YES

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 left-linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x) = s(+(x,0())), p(x) = p(+(x,0())), s(+(0(),x679)) = 
  s(x679), s(+(s(x),x681)) = s(+(x,s(x681))), s(+(p(x),x683)) = p(+(x,s(x683))), 
  p(+(0(),x685)) = p(x685), p(+(s(x),x687)) = s(+(x,p(x687))), p(+(p(x),x689)) = 
  p(+(x,p(x689))), s(y) = s(+(0(),y)), p(y) = p(+(0(),y)), s(+(x693,0())) = 
  s(x693), s(+(x695,s(y))) = s(+(s(x695),y)), s(+(x697,p(y))) = p(+(s(x697),y)), 
  p(+(x699,0())) = p(x699), p(+(x701,s(y))) = s(+(p(x701),y)), p(+(x703,p(y))) = 
  p(+(p(x703),y)), +(x,x705) = s(+(x,p(x705))), +(x706,y) = s(+(p(x706),y)), 
  p(x707) = p(x707), +(x,x708) = p(+(x,s(x708))), +(x709,y) = p(+(s(x709),y)), 
  s(x710) = s(x710)
  
   CP(S,P union P^-1) = 
  +(x,y) = +(x,+(y,0())), +(x,z) = +(x,+(0(),z)), +(x,y) = +(+(x,y),0()), 
  s(+(+(x,y),x763)) = +(x,+(y,s(x763))), +(s(+(x,x765)),z) = +(x,+(s(x765),z)), 
  +(x,s(+(y,x767))) = +(+(x,y),s(x767)), p(+(+(x,y),x769)) = +(x,+(y,p(x769))), 
  +(p(+(x,x771)),z) = +(x,+(p(x771),z)), +(x,p(+(y,x773))) = +(+(x,y),p(x773)), 
  +(y,z) = +(0(),+(y,z)), +(y,z) = +(+(0(),y),z), +(x,z) = +(+(x,0()),z), 
  +(s(+(x777,y)),z) = +(s(x777),+(y,z)), s(+(x779,+(y,z))) = +(+(s(x779),y),z), 
  +(x,s(+(x781,z))) = +(+(x,s(x781)),z), +(p(+(x783,y)),z) = +(p(x783),+(y,z)), 
  p(+(x785,+(y,z))) = +(+(p(x785),y),z), +(x,p(+(x787,z))) = +(+(x,p(x787)),z)
  
  
   PCP_in(P union P^-1,S) = 
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [+](x0, x1) = x0 + x1,
    
    [p](x0) = x0 + 4,
    
    [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 + 4 >= x + y + 4 = 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 + 4 >= x + y + 4 = p(+(x,y))
    
    s(p(x)) = x + 5 >= x = x
    
    p(s(x)) = x + 5 >= 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) = 2x0 + x1,
     
     [p](x0) = x0 + 4,
     
     [s](x0) = x0
    orientation:
     +(x,s(y)) = 2x + y >= 2x + y = s(+(x,y))
     
     +(x,p(y)) = 2x + y + 4 >= 2x + y + 4 = p(+(x,y))
     
     +(s(x),y) = 2x + y >= 2x + y = s(+(x,y))
     
     +(p(x),y) = 2x + y + 8 >= 2x + y + 4 = p(+(x,y))
    problem:
     +(x,s(y)) -> s(+(x,y))
     +(x,p(y)) -> p(+(x,y))
     +(s(x),y) -> s(+(x,y))
    KBO Processor:
     weight function:
      w0 = 1
      w(p) = w(s) = 1
      w(+) = 0
     precedence:
      + > p ~ s
     problem:
      
     Qed