Problem:
 +(x,0()) -> x
 +(x,s(y)) -> s(+(x,y))
 +(0(),y) -> y
 +(s(x),y) -> s(+(x,y))
 dbl(x) -> +(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))
  +(0(),y) -> y
  +(s(x),y) -> s(+(x,y))
  dbl(x) -> +(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))
         +(0(),y) -> y
         +(s(x),y) -> s(+(x,y))
         dbl(x) -> +(x,x)
  
  S is left-linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x) = s(+(x,0())), s(+(0(),x256)) = s(x256), s(+(s(x),x258)) = 
  s(+(x,s(x258))), s(y) = s(+(0(),y)), s(+(x261,0())) = s(x261), s(+(x263,s(y))) = 
  s(+(s(x263),y))
  
   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),x313)) = +(x,+(y,s(x313))), 
  +(s(+(x,x315)),z) = +(x,+(s(x315),z)), s(+(x,x317)) = +(s(x317),x), 
  +(x,s(+(y,x319))) = +(+(x,y),s(x319)), s(+(y,x321)) = +(s(x321),y), 
  +(y,z) = +(0(),+(y,z)), y = +(y,0()), +(y,z) = +(+(0(),y),z), +(x,z) = 
  +(+(x,0()),z), x = +(x,0()), +(s(+(x327,y)),z) = +(s(x327),+(y,z)), 
  s(+(x329,y)) = +(y,s(x329)), s(+(x331,+(y,z))) = +(+(s(x331),y),z), 
  +(x,s(+(x333,z))) = +(+(x,s(x333)),z), s(+(x335,x)) = +(x,s(x335))
  
  
   PCP_in(P union P^-1,S) = 
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [+](x0, x1) = 4x0 + x1 + 1,
    
    [dbl](x0) = 5x0 + 1,
    
    [0] = 2,
    
    [s](x0) = x0
   orientation:
    +(x,0()) = 4x + 3 >= x = x
    
    +(x,s(y)) = 4x + y + 1 >= 4x + y + 1 = s(+(x,y))
    
    +(0(),y) = y + 9 >= y = y
    
    +(s(x),y) = 4x + y + 1 >= 4x + y + 1 = s(+(x,y))
    
    dbl(x) = 5x + 1 >= 5x + 1 = +(x,x)
   problem:
    +(x,s(y)) -> s(+(x,y))
    +(s(x),y) -> s(+(x,y))
    dbl(x) -> +(x,x)
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [+](x0, x1) = x0 + x1,
     
     [dbl](x0) = 2x0 + 1,
     
     [s](x0) = x0
    orientation:
     +(x,s(y)) = x + y >= x + y = s(+(x,y))
     
     +(s(x),y) = x + y >= x + y = s(+(x,y))
     
     dbl(x) = 2x + 1 >= 2x = +(x,x)
    problem:
     +(x,s(y)) -> s(+(x,y))
     +(s(x),y) -> s(+(x,y))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [+](x0, x1) = 2x0 + x1 + 1,
      
      [s](x0) = x0 + 1
     orientation:
      +(x,s(y)) = 2x + y + 2 >= 2x + y + 2 = s(+(x,y))
      
      +(s(x),y) = 2x + y + 3 >= 2x + y + 2 = s(+(x,y))
     problem:
      +(x,s(y)) -> s(+(x,y))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [+](x0, x1) = x0 + 4x1,
       
       [s](x0) = x0 + 5
      orientation:
       +(x,s(y)) = x + 4y + 20 >= x + 4y + 5 = s(+(x,y))
      problem:
       
      Qed