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,z)) -> +(+(x,y),z)

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,z)) -> +(+(x,y),z)
  
   T' = (P union S) with
  
   TRS P:+(+(x,y),z) -> +(x,+(y,z))
         +(x,+(y,z)) -> +(+(x,y),z)
  
   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(),x811)) = 
  s(x811), s(+(s(x),x813)) = s(+(x,s(x813))), s(+(p(x),x815)) = p(+(x,s(x815))), 
  p(+(0(),x817)) = p(x817), p(+(s(x),x819)) = s(+(x,p(x819))), p(+(p(x),x821)) = 
  p(+(x,p(x821))), s(y) = s(+(0(),y)), p(y) = p(+(0(),y)), s(+(x825,0())) = 
  s(x825), s(+(x827,s(y))) = s(+(s(x827),y)), s(+(x829,p(y))) = p(+(s(x829),y)), 
  p(+(x831,0())) = p(x831), p(+(x833,s(y))) = s(+(p(x833),y)), p(+(x835,p(y))) = 
  p(+(p(x835),y)), +(x,x837) = s(+(x,p(x837))), +(x838,y) = s(+(p(x838),y)), 
  p(x839) = p(x839), +(x,x840) = p(+(x,s(x840))), +(x841,y) = p(+(s(x841),y)), 
  s(x842) = s(x842)
  
   CP(S,P union P^-1) = 
  +(x,y) = +(x,+(y,0())), +(x,z) = +(x,+(0(),z)), +(x,y) = +(+(x,y),0()), 
  s(+(+(x,y),x895)) = +(x,+(y,s(x895))), +(s(+(x,x897)),z) = +(x,+(s(x897),z)), 
  +(x,s(+(y,x899))) = +(+(x,y),s(x899)), p(+(+(x,y),x901)) = +(x,+(y,p(x901))), 
  +(p(+(x,x903)),z) = +(x,+(p(x903),z)), +(x,p(+(y,x905))) = +(+(x,y),p(x905)), 
  +(y,z) = +(0(),+(y,z)), +(y,z) = +(+(0(),y),z), +(x,z) = +(+(x,0()),z), 
  +(s(+(x909,y)),z) = +(s(x909),+(y,z)), s(+(x911,+(y,z))) = +(+(s(x911),y),z), 
  +(x,s(+(x913,z))) = +(+(x,s(x913)),z), +(p(+(x915,y)),z) = +(p(x915),+(y,z)), 
  p(+(x917,+(y,z))) = +(+(p(x917),y),z), +(x,p(+(x919,z))) = +(+(x,p(x919)),z)
  
   CP(P union P^-1,S) = 
  +(x1017,+(x1018,0())) = +(x1017,x1018), +(x1020,+(x1021,s(y))) = s(+(+(x1020,x1021),y)), 
  +(x1023,+(x1024,p(y))) = p(+(+(x1023,x1024),y)), +(+(0(),x1027),x1028) = 
  +(x1027,x1028), +(+(s(x),x1030),x1031) = s(+(x,+(x1030,x1031))), +(+(p(x),x1033),x1034) = 
  p(+(x,+(x1033,x1034)))
  
  
  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