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

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(0(),y) -> y
  +(x,s(y)) -> s(+(x,y))
  +(y,0()) -> y
  +(s(y),x) -> s(+(y,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:+(0(),y) -> y
         +(x,s(y)) -> s(+(x,y))
         +(y,0()) -> y
         +(s(y),x) -> s(+(y,x))
  
  S is linear and P is reversible.
  
   CP(S,S) = 
  s(y) = s(+(0(),y)), 0() = 0(), s(+(0(),x342)) = s(x342), s(+(s(y),x344)) = 
  s(+(y,s(x344))), s(y) = s(+(y,0())), s(+(x347,s(y))) = s(+(s(x347),y)), 
  s(+(x349,0())) = s(x349)
  
   CP(S,P union P^-1) = 
  +(y,z) = +(0(),+(y,z)), y = +(y,0()), +(y,z) = +(+(0(),y),z), +(x,z) = 
  +(+(x,0()),z), x = +(x,0()), s(+(+(x,y),x393)) = +(x,+(y,s(x393))), 
  +(s(+(x,x395)),z) = +(x,+(s(x395),z)), s(+(x,x397)) = +(s(x397),x), 
  +(x,s(+(y,x399))) = +(+(x,y),s(x399)), s(+(y,x401)) = +(s(x401),y), 
  +(x,y) = +(x,+(y,0())), +(x,z) = +(x,+(0(),z)), x = +(0(),x), +(x,y) = 
  +(+(x,y),0()), y = +(0(),y), +(s(+(x407,y)),z) = +(s(x407),+(y,z)), 
  s(+(x409,y)) = +(y,s(x409)), s(+(x411,+(y,z))) = +(+(s(x411),y),z), 
  +(x,s(+(x413,z))) = +(+(x,s(x413)),z), s(+(x415,x)) = +(x,s(x415))
  
   CP(P union P^-1,S) = 
  +(x497,+(x498,s(y))) = s(+(+(x497,x498),y)), +(x500,+(x501,0())) = 
  +(x500,x501), +(y,0()) = y, +(s(y),x) = s(+(x,y)), +(0(),y) = y, +(x,s(y)) = 
  s(+(y,x)), +(+(0(),x512),x513) = +(x512,x513), +(+(s(y),x515),x516) = 
  s(+(y,+(x515,x516)))
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [+](x0, x1) = 3x0 + 4x1,
    
    [0] = 6,
    
    [s](x0) = x0 + 6
   orientation:
    +(0(),y) = 4y + 18 >= y = y
    
    +(x,s(y)) = 3x + 4y + 24 >= 3x + 4y + 6 = s(+(x,y))
    
    +(y,0()) = 3y + 24 >= y = y
    
    +(s(y),x) = 4x + 3y + 18 >= 4x + 3y + 6 = s(+(y,x))
   problem:
    
   Qed