YES(?,O(n^1))

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

Strict Trs:
  { rec(rec(x)) -> sent(rec(x))
  , rec(sent(x)) -> sent(rec(x))
  , rec(no(x)) -> sent(rec(x))
  , rec(bot()) -> up(sent(bot()))
  , rec(up(x)) -> up(rec(x))
  , sent(up(x)) -> up(sent(x))
  , no(up(x)) -> up(no(x))
  , top(rec(up(x))) -> top(check(rec(x)))
  , top(sent(up(x))) -> top(check(rec(x)))
  , top(no(up(x))) -> top(check(rec(x)))
  , check(rec(x)) -> rec(check(x))
  , check(sent(x)) -> sent(check(x))
  , check(no(x)) -> no(x)
  , check(no(x)) -> no(check(x))
  , check(up(x)) -> up(check(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

Arguments of following rules are not normal-forms:

{ top(rec(up(x))) -> top(check(rec(x)))
, top(sent(up(x))) -> top(check(rec(x)))
, top(no(up(x))) -> top(check(rec(x))) }

All above mentioned rules can be savely removed.

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

Strict Trs:
  { rec(rec(x)) -> sent(rec(x))
  , rec(sent(x)) -> sent(rec(x))
  , rec(no(x)) -> sent(rec(x))
  , rec(bot()) -> up(sent(bot()))
  , rec(up(x)) -> up(rec(x))
  , sent(up(x)) -> up(sent(x))
  , no(up(x)) -> up(no(x))
  , check(rec(x)) -> rec(check(x))
  , check(sent(x)) -> sent(check(x))
  , check(no(x)) -> no(x)
  , check(no(x)) -> no(check(x))
  , check(up(x)) -> up(check(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 1. The enriched problem is
compatible with the following automaton.
{ rec_0(4) -> 1
, rec_0(5) -> 1
, rec_1(4) -> 8
, rec_1(5) -> 8
, sent_0(4) -> 2
, sent_0(5) -> 2
, sent_1(4) -> 10
, sent_1(5) -> 10
, sent_1(9) -> 8
, no_0(4) -> 3
, no_0(5) -> 3
, no_1(4) -> 11
, no_1(5) -> 11
, bot_0() -> 4
, bot_1() -> 9
, up_0(4) -> 5
, up_0(5) -> 5
, up_1(8) -> 1
, up_1(8) -> 8
, up_1(10) -> 2
, up_1(10) -> 10
, up_1(11) -> 3
, up_1(11) -> 11
, up_1(12) -> 7
, up_1(12) -> 12
, top_0(4) -> 6
, top_0(5) -> 6
, check_0(4) -> 7
, check_0(5) -> 7
, check_1(4) -> 12
, check_1(5) -> 12 }

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