YES(?,O(n^2))

Problem:
 ++(nil(),y) -> y
 ++(x,nil()) -> x
 ++(.(x,y),z) -> .(x,++(y,z))
 ++(++(x,y),z) -> ++(x,++(y,z))

Proof:
 RT Transformation Processor:
  strict:
   ++(nil(),y) -> y
   ++(x,nil()) -> x
   ++(.(x,y),z) -> .(x,++(y,z))
   ++(++(x,y),z) -> ++(x,++(y,z))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [.](x0, x1) = x0 + x1 + 1,
    
    [++](x0, x1) = x0 + x1,
    
    [nil] = 8
   orientation:
    ++(nil(),y) = y + 8 >= y = y
    
    ++(x,nil()) = x + 8 >= x = x
    
    ++(.(x,y),z) = x + y + z + 1 >= x + y + z + 1 = .(x,++(y,z))
    
    ++(++(x,y),z) = x + y + z >= x + y + z = ++(x,++(y,z))
   problem:
    strict:
     ++(.(x,y),z) -> .(x,++(y,z))
     ++(++(x,y),z) -> ++(x,++(y,z))
    weak:
     ++(nil(),y) -> y
     ++(x,nil()) -> x
   Matrix Interpretation Processor:
    dimension: 2
    interpretation:
                   [1 4]          [0]
     [.](x0, x1) = [0 0]x0 + x1 + [5],
     
                    [1 5]          [6]
     [++](x0, x1) = [0 1]x0 + x1 + [3],
     
             [0]
     [nil] = [0]
    orientation:
                    [1 4]    [1 5]        [31]    [1 4]    [1 5]        [6]               
     ++(.(x,y),z) = [0 0]x + [0 1]y + z + [8 ] >= [0 0]x + [0 1]y + z + [8] = .(x,++(y,z))
     
                     [1  10]    [1 5]        [27]    [1 5]    [1 5]        [12]                
     ++(++(x,y),z) = [0  1 ]x + [0 1]y + z + [6 ] >= [0 1]x + [0 1]y + z + [6 ] = ++(x,++(y,z))
     
                       [6]         
     ++(nil(),y) = y + [3] >= y = y
     
                   [1 5]    [6]         
     ++(x,nil()) = [0 1]x + [3] >= x = x
    problem:
     strict:
      
     weak:
      ++(.(x,y),z) -> .(x,++(y,z))
      ++(++(x,y),z) -> ++(x,++(y,z))
      ++(nil(),y) -> y
      ++(x,nil()) -> x
    Qed