(y,x199)) = 
  +(s(x199),y), x = +(0(),x), y = +(0(),y), s(+(x202,y)) = +(y,s(x202)), 
  s(+(x204,x)) = +(x,s(x204))
  
  
   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) = x0 + 4x1 + 4,
    
    [0] = 2
   orientation:
    +(0(),y) = 4y + 6 >= y = y
    
    +(x,s(y)) = x + 4y + 12 >= x + 4y + 6 = s(+(x,y))
    
    +(x,0()) = x + 12 >= x = x
    
    +(s(x21),y) = x21 + 4y + 6 >= x21 + 4y + 6 = s(+(x21,y))
   problem:
    +(s(x21),y) -> s(+(x21,y))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [s](x0) = x0 + 5,
     
     [+](x0, x1) = 2x0 + x1
    orientation:
     +(s(x21),y) = 2x21 + y + 10 >= 2x21 + y + 5 = s(+(x21,y))
    problem:
     
    Qed