Problem:
 +(x,y) -> +(y,x)
 *(+(x,y),z) -> +(*(x,z),*(y,z))
 *(+(y,x),z) -> +(*(x,z),*(y,z))

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  *(+(x,y),z) -> +(*(x,z),*(y,z))
  *(+(y,x),z) -> +(*(x,z),*(y,z))
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(x,y) -> +(y,x)
  
   TRS S:*(+(x,y),z) -> +(*(x,z),*(y,z))
         *(+(y,x),z) -> +(*(x,z),*(y,z))
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  +(*(y,z),*(x,z)) = +(*(x,z),*(y,z))
  
   CP(S,P union P^-1) = 
  
  
   PCP_in(P union P^-1,S) = 
  *(+(y,x),z) = +(*(x,z),*(y,z)), *(+(x,y),z) = +(*(x,z),*(y,z))
  
  
  We have to check termination of S:
  
  LPO Processor:
   precedence:
    * > +
   problem:
    
   Qed