YES(?,O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict Trs:
  { active(zeros()) -> mark(cons(0(), zeros()))
  , active(cons(X1, X2)) -> cons(active(X1), X2)
  , active(and(X1, X2)) -> and(active(X1), X2)
  , active(and(tt(), X)) -> mark(X)
  , active(length(X)) -> length(active(X))
  , active(length(cons(N, L))) -> mark(s(length(L)))
  , active(length(nil())) -> mark(0())
  , active(s(X)) -> s(active(X))
  , active(take(X1, X2)) -> take(X1, active(X2))
  , active(take(X1, X2)) -> take(active(X1), X2)
  , active(take(0(), IL)) -> mark(nil())
  , active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
  , cons(mark(X1), X2) -> mark(cons(X1, X2))
  , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
  , and(mark(X1), X2) -> mark(and(X1, X2))
  , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
  , length(mark(X)) -> mark(length(X))
  , length(ok(X)) -> ok(length(X))
  , s(mark(X)) -> mark(s(X))
  , s(ok(X)) -> ok(s(X))
  , take(X1, mark(X2)) -> mark(take(X1, X2))
  , take(mark(X1), X2) -> mark(take(X1, X2))
  , take(ok(X1), ok(X2)) -> ok(take(X1, X2))
  , proper(zeros()) -> ok(zeros())
  , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
  , proper(0()) -> ok(0())
  , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
  , proper(tt()) -> ok(tt())
  , proper(length(X)) -> length(proper(X))
  , proper(nil()) -> ok(nil())
  , proper(s(X)) -> s(proper(X))
  , proper(take(X1, X2)) -> take(proper(X1), proper(X2))
  , top(mark(X)) -> top(proper(X))
  , top(ok(X)) -> top(active(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 5. The enriched problem is
compatible with the following automaton.
{ active_0(2) -> 1
, active_0(3) -> 1
, active_0(5) -> 1
, active_0(7) -> 1
, active_0(9) -> 1
, active_0(13) -> 1
, active_1(2) -> 23
, active_1(3) -> 23
, active_1(5) -> 23
, active_1(7) -> 23
, active_1(9) -> 23
, active_1(13) -> 23
, active_2(16) -> 24
, active_2(17) -> 24
, active_3(34) -> 30
, active_4(26) -> 36
, active_4(37) -> 38
, active_5(33) -> 39
, zeros_0() -> 2
, zeros_1() -> 17
, zeros_2() -> 27
, zeros_3() -> 35
, mark_0(2) -> 3
, mark_0(3) -> 3
, mark_0(5) -> 3
, mark_0(7) -> 3
, mark_0(9) -> 3
, mark_0(13) -> 3
, mark_1(15) -> 1
, mark_1(15) -> 23
, mark_1(18) -> 4
, mark_1(18) -> 18
, mark_1(19) -> 6
, mark_1(19) -> 19
, mark_1(20) -> 8
, mark_1(20) -> 20
, mark_1(21) -> 10
, mark_1(21) -> 21
, mark_1(22) -> 11
, mark_1(22) -> 22
, mark_2(25) -> 24
, cons_0(2, 2) -> 4
, cons_0(2, 3) -> 4
, cons_0(2, 5) -> 4
, cons_0(2, 7) -> 4
, cons_0(2, 9) -> 4
, cons_0(2, 13) -> 4
, cons_0(3, 2) -> 4
, cons_0(3, 3) -> 4
, cons_0(3, 5) -> 4
, cons_0(3, 7) -> 4
, cons_0(3, 9) -> 4
, cons_0(3, 13) -> 4
, cons_0(5, 2) -> 4
, cons_0(5, 3) -> 4
, cons_0(5, 5) -> 4
, cons_0(5, 7) -> 4
, cons_0(5, 9) -> 4
, cons_0(5, 13) -> 4
, cons_0(7, 2) -> 4
, cons_0(7, 3) -> 4
, cons_0(7, 5) -> 4
, cons_0(7, 7) -> 4
, cons_0(7, 9) -> 4
