YES
Time: 0.006254
TRS:
 {       purge nil() -> nil(),
       purge .(x, y) -> .(x, purge remove(x, y)),
    remove(x, nil()) -> nil(),
  remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))}
 DP:
  DP:
   {     purge# .(x, y) -> purge# remove(x, y),
         purge# .(x, y) -> remove#(x, y),
    remove#(x, .(y, z)) -> remove#(x, z)}
  TRS:
  {       purge nil() -> nil(),
        purge .(x, y) -> .(x, purge remove(x, y)),
     remove(x, nil()) -> nil(),
   remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))}
  UR:
   {  remove(x, nil()) -> nil(),
    remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z))),
               a(w, v) -> w,
               a(w, v) -> v}
   EDG:
    {(remove#(x, .(y, z)) -> remove#(x, z), remove#(x, .(y, z)) -> remove#(x, z))
     (purge# .(x, y) -> remove#(x, y), remove#(x, .(y, z)) -> remove#(x, z))}
    STATUS:
     arrows: 0.777778
     SCCS (1):
      Scc:
       {remove#(x, .(y, z)) -> remove#(x, z)}
      
      
      SCC (1):
       Strict:
        {remove#(x, .(y, z)) -> remove#(x, z)}
       Weak:
       {       purge nil() -> nil(),
             purge .(x, y) -> .(x, purge remove(x, y)),
          remove(x, nil()) -> nil(),
        remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))}
       POLY:
        Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
        Interpretation:
         [if](x0, x1, x2) = x0 + 1,
         
         [.](x0, x1) = x0 + 1,
         
         [remove](x0, x1) = 1,
         
         [=](x0, x1) = 0,
         
         [purge](x0) = x0 + 1,
         
         [nil] = 1,
         
         [remove#](x0, x1) = x0
        Strict:
         remove#(x, .(y, z)) -> remove#(x, z)
         1 + 0x + 0y + 1z >= 0 + 0x + 1z
        Weak:
         remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(x, z)))
         1 + 0x + 0y + 0z >= 2 + 0x + 0y + 0z
         remove(x, nil()) -> nil()
         1 + 0x >= 1
         purge .(x, y) -> .(x, purge remove(x, y))
         2 + 0x + 1y >= 3 + 0x + 0y
         purge nil() -> nil()
         2 >= 1
       Qed