WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            check(no(x)) -> no(x)
            check(no(x)) -> no(check(x))
            check(rec(x)) -> rec(check(x))
            check(sent(x)) -> sent(check(x))
            check(up(x)) -> up(check(x))
            no(up(x)) -> up(no(x))
            rec(bot()) -> up(sent(bot()))
            rec(no(x)) -> sent(rec(x))
            rec(rec(x)) -> sent(rec(x))
            rec(sent(x)) -> sent(rec(x))
            rec(up(x)) -> up(rec(x))
            sent(up(x)) -> up(sent(x))
            top(no(up(x))) -> top(check(rec(x)))
            top(rec(up(x))) -> top(check(rec(x)))
            top(sent(up(x))) -> top(check(rec(x)))
        - Signature:
            {check/1,no/1,rec/1,sent/1,top/1} / {bot/0,up/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {check,no,rec,sent,top} and constructors {bot,up}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          top(no(up(x))) -> top(check(rec(x)))
          top(rec(up(x))) -> top(check(rec(x)))
          top(sent(up(x))) -> top(check(rec(x)))
        All above mentioned rules can be savely removed.
* Step 2: Bounds WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            check(no(x)) -> no(x)
            check(no(x)) -> no(check(x))
            check(rec(x)) -> rec(check(x))
            check(sent(x)) -> sent(check(x))
            check(up(x)) -> up(check(x))
            no(up(x)) -> up(no(x))
            rec(bot()) -> up(sent(bot()))
            rec(no(x)) -> sent(rec(x))
            rec(rec(x)) -> sent(rec(x))
            rec(sent(x)) -> sent(rec(x))
            rec(up(x)) -> up(rec(x))
            sent(up(x)) -> up(sent(x))
        - Signature:
            {check/1,no/1,rec/1,sent/1,top/1} / {bot/0,up/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {check,no,rec,sent,top} and constructors {bot,up}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          bot_0() -> 2
          bot_1() -> 4
          check_0(2) -> 1
          check_1(2) -> 3
          no_0(2) -> 1
          no_1(2) -> 3
          rec_0(2) -> 1
          rec_1(2) -> 3
          sent_0(2) -> 1
          sent_1(2) -> 3
          sent_1(4) -> 3
          top_0(2) -> 1
          up_0(2) -> 2
          up_1(3) -> 1
          up_1(3) -> 3
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            check(no(x)) -> no(x)
            check(no(x)) -> no(check(x))
            check(rec(x)) -> rec(check(x))
            check(sent(x)) -> sent(check(x))
            check(up(x)) -> up(check(x))
            no(up(x)) -> up(no(x))
            rec(bot()) -> up(sent(bot()))
            rec(no(x)) -> sent(rec(x))
            rec(rec(x)) -> sent(rec(x))
            rec(sent(x)) -> sent(rec(x))
            rec(up(x)) -> up(rec(x))
            sent(up(x)) -> up(sent(x))
        - Signature:
            {check/1,no/1,rec/1,sent/1,top/1} / {bot/0,up/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {check,no,rec,sent,top} and constructors {bot,up}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))