YES

TRS:

  f(a()) -> f(c(a()))
  f(c(X)) -> X
  f(c(a())) -> f(d(b()))
  f(a()) -> f(d(a()))
  f(d(X)) -> X
  f(c(b())) -> f(d(a()))
  e(g(X)) -> e(X)

max/plus interpretations on N:

  f_A(x1) = max{14, 9 + x1}
  f#_A(x1) = max{2, -2 + x1}
  a_A = 7
  a#_A = 0
  c_A(x1) = max{1, -9 + x1}
  c#_A(x1) = max{5, -10}
  d_A(x1) = max{5, -2 + x1}
  d#_A(x1) = max{1, -11}
  b_A = 1
  b#_A = 4
  e_A(x1) = max{0, 0}
  e#_A(x1) = max{1, x1}
  g_A(x1) = max{1, -1 + x1}
  g#_A(x1) = max{1, x1}

precedence:

  b > f = e = g > c > a > d