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

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(0(),y) -> y
  +(x,0()) -> x
  +(s(x),y) -> s(+(x,y))
  +(x,s(x109)) -> s(+(x,x109))
  +(x,s(y)) -> +(s(y),x)
  +(x,+(y,z)) -> +(+(x,y),z)
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(x,s(y)) -> +(s(y),x)
         +(x,+(y,z)) -> +(+(x,y),z)
         +(x,y) -> +(y,x)
  
   TRS S:+(0(),y) -> y
         +(x,0()) -> x
         +(s(x),y) -> s(+(x,y))
         +(x,s(x109)) -> s(+(x,x109))
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x109) = s(+(0(),x109)), s(x) = s(+(x,0())), s(+(x379,0())) = 
  s(x379), s(+(x381,s(x109))) = s(+(s(x381),x109)), s(+(0(),x384)) = 
  s(x384), s(+(s(x),x386)) = s(+(x,s(x386)))
  
   CP(S,P union P^-1) = 
  s(y) = +(s(y),0()), +(y,z) = +(+(0(),y),z), +(x,z) = +(+(x,0()),z), 
  y = +(y,0()), +(y,z) = +(0(),+(y,z)), x = +(x,0()), +(x,y) = +(+(x,y),0()), 
  x = +(0(),x), s(y) = +(0(),s(y)), +(x,y) = +(x,+(y,0())), +(x,z) = 
  +(x,+(0(),z)), y = +(0(),y), s(+(x459,s(y))) = +(s(y),s(x459)), s(+(x461,+(y,z))) = 
  +(+(s(x461),y),z), +(x,s(+(x463,z))) = +(+(x,s(x463)),z), s(+(x465,y)) = 
  +(y,s(x465)), s(+(y,x)) = +(x,s(y)), +(s(+(x469,y)),z) = +(s(x469),+(y,z)), 
  s(+(x471,x)) = +(x,s(x471)), s(+(x,y)) = +(s(y),x), +(x,s(+(y,x476))) = 
  +(+(x,y),s(x476)), s(+(x,x478)) = +(s(x478),x), s(+(s(y),x480)) = +(s(x480),s(y)), 
  s(+(+(x,y),x482)) = +(x,+(y,s(x482))), +(s(+(x,x484)),z) = +(x,+(s(x484),z)), 
  s(+(y,x486)) = +(s(x486),y)
  
  
   PCP_in(P union P^-1,S) = 
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [+](x0, x1) = 2x0 + x1 + 2,
    
    [0] = 2,
    
    [s](x0) = x0 + 2
   orientation:
    +(0(),y) = y + 6 >= y = y
    
    +(x,0()) = 2x + 4 >= x = x
    
    +(s(x),y) = 2x + y + 6 >= 2x + y + 4 = s(+(x,y))
    
    +(x,s(x109)) = 2x + x109 + 4 >= 2x + x109 + 4 = s(+(x,x109))
   problem:
    +(x,s(x109)) -> s(+(x,x109))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [+](x0, x1) = x0 + 4x1,
     
     [s](x0) = x0 + 5
    orientation:
     +(x,s(x109)) = x + 4x109 + 20 >= x + 4x109 + 5 = s(+(x,x109))
    problem:
     
    Qed