MAYBE
Input TRS:
      1: RAo(R()) -> R()
      2: RAn(R()) -> R()
      3: WAo(W()) -> W()
      4: WAn(W()) -> W()
      5: Rw() ->= RIn(Rw())
      6: Ww() ->= WIn(Ww())
      7: top(ok(sys_r(read(r,RIo(x)),write(W(),Ww())))) ->= top(check(sys_r(read(RAo(r),x),write(W(),Ww()))))
      8: top(ok(sys_w(read(r,RIo(x)),write(W(),Ww())))) ->= top(check(sys_w(read(RAo(r),x),write(W(),Ww()))))
      9: top(ok(sys_r(read(r,RIn(x)),write(W(),Ww())))) ->= top(check(sys_r(read(RAn(r),x),write(W(),Ww()))))
      10: top(ok(sys_w(read(r,RIn(x)),write(W(),Ww())))) ->= top(check(sys_w(read(RAn(r),x),write(W(),Ww()))))
      11: top(ok(sys_r(read(R(),Rw()),write(W(),WIn(y))))) ->= top(check(sys_r(read(R(),Rw()),write(WAn(W()),y))))
      12: top(ok(sys_w(read(R(),Rw()),write(W(),WIn(y))))) ->= top(check(sys_w(read(R(),Rw()),write(WAn(W()),y))))
      13: top(ok(sys_r(read(R(),Rw()),write(W(),WIo(y))))) ->= top(check(sys_r(read(R(),Rw()),write(WAo(W()),y))))
      14: top(ok(sys_w(read(R(),Rw()),write(W(),WIo(y))))) ->= top(check(sys_w(read(R(),Rw()),write(WAo(W()),y))))
      15: top(ok(sys_r(read(r,RIo(x)),write(W(),y)))) ->= top(check(sys_w(read(RAo(r),x),write(W(),y))))
      16: top(ok(sys_r(read(r,RIn(x)),write(W(),y)))) ->= top(check(sys_w(read(RAn(r),x),write(W(),y))))
      17: top(ok(sys_w(read(R(),x),write(W(),WIo(y))))) ->= top(check(sys_r(read(R(),x),write(WAo(W()),y))))
      18: top(ok(sys_w(read(R(),x),write(W(),WIn(y))))) ->= top(check(sys_r(read(R(),x),write(WAn(W()),y))))
      19: check(RIo(x)) ->= ok(RIo(x))
      20: check(RAo(x)) ->= RAo(check(x))
      21: check(RAn(x)) ->= RAn(check(x))
      22: check(WAo(x)) ->= WAo(check(x))
      23: check(WAn(x)) ->= WAn(check(x))
      24: check(RIo(x)) ->= RIo(check(x))
      25: check(RIn(x)) ->= RIn(check(x))
      26: check(WIo(x)) ->= WIo(check(x))
      27: check(WIn(x)) ->= WIn(check(x))
      28: check(sys_r(x,y)) ->= sys_r(check(x),y)
      29: check(sys_r(x,y)) ->= sys_r(x,check(y))
      30: check(sys_w(x,y)) ->= sys_w(check(x),y)
      31: check(sys_w(x,y)) ->= sys_w(x,check(y))
      32: RAo(ok(x)) ->= ok(RAo(x))
      33: RAn(ok(x)) ->= ok(RAn(x))
      34: WAo(ok(x)) ->= ok(WAo(x))
      35: WAn(ok(x)) ->= ok(WAn(x))
      36: RIn(ok(x)) ->= ok(RIn(x))
      37: WIo(ok(x)) ->= ok(WIo(x))
      38: WIn(ok(x)) ->= ok(WIn(x))
      39: sys_r(ok(x),y) ->= ok(sys_r(x,y))
      40: sys_r(x,ok(y)) ->= ok(sys_r(x,y))
      41: sys_w(ok(x),y) ->= ok(sys_w(x,y))
      42: sys_w(x,ok(y)) ->= ok(sys_w(x,y))
Number of strict rules: 4
Direct POLO(bPol) ... removes: 15 8 1 3 17 7 14 13
      sys_w	w: x1 + 2 * x2 + 31103
      RAo	w: x1 + 3107
      RIo	w: x1 + 1920
      read	w: x1 + 2 * x2 + 16019
      W 	w: 1
      WIn	w: 2 * x1
      top	w: x1
      WAo	w: x1 + 7722
      Ww	w: 0
      WIo	w: x1 + 7723
      check	w: x1
      ok	w: x1
      Rw	w: 14861
      R 	w: 0
      WAn	w: x1
      RIn	w: x1
      write	w: x1 + x2 + 13861
      RAn	w: x1
      sys_r	w: x1 + 2 * x2 + 31103
Number of strict rules: 2
Direct POLO(bPol) ... failed.
Uncurrying ... failed.