, cons_0(7, 13) -> 4
, cons_0(9, 2) -> 4
, cons_0(9, 3) -> 4
, cons_0(9, 5) -> 4
, cons_0(9, 7) -> 4
, cons_0(9, 9) -> 4
, cons_0(9, 13) -> 4
, cons_0(13, 2) -> 4
, cons_0(13, 3) -> 4
, cons_0(13, 5) -> 4
, cons_0(13, 7) -> 4
, cons_0(13, 9) -> 4
, cons_0(13, 13) -> 4
, cons_1(2, 2) -> 18
, cons_1(2, 3) -> 18
, cons_1(2, 5) -> 18
, cons_1(2, 7) -> 18
, cons_1(2, 9) -> 18
, cons_1(2, 13) -> 18
, cons_1(3, 2) -> 18
, cons_1(3, 3) -> 18
, cons_1(3, 5) -> 18
, cons_1(3, 7) -> 18
, cons_1(3, 9) -> 18
, cons_1(3, 13) -> 18
, cons_1(5, 2) -> 18
, cons_1(5, 3) -> 18
, cons_1(5, 5) -> 18
, cons_1(5, 7) -> 18
, cons_1(5, 9) -> 18
, cons_1(5, 13) -> 18
, cons_1(7, 2) -> 18
, cons_1(7, 3) -> 18
, cons_1(7, 5) -> 18
, cons_1(7, 7) -> 18
, cons_1(7, 9) -> 18
, cons_1(7, 13) -> 18
, cons_1(9, 2) -> 18
, cons_1(9, 3) -> 18
, cons_1(9, 5) -> 18
, cons_1(9, 7) -> 18
, cons_1(9, 9) -> 18
, cons_1(9, 13) -> 18
, cons_1(13, 2) -> 18
, cons_1(13, 3) -> 18
, cons_1(13, 5) -> 18
, cons_1(13, 7) -> 18
, cons_1(13, 9) -> 18
, cons_1(13, 13) -> 18
, cons_1(16, 17) -> 15
, cons_2(26, 27) -> 25
, cons_2(28, 29) -> 24
, cons_3(26, 27) -> 34
, cons_3(31, 32) -> 30
, cons_4(33, 35) -> 37
, cons_4(36, 27) -> 30
, cons_5(39, 35) -> 38
, 0_0() -> 5
, 0_1() -> 16
, 0_2() -> 26
, 0_3() -> 33
, and_0(2, 2) -> 6
, and_0(2, 3) -> 6
, and_0(2, 5) -> 6
, and_0(2, 7) -> 6
, and_0(2, 9) -> 6
, and_0(2, 13) -> 6
, and_0(3, 2) -> 6
, and_0(3, 3) -> 6
, and_0(3, 5) -> 6
, and_0(3, 7) -> 6
, and_0(3, 9) -> 6
, and_0(3, 13) -> 6
, and_0(5, 2) -> 6
, and_0(5, 3) -> 6
, and_0(5, 5) -> 6
, and_0(5, 7) -> 6
, and_0(5, 9) -> 6
, and_0(5, 13) -> 6
, and_0(7, 2) -> 6
, and_0(7, 3) -> 6
, and_0(7, 5) -> 6
, and_0(7, 7) -> 6
, and_0(7, 9) -> 6
, and_0(7, 13) -> 6
, and_0(9, 2) -> 6
, and_0(9, 3) -> 6
, and_0(9, 5) -> 6
, and_0(9, 7) -> 6
, and_0(9, 9) -> 6
, and_0(9, 13) -> 6
, and_0(13, 2) -> 6
, and_0(13, 3) -> 6
, and_0(13, 5) -> 6
, and_0(13, 7) -> 6
, and_0(13, 9) -> 6
, and_0(13, 13) -> 6
, and_1(2, 2) -> 19
, and_1(2, 3) -> 19
, and_1(2, 5) -> 19
, and_1(2, 7) -> 19
, and_1(2, 9) -> 19
, and_1(2, 13) -> 19
, and_1(3, 2) -> 19
, and_1(3, 3) -> 19
, and_1(3, 5) -> 19
, and_1(3, 7) -> 19
, and_1(3, 9) -> 19
, and_1(3, 13) -> 19
, and_1(5, 2) -> 19
, and_1(5, 3) -> 19
, and_1(5, 5) -> 19
, and_1(5, 7) -> 19
, and_1(5, 9) -> 19
, and_1(5, 13) -> 19
, and_1(7, 2) -> 19
, and_1(7, 3) -> 19
, and_1(7, 5) -> 19
, and_1(7, 7) -> 19
, and_1(7, 9) -> 19
, and_1(7, 13) -> 19
, and_1(9, 2) -> 19
, and_1(9, 3) -> 19
, and_1(9, 5) -> 19
, and_1(9, 7) -> 19
, and_1(9, 9) -> 19
, and_1(9, 13) -> 19
, and_1(13, 2) -> 19
, and_1(13, 3) -> 19
, and_1(13, 5) -> 19
, and_1(13, 7) -> 19
, and_1(13, 9) -> 19
, and_1(13, 13) -> 19
, tt_0() -> 7
, tt_1() -> 16
, tt_2() -> 26
, tt_3() -> 33
, length_0(2) -> 8
, length_0(3) -> 8
, length_0(5) -> 8
, length_0(7) -> 8
, length_0(9) -> 8
, length_0(13) -> 8
, length_1(2) -> 20
