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

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(0(),y) -> y
  +(s(x),y) -> s(+(x,y))
  inc(x) -> s(x)
  inc(+(x,y)) -> +(inc(x),y)
  +(x,0()) -> x
  +(x,s(x34)) -> s(+(x,x34))
  +(x,y) -> +(y,x)
  
   T' = (P union S) with
  
   TRS P:+(x,y) -> +(y,x)
  
   TRS S:+(0(),y) -> y
         +(s(x),y) -> s(+(x,y))
         inc(x) -> s(x)
         inc(+(x,y)) -> +(inc(x),y)
         +(x,0()) -> x
         +(x,s(x34)) -> s(+(x,x34))
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  inc(y) = +(inc(0()),y), 0() = 0(), s(x34) = s(+(0(),x34)), inc(s(+(x312,y))) = 
  +(inc(s(x312)),y), s(+(x314,0())) = s(x314), s(+(x316,s(x34))) = s(+(s(x316),x34)), 
  s(+(x,y)) = +(inc(x),y), +(inc(x319),x320) = s(+(x319,x320)), s(x) = 
  s(+(x,0())), inc(x) = +(inc(x),0()), s(+(0(),x325)) = s(x325), s(+(s(x),x327)) = 
  s(+(x,s(x327))), inc(s(+(x,x329))) = +(inc(x),s(x329))
  
   CP(S,P union P^-1) = 
  y = +(y,0()), x = +(x,0()), s(+(x350,y)) = +(y,s(x350)), s(+(x352,x)) = 
  +(x,s(x352)), x = +(0(),x), y = +(0(),y), s(+(x,x357)) = +(s(x357),x), 
  s(+(y,x359)) = +(s(x359),y)
  
  
   PCP_in(P union P^-1,S) = 
  inc(+(y,x)) = +(inc(x),y)
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [inc](x0) = x0 + 6,
    
    [s](x0) = x0 + 6,
    
    [+](x0, x1) = x0 + x1 + 2,
    
    [0] = 1
   orientation:
    +(0(),y) = y + 3 >= y = y
    
    +(s(x),y) = x + y + 8 >= x + y + 8 = s(+(x,y))
    
    inc(x) = x + 6 >= x + 6 = s(x)
    
    inc(+(x,y)) = x + y + 8 >= x + y + 8 = +(inc(x),y)
    
    +(x,0()) = x + 3 >= x = x
    
    +(x,s(x34)) = x + x34 + 8 >= x + x34 + 8 = s(+(x,x34))
   problem:
    +(s(x),y) -> s(+(x,y))
    inc(x) -> s(x)
    inc(+(x,y)) -> +(inc(x),y)
    +(x,s(x34)) -> s(+(x,x34))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [inc](x0) = x0 + 5,
     
     [s](x0) = x0 + 5,
     
     [+](x0, x1) = x0 + 6x1 + 1
    orientation:
     +(s(x),y) = x + 6y + 6 >= x + 6y + 6 = s(+(x,y))
     
     inc(x) = x + 5 >= x + 5 = s(x)
     
     inc(+(x,y)) = x + 6y + 6 >= x + 6y + 6 = +(inc(x),y)
     
     +(x,s(x34)) = x + 6x34 + 31 >= x + 6x34 + 6 = s(+(x,x34))
    problem:
     +(s(x),y) -> s(+(x,y))
     inc(x) -> s(x)
     inc(+(x,y)) -> +(inc(x),y)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [inc](x0) = 2x0 + 2,
      
      [s](x0) = x0,
      
      [+](x0, x1) = 2x0 + x1 + 2
     orientation:
      +(s(x),y) = 2x + y + 2 >= 2x + y + 2 = s(+(x,y))
      
      inc(x) = 2x + 2 >= x = s(x)
      
      inc(+(x,y)) = 4x + 2y + 6 >= 4x + y + 6 = +(inc(x),y)
     problem:
      +(s(x),y) -> s(+(x,y))
      inc(+(x,y)) -> +(inc(x),y)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [inc](x0) = 6x0 + 1,
       
       [s](x0) = x0,
       
       [+](x0, x1) = 2x0 + 5x1 + 2
      orientation:
       +(s(x),y) = 2x + 5y + 2 >= 2x + 5y + 2 = s(+(x,y))
       
       inc(+(x,y)) = 12x + 30y + 13 >= 12x + 5y + 4 = +(inc(x),y)
      problem:
       +(s(x),y) -> s(+(x,y))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [s](x0) = x0 + 1,
        
        [+](x0, x1) = 2x0 + 4x1 + 5
       orientation:
        +(s(x),y) = 2x + 4y + 7 >= 2x + 4y + 6 = s(+(x,y))
       problem:
        
       Qed