Problem:
 +(x,0()) -> x
 +(x,s(y)) -> s(+(x,y))
 +(0(),y) -> y
 +(s(x),y) -> s(+(x,y))
 dbl(0()) -> 0()
 dbl(s(x)) -> s(s(dbl(x)))
 +(+(x,y),z) -> +(x,+(y,z))
 +(x,y) -> +(y,x)
 dbl(+(x,y)) -> +(dbl(x),dbl(y))

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(0()) -> 0()
  dbl(s(x)) -> s(s(dbl(x)))
  dbl(+(x,y)) -> +(dbl(x),dbl(y))
  +(+(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(0()) -> 0()
         dbl(s(x)) -> s(s(dbl(x)))
         dbl(+(x,y)) -> +(dbl(x),dbl(y))
  
  S is linear and P is reversible.
  
   CP(S,S) = 
  0() = 0(), s(x) = s(+(x,0())), dbl(x) = +(dbl(x),dbl(0())), s(+(0(),x539)) = 
  s(x539), s(+(s(x),x541)) = s(+(x,s(x541))), dbl(s(+(x,x543))) = +(dbl(x),dbl(s(x543))), 
  s(y) = s(+(0(),y)), dbl(y) = +(dbl(0()),dbl(y)), s(+(x547,0())) = s(x547), 
  s(+(x549,s(y))) = s(+(s(x549),y)), dbl(s(+(x551,y))) = +(dbl(s(x551)),dbl(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),x613)) = +(x,+(y,s(x613))), 
  +(s(+(x,x615)),z) = +(x,+(s(x615),z)), s(+(x,x617)) = +(s(x617),x), 
  +(x,s(+(y,x619))) = +(+(x,y),s(x619)), s(+(y,x621)) = +(s(x621),y), 
  +(y,z) = +(0(),+(y,z)), y = +(y,0()), +(y,z) = +(+(0(),y),z), +(x,z) = 
  +(+(x,0()),z), x = +(x,0()), +(s(+(x627,y)),z) = +(s(x627),+(y,z)), 
  s(+(x629,y)) = +(y,s(x629)), s(+(x631,+(y,z))) = +(+(s(x631),y),z), 
  +(x,s(+(x633,z))) = +(+(x,s(x633)),z), s(+(x635,x)) = +(x,s(x635))
  
   CP(P union P^-1,S) = 
  +(x777,+(x778,0())) = +(x777,x778), +(x780,+(x781,s(y))) = s(+(+(x780,x781),y)), 
  dbl(+(x783,+(x784,y))) = +(dbl(+(x783,x784)),dbl(y)), +(0(),x) = x, 
  +(s(y),x) = s(+(x,y)), +(y,0()) = y, +(y,s(x)) = s(+(x,y)), dbl(+(y,x)) = 
  +(dbl(x),dbl(y)), +(+(0(),x797),x798) = +(x797,x798), +(+(s(x),x800),x801) = 
  s(+(x,+(x800,x801))), dbl(+(+(x,x803),x804)) = +(dbl(x),dbl(+(x803,x804)))
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [+](x0, x1) = 5x0 + x1 + 3,
    
    [dbl](x0) = x0,
    
    [0] = 2,
    
    [s](x0) = x0
   orientation:
    +(x,0()) = 5x + 5 >= x = x
    
    +(x,s(y)) = 5x + y + 3 >= 5x + y + 3 = s(+(x,y))
    
    +(0(),y) = y + 13 >= y = y
    
    +(s(x),y) = 5x + y + 3 >= 5x + y + 3 = s(+(x,y))
    
    dbl(0()) = 2 >= 2 = 0()
    
    dbl(s(x)) = x >= x = s(s(dbl(x)))
    
    dbl(+(x,y)) = 5x + y + 3 >= 5x + y + 3 = +(dbl(x),dbl(y))
   problem:
    +(x,s(y)) -> s(+(x,y))
    +(s(x),y) -> s(+(x,y))
    dbl(0()) -> 0()
    dbl(s(x)) -> s(s(dbl(x)))
    dbl(+(x,y)) -> +(dbl(x),dbl(y))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [+](x0, x1) = 2x0 + 2x1 + 4,
     
     [dbl](x0) = 4x0 + 4,
     
     [0] = 1,
     
     [s](x0) = x0
    orientation:
     +(x,s(y)) = 2x + 2y + 4 >= 2x + 2y + 4 = s(+(x,y))
     
     +(s(x),y) = 2x + 2y + 4 >= 2x + 2y + 4 = s(+(x,y))
     
     dbl(0()) = 8 >= 1 = 0()
     
     dbl(s(x)) = 4x + 4 >= 4x + 4 = s(s(dbl(x)))
     
     dbl(+(x,y)) = 8x + 8y + 20 >= 8x + 8y + 20 = +(dbl(x),dbl(y))
    problem:
     +(x,s(y)) -> s(+(x,y))
     +(s(x),y) -> s(+(x,y))
     dbl(s(x)) -> s(s(dbl(x)))
     dbl(+(x,y)) -> +(dbl(x),dbl(y))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [+](x0, x1) = x0 + x1 + 4,
      
      [dbl](x0) = 2x0 + 3,
      
      [s](x0) = x0
     orientation:
      +(x,s(y)) = x + y + 4 >= x + y + 4 = s(+(x,y))
      
      +(s(x),y) = x + y + 4 >= x + y + 4 = s(+(x,y))
      
      dbl(s(x)) = 2x + 3 >= 2x + 3 = s(s(dbl(x)))
      
      dbl(+(x,y)) = 2x + 2y + 11 >= 2x + 2y + 10 = +(dbl(x),dbl(y))
     problem:
      +(x,s(y)) -> s(+(x,y))
      +(s(x),y) -> s(+(x,y))
      dbl(s(x)) -> s(s(dbl(x)))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [+](x0, x1) = x0 + x1 + 4,
       
       [dbl](x0) = 3x0 + 4,
       
       [s](x0) = x0 + 1
      orientation:
       +(x,s(y)) = x + y + 5 >= x + y + 5 = s(+(x,y))
       
       +(s(x),y) = x + y + 5 >= x + y + 5 = s(+(x,y))
       
       dbl(s(x)) = 3x + 7 >= 3x + 6 = s(s(dbl(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