*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {check/1,no/1,rec/1,sent/1,top/1} / {bot/0,up/1}
      Obligation:
        Innermost
        basic terms: {check,no,rec,sent,top}/{bot,up}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      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.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {check/1,no/1,rec/1,sent/1,top/1} / {bot/0,up/1}
      Obligation:
        Innermost
        basic terms: {check,no,rec,sent,top}/{bot,up}
    Applied Processor:
      Bounds {initialAutomaton = perSymbol, enrichment = match}
    Proof:
      The problem is match-bounded by 1.
      The enriched problem is compatible with follwoing automaton.
        bot_0() -> 1
        bot_1() -> 11
        check_0(1) -> 2
        check_0(7) -> 2
        check_1(1) -> 8
        check_1(7) -> 8
        no_0(1) -> 3
        no_0(7) -> 3
        no_1(1) -> 9
        no_1(7) -> 9
        rec_0(1) -> 4
        rec_0(7) -> 4
        rec_1(1) -> 10
        rec_1(7) -> 10
        sent_0(1) -> 5
        sent_0(7) -> 5
        sent_1(1) -> 12
        sent_1(7) -> 12
        sent_1(11) -> 10
        top_0(1) -> 6
        top_0(7) -> 6
        up_0(1) -> 7
        up_0(7) -> 7
        up_1(8) -> 2
        up_1(8) -> 8
        up_1(9) -> 3
        up_1(9) -> 9
        up_1(10) -> 4
        up_1(10) -> 10
        up_1(12) -> 5
        up_1(12) -> 12
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {check,no,rec,sent,top}/{bot,up}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).