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

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  +(x,p(y)) -> p(+(x,y))
  +(0(),y) -> y
  +(s(x),y) -> s(+(x,y))
  +(p(x),y) -> p(+(x,y))
  s(p(x)) -> x
  p(s(x)) -> 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:+(x,0()) -> x
         +(x,s(y)) -> s(+(x,y))
         +(x,p(y)) -> p(+(x,y))
         +(0(),y) -> y
         +(s(x),y) -> s(+(x,y))
         +(p(x),y) -> p(+(x,y))
         s(p(x)) -> x
         p(s(x)) -> x
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x) = s(+(x,0())), p(x) = p(+(x,0())), s(+(0(),x629)) = 
  s(x629), s(+(s(x),x631)) = s(+(x,s(x631))), s(+(p(x),x633)) = p(+(x,s(x633))), 
  p(+(0(),x635)) = p(x635), p(+(s(x),x637)) = s(+(x,p(x637))), p(+(p(x),x639)) = 
  p(+(x,p(x639))), s(y) = s(+(0(),y)), p(y) = p(+(0(),y)), s(+(x643,0())) = 
  s(x643), s(+(x645,s(y))) = s(+(s(x645),y)), s(+(x647,p(y))) = p(+(s(x647),y)), 
  p(+(x649,0())) = p(x649), p(+(x651,s(y))) = s(+(p(x651),y)), p(+(x653,p(y))) = 
  p(+(p(x653),y)), +(x,x655) = s(+(x,p(x655))), +(x656,y) = s(+(p(x656),y)), 
  p(x657) = p(x657), +(x,x658) = p(+(x,s(x658))), +(x659,y) = p(+(s(x659),y)), 
  s(x660) = s(x660)
  
   CP(S,P union P^-1) = 
  +(x,y) = +(x,+(y,0())), +(x,z) = +(x,+(0(),z)), x = +(0(),x), +(x,y) = 
  +(+(x,y),0()), y = +(0(),y), s(+(+(x,y),x739)) = +(x,+(y,s(x739))), 
  +(s(+(x,x741)),z) = +(x,+(s(x741),z)), s(+(x,x743)) = +(s(x743),x), 
  +(x,s(+(y,x745))) = +(+(x,y),s(x745)), s(+(y,x747)) = +(s(x747),y), 
  p(+(+(x,y),x749)) = +(x,+(y,p(x749))), +(p(+(x,x751)),z) = +(x,+(p(x751),z)), 
  p(+(x,x753)) = +(p(x753),x), +(x,p(+(y,x755))) = +(+(x,y),p(x755)), 
  p(+(y,x757)) = +(p(x757),y), +(y,z) = +(0(),+(y,z)), y = +(y,0()), 
  +(y,z) = +(+(0(),y),z), +(x,z) = +(+(x,0()),z), x = +(x,0()), +(s(+(x763,y)),z) = 
  +(s(x763),+(y,z)), s(+(x765,y)) = +(y,s(x765)), s(+(x767,+(y,z))) = 
  +(+(s(x767),y),z), +(x,s(+(x769,z))) = +(+(x,s(x769)),z), s(+(x771,x)) = 
  +(x,s(x771)), +(p(+(x773,y)),z) = +(p(x773),+(y,z)), p(+(x775,y)) = 
  +(y,p(x775)), p(+(x777,+(y,z))) = +(+(p(x777),y),z), +(x,p(+(x779,z))) = 
  +(+(x,p(x779)),z), p(+(x781,x)) = +(x,p(x781))
  
  
   PCP_in(P union P^-1,S) = 
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [p](x0) = x0,
    
    [s](x0) = x0 + 1,
    
    [+](x0, x1) = x0 + x1,
    
    [0] = 1
   orientation:
    +(x,0()) = x + 1 >= x = x
    
    +(x,s(y)) = x + y + 1 >= x + y + 1 = s(+(x,y))
    
    +(x,p(y)) = x + y >= x + y = p(+(x,y))
    
    +(0(),y) = y + 1 >= y = y
    
    +(s(x),y) = x + y + 1 >= x + y + 1 = s(+(x,y))
    
    +(p(x),y) = x + y >= x + y = p(+(x,y))
    
    s(p(x)) = x + 1 >= x = x
    
    p(s(x)) = x + 1 >= x = x
   problem:
    +(x,s(y)) -> s(+(x,y))
    +(x,p(y)) -> p(+(x,y))
    +(s(x),y) -> s(+(x,y))
    +(p(x),y) -> p(+(x,y))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [p](x0) = x0 + 3,
     
     [s](x0) = x0,
     
     [+](x0, x1) = 4x0 + x1 + 1
    orientation:
     +(x,s(y)) = 4x + y + 1 >= 4x + y + 1 = s(+(x,y))
     
     +(x,p(y)) = 4x + y + 4 >= 4x + y + 4 = p(+(x,y))
     
     +(s(x),y) = 4x + y + 1 >= 4x + y + 1 = s(+(x,y))
     
     +(p(x),y) = 4x + y + 13 >= 4x + y + 4 = p(+(x,y))
    problem:
     +(x,s(y)) -> s(+(x,y))
     +(x,p(y)) -> p(+(x,y))
     +(s(x),y) -> s(+(x,y))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [p](x0) = x0,
      
      [s](x0) = x0 + 5,
      
      [+](x0, x1) = 2x0 + x1 + 3
     orientation:
      +(x,s(y)) = 2x + y + 8 >= 2x + y + 8 = s(+(x,y))
      
      +(x,p(y)) = 2x + y + 3 >= 2x + y + 3 = p(+(x,y))
      
      +(s(x),y) = 2x + y + 13 >= 2x + y + 8 = s(+(x,y))
     problem:
      +(x,s(y)) -> s(+(x,y))
      +(x,p(y)) -> p(+(x,y))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [p](x0) = x0 + 2,
       
       [s](x0) = x0 + 1,
       
       [+](x0, x1) = x0 + 2x1 + 6
      orientation:
       +(x,s(y)) = x + 2y + 8 >= x + 2y + 7 = s(+(x,y))
       
       +(x,p(y)) = x + 2y + 10 >= x + 2y + 8 = p(+(x,y))
      problem:
       
      Qed