YES

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)))

linear polynomial interpretations on N:

  purge_A(x1) = x1 + 1
  purge#_A(x1) = x1
  nil_A = 1
  nil#_A = 2
  ._A(x1,x2) = x1 + x2 + 4
  .#_A(x1,x2) = 3
  remove_A(x1,x2) = x2
  remove#_A(x1,x2) = x1 + x2 + 2
  if_A(x1,x2,x3) = 0
  if#_A(x1,x2,x3) = 0
  =_A(x1,x2) = x1 + 1
  =#_A(x1,x2) = 0

precedence:

  . > remove = if > nil = = > purge