YES

TRS:

  t(N) -> cs(r(q(N)),nt(ns(N)))
  q(0()) -> 0()
  q(s(X)) -> s(p(q(X),d(X)))
  d(0()) -> 0()
  d(s(X)) -> s(s(d(X)))
  p(0(),X) -> X
  p(X,0()) -> X
  p(s(X),s(Y)) -> s(s(p(X,Y)))
  f(0(),X) -> nil()
  f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
  t(X) -> nt(X)
  s(X) -> ns(X)
  f(X1,X2) -> nf(X1,X2)
  a(nt(X)) -> t(a(X))
  a(ns(X)) -> s(a(X))
  a(nf(X1,X2)) -> f(a(X1),a(X2))
  a(X) -> X

max/plus interpretations on N:

  t_A(x1) = max{1, 7 + x1}
  t#_A(x1) = max{23, 22 + x1}
  cs_A(x1,x2) = max{3, 5, -5 + x2}
  cs#_A(x1,x2) = max{1, 0, -10 + x2}
  r_A(x1) = max{1, 4 + x1}
  r#_A(x1) = max{23, 9 + x1}
  q_A(x1) = max{14, -3}
  q#_A(x1) = max{1, 21}
  nt_A(x1) = max{0, 7 + x1}
  nt#_A(x1) = max{21, 22}
  ns_A(x1) = max{2, x1}
  ns#_A(x1) = max{10, 1}
  0_A = 3
  0#_A = 13
  s_A(x1) = max{2, x1}
  s#_A(x1) = max{11, 3 + x1}
  p_A(x1,x2) = max{7, x1, 1 + x2}
  p#_A(x1,x2) = max{15, 7 + x1, 14 + x2}
  d_A(x1) = max{7, -2}
  d#_A(x1) = max{12, 0}
  f_A(x1,x2) = max{44, 5 + x1, 7 + x2}
  f#_A(x1,x2) = max{24, -2 + x1, 30 + x2}
  nil_A = 8
  nil#_A = 0
  nf_A(x1,x2) = max{44, 5 + x1, 7 + x2}
  nf#_A(x1,x2) = max{30, -1, 24 + x2}
  a_A(x1) = max{0, x1}
  a#_A(x1) = max{2, 24 + x1}

precedence:

  q = f > cs = 0 = p = nf > d = nil = a > t = s > r = nt = ns