We consider the following Problem:

  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(up(x)) -> up(check(x))
     , check(sent(x)) -> sent(check(x))
     , check(rec(x)) -> rec(check(x))
     , check(no(x)) -> no(check(x))
     , check(no(x)) -> no(x)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Proof:
  Arguments of following rules are not normal-forms:
  {  top(no(up(x))) -> top(check(rec(x)))
   , top(sent(up(x))) -> top(check(rec(x)))
   , top(rec(up(x))) -> top(check(rec(x)))}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    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(up(x)) -> up(check(x))
       , check(sent(x)) -> sent(check(x))
       , check(rec(x)) -> rec(check(x))
       , check(no(x)) -> no(check(x))
       , check(no(x)) -> no(x)}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {check(no(x)) -> no(x)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(rec) = {1}, Uargs(sent) = {1}, Uargs(no) = {1},
        Uargs(up) = {1}, Uargs(top) = {}, Uargs(check) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       rec(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
       sent(x1) = [1 0] x1 + [1]
                  [0 0]      [1]
       no(x1) = [1 0] x1 + [0]
                [0 0]      [1]
       bot() = [0]
               [0]
       up(x1) = [1 0] x1 + [0]
                [0 0]      [1]
       top(x1) = [0 0] x1 + [0]
                 [0 0]      [0]
       check(x1) = [1 0] x1 + [1]
                   [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      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(up(x)) -> up(check(x))
         , check(sent(x)) -> sent(check(x))
         , check(rec(x)) -> rec(check(x))
         , check(no(x)) -> no(check(x))}
      Weak Trs: {check(no(x)) -> no(x)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {rec(no(x)) -> sent(rec(x))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(rec) = {1}, Uargs(sent) = {1}, Uargs(no) = {1},
          Uargs(up) = {1}, Uargs(top) = {}, Uargs(check) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         rec(x1) = [1 0] x1 + [0]
                   [0 0]      [1]
         sent(x1) = [1 0] x1 + [1]
                    [0 0]      [1]
         no(x1) = [1 0] x1 + [2]
                  [0 0]      [0]
         bot() = [0]
                 [0]
         up(x1) = [1 0] x1 + [0]
                  [0 0]      [0]
         top(x1) = [0 0] x1 + [0]
                   [0 0]      [0]
         check(x1) = [1 0] x1 + [0]
                     [0 0]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  rec(rec(x)) -> sent(rec(x))
           , rec(sent(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(up(x)) -> up(check(x))
           , check(sent(x)) -> sent(check(x))
           , check(rec(x)) -> rec(check(x))
           , check(no(x)) -> no(check(x))}
        Weak Trs:
          {  rec(no(x)) -> sent(rec(x))
           , check(no(x)) -> no(x)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {rec(rec(x)) -> sent(rec(x))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(rec) = {1}, Uargs(sent) = {1}, Uargs(no) = {1},
            Uargs(up) = {1}, Uargs(top) = {}, Uargs(check) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           rec(x1) = [1 0] x1 + [1]
                     [0 1]      [1]
           sent(x1) = [1 0] x1 + [0]
                      [0 0]      [1]
           no(x1) = [1 0] x1 + [0]
                    [0 0]      [0]
           bot() = [0]
                   [0]
           up(x1) = [1 0] x1 + [0]
                    [0 1]      [1]
           top(x1) = [0 0] x1 + [0]
                     [0 0]      [0]
           check(x1) = [1 0] x1 + [0]
                       [0 0]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  rec(sent(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(up(x)) -> up(check(x))
             , check(sent(x)) -> sent(check(x))
             , check(rec(x)) -> rec(check(x))
             , check(no(x)) -> no(check(x))}
          Weak Trs:
            {  rec(rec(x)) -> sent(rec(x))
             , rec(no(x)) -> sent(rec(x))
             , check(no(x)) -> no(x)}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {rec(bot()) -> up(sent(bot()))}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(rec) = {1}, Uargs(sent) = {1}, Uargs(no) = {1},
              Uargs(up) = {1}, Uargs(top) = {}, Uargs(check) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             rec(x1) = [1 0] x1 + [1]
                       [0 0]      [1]
             sent(x1) = [1 0] x1 + [0]
                        [0 0]      [1]
             no(x1) = [1 0] x1 + [0]
                      [0 0]      [1]
             bot() = [0]
                     [0]
             up(x1) = [1 0] x1 + [0]
                      [0 1]      [0]
             top(x1) = [0 0] x1 + [0]
                       [0 0]      [0]
             check(x1) = [1 0] x1 + [0]
                         [0 0]      [1]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  rec(sent(x)) -> sent(rec(x))
               , rec(up(x)) -> up(rec(x))
               , sent(up(x)) -> up(sent(x))
               , no(up(x)) -> up(no(x))
               , check(up(x)) -> up(check(x))
               , check(sent(x)) -> sent(check(x))
               , check(rec(x)) -> rec(check(x))
               , check(no(x)) -> no(check(x))}
            Weak Trs:
              {  rec(bot()) -> up(sent(bot()))
               , rec(rec(x)) -> sent(rec(x))
               , rec(no(x)) -> sent(rec(x))
               , check(no(x)) -> no(x)}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            We consider the following Problem:
            
              Strict Trs:
                {  rec(sent(x)) -> sent(rec(x))
                 , rec(up(x)) -> up(rec(x))
                 , sent(up(x)) -> up(sent(x))
                 , no(up(x)) -> up(no(x))
                 , check(up(x)) -> up(check(x))
                 , check(sent(x)) -> sent(check(x))
                 , check(rec(x)) -> rec(check(x))
                 , check(no(x)) -> no(check(x))}
              Weak Trs:
                {  rec(bot()) -> up(sent(bot()))
                 , rec(rec(x)) -> sent(rec(x))
                 , rec(no(x)) -> sent(rec(x))
                 , check(no(x)) -> no(x)}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^1))
            
            Proof:
              The problem is match-bounded by 1.
              The enriched problem is compatible with the following automaton:
              {  rec_0(2) -> 1
               , rec_1(2) -> 3
               , sent_0(2) -> 1
               , sent_1(2) -> 3
               , no_0(2) -> 1
               , no_1(2) -> 3
               , bot_0() -> 2
               , bot_1() -> 2
               , up_0(1) -> 1
               , up_0(2) -> 2
               , up_1(3) -> 1
               , up_1(3) -> 3
               , check_0(2) -> 1
               , check_1(2) -> 3}

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