YES

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

Proof:
 Church Rosser Transformation Processor (critical pair closing system, Thm 2.11):
  +(x129,p(x128)) -> p(+(x129,x128))
  s(p(x132)) -> x132
  -(x134,p(x133)) -> s(-(x134,x133))
  p(s(x137)) -> x137
  +(x139,s(x138)) -> s(+(x139,x138))
  -(x144,s(x143)) -> p(-(x144,x143))
  critical peaks: joinable
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [s](x0) = x0 + 2,
    
    [-](x0, x1) = 4x0 + 2x1 + 2,
    
    [p](x0) = x0 + 1,
    
    [+](x0, x1) = x0 + x1 + 3
   orientation:
    +(x129,p(x128)) = x128 + x129 + 4 >= x128 + x129 + 4 = p(+(x129,x128))
    
    s(p(x132)) = x132 + 3 >= x132 = x132
    
    -(x134,p(x133)) = 2x133 + 4x134 + 4 >= 2x133 + 4x134 + 4 = s(-(x134,x133))
    
    p(s(x137)) = x137 + 3 >= x137 = x137
    
    +(x139,s(x138)) = x138 + x139 + 5 >= x138 + x139 + 5 = s(+(x139,x138))
    
    -(x144,s(x143)) = 2x143 + 4x144 + 6 >= 2x143 + 4x144 + 3 = p(-(x144,x143))
   problem:
    +(x129,p(x128)) -> p(+(x129,x128))
    -(x134,p(x133)) -> s(-(x134,x133))
    +(x139,s(x138)) -> s(+(x139,x138))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [s](x0) = x0,
     
     [-](x0, x1) = 2x0 + x1 + 2,
     
     [p](x0) = x0 + 4,
     
     [+](x0, x1) = x0 + 4x1 + 4
    orientation:
     +(x129,p(x128)) = 4x128 + x129 + 20 >= 4x128 + x129 + 8 = p(+(x129,x128))
     
     -(x134,p(x133)) = x133 + 2x134 + 6 >= x133 + 2x134 + 2 = s(-(x134,x133))
     
     +(x139,s(x138)) = 4x138 + x139 + 4 >= 4x138 + x139 + 4 = s(+(x139,x138))
    problem:
     +(x139,s(x138)) -> s(+(x139,x138))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [s](x0) = x0 + 1,
      
      [+](x0, x1) = x0 + 4x1 + 7
     orientation:
      +(x139,s(x138)) = 4x138 + x139 + 11 >= 4x138 + x139 + 8 = s(+(x139,x138))
     problem:
      
     Qed