YES

TRS:

  eq(0(),0()) -> true()
  eq(0(),s(X)) -> false()
  eq(s(X),0()) -> false()
  eq(s(X),s(Y)) -> eq(X,Y)
  rm(N,nil()) -> nil()
  rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
  ifrm(true(),N,add(M,X)) -> rm(N,X)
  ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
  purge(nil()) -> nil()
  purge(add(N,X)) -> add(N,purge(rm(N,X)))

linear polynomial interpretations on N:

  eq_A(x1,x2) = x2 + 1
  eq#_A(x1,x2) = x1 + x2 + 1
  0_A = 1
  0#_A = 1
  true_A = 1
  true#_A = 0
  s_A(x1) = x1 + 1
  s#_A(x1) = 3
  false_A = 1
  false#_A = 2
  rm_A(x1,x2) = x2
  rm#_A(x1,x2) = x1 + x2 + 1
  nil_A = 1
  nil#_A = 0
  add_A(x1,x2) = x1 + x2 + 3
  add#_A(x1,x2) = 2
  ifrm_A(x1,x2,x3) = x3
  ifrm#_A(x1,x2,x3) = x2 + x3
  purge_A(x1) = x1 + 1
  purge#_A(x1) = x1 + 3

precedence:

  ifrm > nil = add > purge > s = rm > eq = 0 > true = false