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)))

max/plus interpretations on N:

  eq_A(x1,x2) = max{1, 1 + x1, 2 + x2}
  eq#_A(x1,x2) = max{7, 1, -2 + x2}
  0_A = 1
  0#_A = 9
  true_A = 3
  true#_A = 8
  s_A(x1) = max{12, 10 + x1}
  s#_A(x1) = max{1, -1}
  false_A = 13
  false#_A = 0
  rm_A(x1,x2) = max{6, 12, x2}
  rm#_A(x1,x2) = max{2, 4, -5 + x2}
  nil_A = 13
  nil#_A = 0
  add_A(x1,x2) = max{20, 11 + x1, 8 + x2}
  add#_A(x1,x2) = max{9, 8, 7}
  ifrm_A(x1,x2,x3) = max{13, 7, -1, x3}
  ifrm#_A(x1,x2,x3) = max{5, -6, 15, -5 + x3}
  purge_A(x1) = max{8, 1 + x1}
  purge#_A(x1) = max{17, 10 + x1}

precedence:

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