Dependency Pairs:
     #1: #check(sys_r(x,y)) ->? #check(y)
     #2: #WAn(ok(x)) ->? #WAn(x)
     #3: #top(ok(sys_r(read(r,RIn(x)),write(W(),Ww())))) ->? #top(check(sys_r(read(RAn(r),x),write(W(),Ww()))))
     #4: #top(ok(sys_r(read(r,RIn(x)),write(W(),Ww())))) ->? #check(sys_r(read(RAn(r),x),write(W(),Ww())))
     #5: #top(ok(sys_r(read(r,RIn(x)),write(W(),Ww())))) ->? #RAn(r)
     #6: #top(ok(sys_r(read(R(),Rw()),write(W(),WIn(y))))) ->? #top(check(sys_r(read(R(),Rw()),write(WAn(W()),y))))
     #7: #top(ok(sys_r(read(R(),Rw()),write(W(),WIn(y))))) ->? #check(sys_r(read(R(),Rw()),write(WAn(W()),y)))
     #8: #top(ok(sys_r(read(R(),Rw()),write(W(),WIn(y))))) ->? #WAn(W())
     #9: #check(RIo(x)) ->? #check(x)
     #10: #check(WAn(x)) ->? #WAn(check(x))
     #11: #check(WAn(x)) ->? #check(x)
     #12: #top(ok(sys_w(read(R(),Rw()),write(W(),WIn(y))))) ->? #top(check(sys_w(read(R(),Rw()),write(WAn(W()),y))))
     #13: #top(ok(sys_w(read(R(),Rw()),write(W(),WIn(y))))) ->? #check(sys_w(read(R(),Rw()),write(WAn(W()),y)))
     #14: #top(ok(sys_w(read(R(),Rw()),write(W(),WIn(y))))) ->? #WAn(W())
     #15: #check(sys_w(x,y)) ->? #check(y)
     #16: #check(sys_w(x,y)) ->? #check(x)
     #17: #check(RIn(x)) ->? #check(x)
     #18: #check(RAo(x)) ->? #check(x)
     #19: #top(ok(sys_w(read(r,RIn(x)),write(W(),Ww())))) ->? #top(check(sys_w(read(RAn(r),x),write(W(),Ww()))))
     #20: #top(ok(sys_w(read(r,RIn(x)),write(W(),Ww())))) ->? #check(sys_w(read(RAn(r),x),write(W(),Ww())))
     #21: #top(ok(sys_w(read(r,RIn(x)),write(W(),Ww())))) ->? #RAn(r)
     #22: #RAn(ok(x)) ->? #RAn(x)
     #23: #check(sys_r(x,y)) ->? #check(x)
     #24: #check(WAo(x)) ->? #check(x)
     #25: #check(WIn(x)) ->? #check(x)
     #26: #check(WIo(x)) ->? #check(x)
     #27: #check(RAn(x)) ->? #RAn(check(x))
     #28: #check(RAn(x)) ->? #check(x)
     #29: #top(ok(sys_r(read(r,RIn(x)),write(W(),y)))) ->? #top(check(sys_w(read(RAn(r),x),write(W(),y))))
     #30: #top(ok(sys_r(read(r,RIn(x)),write(W(),y)))) ->? #check(sys_w(read(RAn(r),x),write(W(),y)))
     #31: #top(ok(sys_r(read(r,RIn(x)),write(W(),y)))) ->? #RAn(r)
     #32: #top(ok(sys_w(read(R(),x),write(W(),WIn(y))))) ->? #top(check(sys_r(read(R(),x),write(WAn(W()),y))))
     #33: #top(ok(sys_w(read(R(),x),write(W(),WIn(y))))) ->? #check(sys_r(read(R(),x),write(WAn(W()),y)))
     #34: #top(ok(sys_w(read(R(),x),write(W(),WIn(y))))) ->? #WAn(W())
Number of SCCs: 4, DPs: 20
  SCC { #2 }
POLO(Sum)... succeeded.
      sys_w	w: x1 + x2 + 1
      RAo	w: x1 + 1
      RIo	w: x1 + 40436
      read	w: 1
      W 	w: 1
      #RAn	w: 0
      WIn	w: x1
      top	w: x1
      #check	w: 0
      #top	w: 0
      WAo	w: x1 + 1
      #WAn	w: x1
      Ww	w: 17888
      WIo	w: x1 + 1
      check	w: x1 + 1
      ok	w: x1 + 1
      Rw	w: 31598
      R 	w: 1
      WAn	w: x1 + 1
      RIn	w: x1
      write	w: 1
      RAn	w: x1 + 20585
      sys_r	w: x1 + x2 + 1
    Removed DPs: #2
Number of SCCs: 3, DPs: 19
  SCC { #22 }
POLO(Sum)... succeeded.
      sys_w	w: x1 + x2 + 1
      RAo	w: x1 + 1
      RIo	w: x1 + 40436
      read	w: 1
      W 	w: 1
      #RAn	w: x1
      WIn	w: x1
      top	w: x1
      #check	w: 0
      #top	w: 0
      WAo	w: x1 + 1
      #WAn	w: 0
      Ww	w: 1
      WIo	w: x1 + 1
      check	w: x1 + 1
      ok	w: x1 + 1
      Rw	w: 1
      R 	w: 1
      WAn	w: x1 + 29824
      RIn	w: x1
      write	w: 1
      RAn	w: x1 + 19826
      sys_r	w: x1 + x2 + 1
    Removed DPs: #22
Number of SCCs: 2, DPs: 18
  SCC { #1 #9 #11 #15..18 #23..26 #28 }
POLO(Sum)... succeeded.
      sys_w	w: x1 + x2 + 1
      RAo	w: x1 + 1
      RIo	w: x1 + 40436
      read	w: 1
      W 	w: 23637
      #RAn	w: 0
      WIn	w: x1
      top	w: x1
      #check	w: x1
      #top	w: 0
      WAo	w: x1 + 1
      #WAn	w: 0
      Ww	w: 1
      WIo	w: x1 + 1
      check	w: x1 + 1
      ok	w: x1 + 1
      Rw	w: 41347
      R 	w: 1
      WAn	w: x1 + 1
      RIn	w: x1
      write	w: 1
      RAn	w: x1 + 19826
      sys_r	w: x1 + x2 + 1
    Removed DPs: #1 #9 #11 #15 #16 #18 #23 #24 #26 #28
Number of SCCs: 2, DPs: 8
  SCC { #17 #25 }
POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... QWPOpS(mSum)... Mat2b... failed.
Finding a loop...  failed.