YES
Input TRS:
      1: T(ok(sys(x,P(d,b),R(b),y))) -> T(check(sys(x,bot(),R(not(b)),y)))
      2: T(ok(sys(S(b,c(d,ds)),bot(),y,z))) ->= T(check(sys(S(b,c(d,ds)),P(d,b),y,z)))
      3: T(ok(sys(S(b,c(d,ds)),x,y,F(b)))) ->= T(check(sys(S(not(b),ds),x,y,bot())))
      4: T(ok(sys(x,y,R(b),bot()))) ->= T(check(sys(x,y,R(b),F(not(b)))))
      5: not(1()) ->= 0()
      6: not(0()) ->= 1()
      7: nils() ->= c(new(),nils())
      8: p(d,b) ->= bot()
      9: f(b) ->= bot()
      10: check(old()) ->= ok(old())
      11: check(f(v1)) ->= f(check(v1))
      12: f(ok(v1)) ->= ok(f(v1))
      13: check(p(v1,v2())) ->= p(v1,check(v2()))
      14: check(p(v1,v2())) ->= p(check(v1),v2())
      15: p(v1,ok(v2())) ->= ok(p(v1,v2()))
      16: p(ok(v1),v2()) ->= ok(p(v1,v2()))
      17: check(R(v1)) ->= R(check(v1))
      18: R(ok(v1)) ->= ok(R(v1))
      19: check(not(v1)) ->= not(check(v1))
      20: not(ok(v1)) ->= ok(not(v1))
      21: check(F(v1)) ->= F(check(v1))
      22: F(ok(v1)) ->= ok(F(v1))
      23: check(P(v1,v2())) ->= P(v1,check(v2()))
      24: check(P(v1,v2())) ->= P(check(v1),v2())
      25: P(v1,ok(v2())) ->= ok(P(v1,v2()))
      26: P(ok(v1),v2()) ->= ok(P(v1,v2()))
      27: check(c(v1,v2())) ->= c(v1,check(v2()))
      28: check(c(v1,v2())) ->= c(check(v1),v2())
      29: c(v1,ok(v2())) ->= ok(c(v1,v2()))
      30: c(ok(v1),v2()) ->= ok(c(v1,v2()))
      31: check(S(v1,v2())) ->= S(v1,check(v2()))
      32: check(S(v1,v2())) ->= S(check(v1),v2())
      33: S(v1,ok(v2())) ->= ok(S(v1,v2()))
      34: S(ok(v1),v2()) ->= ok(S(v1,v2()))
      35: check(sys(v1,v2(),v3(),v4)) ->= sys(v1,v2(),v3(),check(v4))
      36: check(sys(v1,v2(),v3(),v4)) ->= sys(v1,v2(),check(v3()),v4)
      37: check(sys(v1,v2(),v3(),v4)) ->= sys(v1,check(v2()),v3(),v4)
      38: check(sys(v1,v2(),v3(),v4)) ->= sys(check(v1),v2(),v3(),v4)
      39: sys(v1,v2(),v3(),ok(v4)) ->= ok(sys(v1,v2(),v3(),v4))
      40: sys(v1,v2(),ok(v3()),v4) ->= ok(sys(v1,v2(),v3(),v4))
      41: sys(v1,ok(v2()),v3(),v4) ->= ok(sys(v1,v2(),v3(),v4))
      42: sys(ok(v1),v2(),v3(),v4) ->= ok(sys(v1,v2(),v3(),v4))
Number of strict rules: 1
Direct POLO(bPol) ... failed.
Uncurrying ... failed.
Dependency Pairs:
     #1: #T(ok(sys(S(b,c(d,ds)),bot(),y,z))) ->? #T(check(sys(S(b,c(d,ds)),P(d,b),y,z)))
     #2: #T(ok(sys(S(b,c(d,ds)),x,y,F(b)))) ->? #T(check(sys(S(not(b),ds),x,y,bot())))
     #3: #T(ok(sys(x,P(d,b),R(b),y))) -> #T(check(sys(x,bot(),R(not(b)),y)))
     #4: #T(ok(sys(x,y,R(b),bot()))) ->? #T(check(sys(x,y,R(b),F(not(b)))))
Number of SCCs: 1, DPs: 1
  SCC { #2 }
POLO(Sum)... succeeded.
      1 	w: 1
      new	w: 1
      S 	w: 22115
      #T 	w: x1
      v2	w: 2332
      T 	w: 0
      F 	w: 12332
      v3	w: 1
      bot	w: 1
      nils	w: 1
      c 	w: 1
      f 	w: 13506
      check	w: x1 + 1
      sys	w: x2 + x3 + x4
      p 	w: x2 + 7177
      ok	w: x1
      0 	w: 1
      R 	w: 1
      old	w: 38525
      P 	w: 30914
      not	w: 1
    Removed DPs: #2
Number of SCCs: 0, DPs: 0