, length_1(3) -> 20
, length_1(5) -> 20
, length_1(7) -> 20
, length_1(9) -> 20
, length_1(13) -> 20
, nil_0() -> 9
, nil_1() -> 16
, nil_2() -> 26
, nil_3() -> 33
, s_0(2) -> 10
, s_0(3) -> 10
, s_0(5) -> 10
, s_0(7) -> 10
, s_0(9) -> 10
, s_0(13) -> 10
, s_1(2) -> 21
, s_1(3) -> 21
, s_1(5) -> 21
, s_1(7) -> 21
, s_1(9) -> 21
, s_1(13) -> 21
, take_0(2, 2) -> 11
, take_0(2, 3) -> 11
, take_0(2, 5) -> 11
, take_0(2, 7) -> 11
, take_0(2, 9) -> 11
, take_0(2, 13) -> 11
, take_0(3, 2) -> 11
, take_0(3, 3) -> 11
, take_0(3, 5) -> 11
, take_0(3, 7) -> 11
, take_0(3, 9) -> 11
, take_0(3, 13) -> 11
, take_0(5, 2) -> 11
, take_0(5, 3) -> 11
, take_0(5, 5) -> 11
, take_0(5, 7) -> 11
, take_0(5, 9) -> 11
, take_0(5, 13) -> 11
, take_0(7, 2) -> 11
, take_0(7, 3) -> 11
, take_0(7, 5) -> 11
, take_0(7, 7) -> 11
, take_0(7, 9) -> 11
, take_0(7, 13) -> 11
, take_0(9, 2) -> 11
, take_0(9, 3) -> 11
, take_0(9, 5) -> 11
, take_0(9, 7) -> 11
, take_0(9, 9) -> 11
, take_0(9, 13) -> 11
, take_0(13, 2) -> 11
, take_0(13, 3) -> 11
, take_0(13, 5) -> 11
, take_0(13, 7) -> 11
, take_0(13, 9) -> 11
, take_0(13, 13) -> 11
, take_1(2, 2) -> 22
, take_1(2, 3) -> 22
, take_1(2, 5) -> 22
, take_1(2, 7) -> 22
, take_1(2, 9) -> 22
, take_1(2, 13) -> 22
, take_1(3, 2) -> 22
, take_1(3, 3) -> 22
, take_1(3, 5) -> 22
, take_1(3, 7) -> 22
, take_1(3, 9) -> 22
, take_1(3, 13) -> 22
, take_1(5, 2) -> 22
, take_1(5, 3) -> 22
, take_1(5, 5) -> 22
, take_1(5, 7) -> 22
, take_1(5, 9) -> 22
, take_1(5, 13) -> 22
, take_1(7, 2) -> 22
, take_1(7, 3) -> 22
, take_1(7, 5) -> 22
, take_1(7, 7) -> 22
, take_1(7, 9) -> 22
, take_1(7, 13) -> 22
, take_1(9, 2) -> 22
, take_1(9, 3) -> 22
, take_1(9, 5) -> 22
, take_1(9, 7) -> 22
, take_1(9, 9) -> 22
, take_1(9, 13) -> 22
, take_1(13, 2) -> 22
, take_1(13, 3) -> 22
, take_1(13, 5) -> 22
, take_1(13, 7) -> 22
, take_1(13, 9) -> 22
, take_1(13, 13) -> 22
, proper_0(2) -> 12
, proper_0(3) -> 12
, proper_0(5) -> 12
, proper_0(7) -> 12
, proper_0(9) -> 12
, proper_0(13) -> 12
, proper_1(2) -> 23
, proper_1(3) -> 23
, proper_1(5) -> 23
, proper_1(7) -> 23
, proper_1(9) -> 23
, proper_1(13) -> 23
, proper_2(15) -> 24
, proper_2(16) -> 28
, proper_2(17) -> 29
, proper_3(25) -> 30
, proper_3(26) -> 31
, proper_3(27) -> 32
, ok_0(2) -> 13
, ok_0(3) -> 13
, ok_0(5) -> 13
, ok_0(7) -> 13
, ok_0(9) -> 13
, ok_0(13) -> 13
, ok_1(16) -> 12
, ok_1(16) -> 23
, ok_1(17) -> 12
, ok_1(17) -> 23
, ok_1(18) -> 4
, ok_1(18) -> 18
, ok_1(19) -> 6
, ok_1(19) -> 19
, ok_1(20) -> 8
, ok_1(20) -> 20
, ok_1(21) -> 10
, ok_1(21) -> 21
, ok_1(22) -> 11
, ok_1(22) -> 22
, ok_2(26) -> 28
, ok_2(27) -> 29
, ok_3(33) -> 31
, ok_3(34) -> 24
, ok_3(35) -> 32
, ok_4(37) -> 30
, top_0(2) -> 14
, top_0(3) -> 14
, top_0(5) -> 14
, top_0(7) -> 14
, top_0(9) -> 14
, top_0(13) -> 14
, top_1(23) -> 14
, top_2(24) -> 14
, top_3(30) -> 14
, top_4(38) -> 14 }

Hurray, we answered YES(?,O(n^1))