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 linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x) = s(+(x,0())), p(x) = p(+(x,0())), s(+(0(),x751)) = 
  s(x751), s(+(s(x),x753)) = s(+(x,s(x753))), s(+(p(x),x755)) = p(+(x,s(x755))), 
  p(+(0(),x757)) = p(x757), p(+(s(x),x759)) = s(+(x,p(x759))), p(+(p(x),x761)) = 
  p(+(x,p(x761))), s(y) = s(+(0(),y)), p(y) = p(+(0(),y)), s(+(x765,0())) = 
  s(x765), s(+(x767,s(y))) = s(+(s(x767),y)), s(+(x769,p(y))) = p(+(s(x769),y)), 
  p(+(x771,0())) = p(x771), p(+(x773,s(y))) = s(+(p(x773),y)), p(+(x775,p(y))) = 
  p(+(p(x775),y)), +(x,x777) = s(+(x,p(x777))), +(x778,y) = s(+(p(x778),y)), 
  p(x779) = p(x779), +(x,x780) = p(+(x,s(x780))), +(x781,y) = p(+(s(x781),y)), 
  s(x782) = s(x782)
  
   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),x861)) = +(x,+(y,s(x861))), 
  +(s(+(x,x863)),z) = +(x,+(s(x863),z)), s(+(x,x865)) = +(s(x865),x), 
  +(x,s(+(y,x867))) = +(+(x,y),s(x867)), s(+(y,x869)) = +(s(x869),y), 
  p(+(+(x,y),x871)) = +(x,+(y,p(x871))), +(p(+(x,x873)),z) = +(x,+(p(x873),z)), 
  p(+(x,x875)) = +(p(x875),x), +(x,p(+(y,x877))) = +(+(x,y),p(x877)), 
  p(+(y,x879)) = +(p(x879),y), +(y,z) = +(0(),+(y,z)), y = +(y,0()), 
  +(y,z) = +(+(0(),y),z), +(x,z) = +(+(x,0()),z), x = +(x,0()), +(s(+(x885,y)),z) = 
  +(s(x885),+(y,z)), s(+(x887,y)) = +(y,s(x887)), s(+(x889,+(y,z))) = 
  +(+(s(x889),y),z), +(x,s(+(x891,z))) = +(+(x,s(x891)),z), s(+(x893,x)) = 
  +(x,s(x893)), +(p(+(x895,y)),z) = +(p(x895),+(y,z)), p(+(x897,y)) = 
  +(y,p(x897)), p(+(x899,+(y,z))) = +(+(p(x899),y),z), +(x,p(+(x901,z))) = 
  +(+(x,p(x901)),z), p(+(x903,x)) = +(x,p(x903))
  
   CP(P union P^-1,S) = 
  +(x1065,+(x1066,0())) = +(x1065,x1066), +(x1068,+(x1069,s(y))) = s(+(+(x1068,x1069),y)), 
  +(x1071,+(x1072,p(y))) = p(+(+(x1071,x1072),y)), +(0(),x) = x, +(s(y),x) = 
  s(+(x,y)), +(p(y),x) = p(+(x,y)), +(y,0()) = y, +(y,s(x)) = s(+(x,y)), 
  +(y,p(x)) = p(+(x,y)), +(+(0(),x1087),x1088) = +(x1087,x1088), +(+(s(x),x1090),x1091) = 
  s(+(x,+(x1090,x1091))), +(+(p(x),x1093),x1094) = p(+(x,+(x1093,x1094)))
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [+](x0, x1) = x0 + x1,
    
    [p](x0) = x0 + 6,
    
    [0] = 3,
    
    [s](x0) = x0 + 1
   orientation:
    +(x,0()) = x + 3 >= x = x
    
    +(x,s(y)) = x + y + 1 >= x + y + 1 = s(+(x,y))
    
    +(x,p(y)) = x + y + 6 >= x + y + 6 = p(+(x,y))
    
    +(0(),y) = y + 3 >= y = y
    
    +(s(x),y) = x + y + 1 >= x + y + 1 = s(+(x,y))
    
    +(p(x),y) = x + y + 6 >= x + y + 6 = p(+(x,y))
    
    s(p(x)) = x + 7 >= x = x
    
    p(s(x)) = x + 7 >= 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:
     [+](x0, x1) = 4x0 + x1 + 7,
     
     [p](x0) = x0 + 1,
     
     [s](x0) = x0
    orientation:
     +(x,s(y)) = 4x + y + 7 >= 4x + y + 7 = s(+(x,y))
     
     +(x,p(y)) = 4x + y + 8 >= 4x + y + 8 = p(+(x,y))
     
     +(s(x),y) = 4x + y + 7 >= 4x + y + 7 = s(+(x,y))
     
     +(p(x),y) = 4x + y + 11 >= 4x + y + 8 = 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:
      [+](x0, x1) = 2x0 + 6x1,
      
      [p](x0) = x0,
      
      [s](x0) = x0 + 2
     orientation:
      +(x,s(y)) = 2x + 6y + 12 >= 2x + 6y + 2 = s(+(x,y))
      
      +(x,p(y)) = 2x + 6y >= 2x + 6y = p(+(x,y))
      
      +(s(x),y) = 2x + 6y + 4 >= 2x + 6y + 2 = s(+(x,y))
     problem:
      +(x,p(y)) -> p(+(x,y))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [+](x0, x1) = x0 + 4x1,
       
       [p](x0) = x0 + 5
      orientation:
       +(x,p(y)) = x + 4y + 20 >= x + 4y + 5 = p(+(x,y))
      problem:
       
      Qed