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

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  *(x,0()) -> 0()
  *(x,s(y)) -> +(*(x,y),x)
  +(0(),y) -> y
  +(s(x35),y) -> s(+(x35,y))
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(x,y) -> +(y,x)
  
   TRS S:+(x,0()) -> x
         +(x,s(y)) -> s(+(x,y))
         *(x,0()) -> 0()
         *(x,s(y)) -> +(*(x,y),x)
         +(0(),y) -> y
         +(s(x35),y) -> s(+(x35,y))
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x35) = s(+(x35,0())), s(+(0(),x321)) = s(x321), s(+(s(x35),x323)) = 
  s(+(x35,s(x323))), s(y) = s(+(0(),y)), s(+(x326,0())) = s(x326), s(+(x328,s(y))) = 
  s(+(s(x328),y))
  
   CP(S,P union P^-1) = 
  x = +(0(),x), y = +(0(),y), s(+(x,x351)) = +(s(x351),x), s(+(y,x353)) = 
  +(s(x353),y), y = +(y,0()), x = +(x,0()), s(+(x356,y)) = +(y,s(x356)), 
  s(+(x358,x)) = +(x,s(x358))
  
  
   PCP_in(P union P^-1,S) = 
  
  
  We have to check termination of S:
  
  LPO Processor:
   precedence:
    * > + > s ~ 0
   problem:
    
   Qed