YES

TRS:

  active(f(f(X))) -> mark(c(f(g(f(X)))))
  active(c(X)) -> mark(d(X))
  active(h(X)) -> mark(c(d(X)))
  active(f(X)) -> f(active(X))
  active(h(X)) -> h(active(X))
  f(mark(X)) -> mark(f(X))
  h(mark(X)) -> mark(h(X))
  proper(f(X)) -> f(proper(X))
  proper(c(X)) -> c(proper(X))
  proper(g(X)) -> g(proper(X))
  proper(d(X)) -> d(proper(X))
  proper(h(X)) -> h(proper(X))
  f(ok(X)) -> ok(f(X))
  c(ok(X)) -> ok(c(X))
  g(ok(X)) -> ok(g(X))
  d(ok(X)) -> ok(d(X))
  h(ok(X)) -> ok(h(X))
  top(mark(X)) -> top(proper(X))
  top(ok(X)) -> top(active(X))

max/plus interpretations on N:

  active_A(x1) = max{1, 17}
  active#_A(x1) = max{46, 37 + x1}
  f_A(x1) = max{5, x1}
  f#_A(x1) = max{33, 43}
  mark_A(x1) = max{3, 17}
  mark#_A(x1) = max{32, 34}
  c_A(x1) = max{5, x1}
  c#_A(x1) = max{27, 27}
  g_A(x1) = max{4, x1}
  g#_A(x1) = max{34, 17}
  d_A(x1) = max{1, x1}
  d#_A(x1) = max{0, 1 + x1}
  h_A(x1) = max{8, x1}
  h#_A(x1) = max{33, 36}
  proper_A(x1) = max{8, 4}
  proper#_A(x1) = max{35, 44}
  ok_A(x1) = max{26, 18 + x1}
  ok#_A(x1) = max{27, 18}
  top_A(x1) = max{0, 1}
  top#_A(x1) = max{29, 27 + x1}

precedence:

  top > active = proper > f > mark = g > c = d = h > ok