Problem:
 +(x,0()) -> x
 +(x,s(y)) -> s(+(x,y))
 +(0(),y) -> y
 +(s(x),y) -> s(+(x,y))
 *(1(),0()) -> 0()
 *(1(),s(y)) -> s(*(1(),y))
 +(x,y) -> +(y,x)
 +(+(x,y),z) -> +(x,+(y,z))

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  +(0(),y) -> y
  +(s(x),y) -> s(+(x,y))
  *(1(),0()) -> 0()
  *(1(),s(y)) -> s(*(1(),y))
  +(x,y) -> +(y,x)
  +(+(x,y),z) -> +(x,+(y,z))
  
   T' = (P union S) with
  
   TRS P:+(x,y) -> +(y,x)
         +(+(x,y),z) -> +(x,+(y,z))
  
   TRS S:+(x,0()) -> x
         +(x,s(y)) -> s(+(x,y))
         +(0(),y) -> y
         +(s(x),y) -> s(+(x,y))
         *(1(),0()) -> 0()
         *(1(),s(y)) -> s(*(1(),y))
  
  S is linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x) = s(+(x,0())), s(+(0(),x454)) = s(x454), s(+(s(x),x456)) = 
  s(+(x,s(x456))), s(y) = s(+(0(),y)), s(+(x459,0())) = s(x459), s(+(x461,s(y))) = 
  s(+(s(x461),y))
  
   CP(S,P union P^-1) = 
  x = +(0(),x), +(x,y) = +(x,+(y,0())), +(x,z) = +(x,+(0(),z)), y = +(0(),y), 
  +(x,y) = +(+(x,y),0()), s(+(x,x511)) = +(s(x511),x), s(+(+(x,y),x513)) = 
  +(x,+(y,s(x513))), +(s(+(x,x515)),z) = +(x,+(s(x515),z)), s(+(y,x517)) = 
  +(s(x517),y), +(x,s(+(y,x519))) = +(+(x,y),s(x519)), y = +(y,0()), 
  +(y,z) = +(0(),+(y,z)), x = +(x,0()), +(y,z) = +(+(0(),y),z), +(x,z) = 
  +(+(x,0()),z), s(+(x525,y)) = +(y,s(x525)), +(s(+(x527,y)),z) = +(s(x527),+(y,z)), 
  s(+(x529,x)) = +(x,s(x529)), s(+(x531,+(y,z))) = +(+(s(x531),y),z), 
  +(x,s(+(x533,z))) = +(+(x,s(x533)),z)
  
   CP(P union P^-1,S) = 
  +(0(),x) = x, +(s(y),x) = s(+(x,y)), +(y,0()) = y, +(y,s(x)) = s(+(x,y)), 
  +(x683,+(x684,0())) = +(x683,x684), +(x686,+(x687,s(y))) = s(+(+(x686,x687),y)), 
  +(+(0(),x698),x699) = +(x698,x699), +(+(s(x),x701),x702) = s(+(x,+(x701,x702)))
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [*](x0, x1) = x0 + x1,
    
    [+](x0, x1) = x0 + x1,
    
    [1] = 0,
    
    [0] = 1,
    
    [s](x0) = x0 + 2
   orientation:
    +(x,0()) = x + 1 >= x = x
    
    +(x,s(y)) = x + y + 2 >= x + y + 2 = s(+(x,y))
    
    +(0(),y) = y + 1 >= y = y
    
    +(s(x),y) = x + y + 2 >= x + y + 2 = s(+(x,y))
    
    *(1(),0()) = 1 >= 1 = 0()
    
    *(1(),s(y)) = y + 2 >= y + 2 = s(*(1(),y))
   problem:
    +(x,s(y)) -> s(+(x,y))
    +(s(x),y) -> s(+(x,y))
    *(1(),0()) -> 0()
    *(1(),s(y)) -> s(*(1(),y))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [*](x0, x1) = 4x0 + x1 + 1,
     
     [+](x0, x1) = x0 + 4x1 + 1,
     
     [1] = 4,
     
     [0] = 4,
     
     [s](x0) = x0 + 3
    orientation:
     +(x,s(y)) = x + 4y + 13 >= x + 4y + 4 = s(+(x,y))
     
     +(s(x),y) = x + 4y + 4 >= x + 4y + 4 = s(+(x,y))
     
     *(1(),0()) = 21 >= 4 = 0()
     
     *(1(),s(y)) = y + 20 >= y + 20 = s(*(1(),y))
    problem:
     +(s(x),y) -> s(+(x,y))
     *(1(),s(y)) -> s(*(1(),y))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [*](x0, x1) = x0 + 5x1 + 4,
      
      [+](x0, x1) = x0 + 4x1,
      
      [1] = 4,
      
      [s](x0) = x0 + 1
     orientation:
      +(s(x),y) = x + 4y + 1 >= x + 4y + 1 = s(+(x,y))
      
      *(1(),s(y)) = 5y + 13 >= 5y + 9 = s(*(1(),y))
     problem:
      +(s(x),y) -> s(+(x,y))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [+](x0, x1) = 2x0 + 4x1 + 7,
       
       [s](x0) = x0 + 1
      orientation:
       +(s(x),y) = 2x + 4y + 9 >= 2x + 4y + 8 = s(+(x,y))
      problem:
       
      Qed