YES

TRS:

  active(eq(0(),0())) -> mark(true())
  active(eq(s(X),s(Y))) -> mark(eq(X,Y))
  active(eq(X,Y)) -> mark(false())
  active(inf(X)) -> mark(cons(X,inf(s(X))))
  active(take(0(),X)) -> mark(nil())
  active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L)))
  active(length(nil())) -> mark(0())
  active(length(cons(X,L))) -> mark(s(length(L)))
  mark(eq(X1,X2)) -> active(eq(X1,X2))
  mark(0()) -> active(0())
  mark(true()) -> active(true())
  mark(s(X)) -> active(s(X))
  mark(false()) -> active(false())
  mark(inf(X)) -> active(inf(mark(X)))
  mark(cons(X1,X2)) -> active(cons(X1,X2))
  mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
  mark(nil()) -> active(nil())
  mark(length(X)) -> active(length(mark(X)))
  eq(mark(X1),X2) -> eq(X1,X2)
  eq(X1,mark(X2)) -> eq(X1,X2)
  eq(active(X1),X2) -> eq(X1,X2)
  eq(X1,active(X2)) -> eq(X1,X2)
  s(mark(X)) -> s(X)
  s(active(X)) -> s(X)
  inf(mark(X)) -> inf(X)
  inf(active(X)) -> inf(X)
  cons(mark(X1),X2) -> cons(X1,X2)
  cons(X1,mark(X2)) -> cons(X1,X2)
  cons(active(X1),X2) -> cons(X1,X2)
  cons(X1,active(X2)) -> cons(X1,X2)
  take(mark(X1),X2) -> take(X1,X2)
  take(X1,mark(X2)) -> take(X1,X2)
  take(active(X1),X2) -> take(X1,X2)
  take(X1,active(X2)) -> take(X1,X2)
  length(mark(X)) -> length(X)
  length(active(X)) -> length(X)

linear polynomial interpretations on N:

  active_A(x1) = x1
  active#_A(x1) = x1
  eq_A(x1,x2) = 4
  eq#_A(x1,x2) = 7
  0_A = 4
  0#_A = 0
  mark_A(x1) = x1
  mark#_A(x1) = x1 + 2
  true_A = 4
  true#_A = 1
  s_A(x1) = 4
  s#_A(x1) = 1
  false_A = 1
  false#_A = 0
  inf_A(x1) = x1 + 9
  inf#_A(x1) = 3
  cons_A(x1,x2) = x1 + 4
  cons#_A(x1,x2) = x1 + 8
  take_A(x1,x2) = x1 + x2 + 1
  take#_A(x1,x2) = 0
  nil_A = 1
  nil#_A = 4
  length_A(x1) = x1 + 6
  length#_A(x1) = 7

precedence:

  0 = cons > active > eq = nil = length > mark > true = false = inf = take > s