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(system(r,W(),RIo(x),Ww(),p))) ->= top(check(system(RAo(r),W(),x,Ww(),p)))
      8: top(ok(system(r,W(),RIn(x),Ww(),p))) ->= top(check(system(RAn(r),W(),x,Ww(),p)))
      9: top(ok(system(R(),W(),Rw(),WIn(y),p))) ->= top(check(system(R(),WAn(W()),Rw(),y,p)))
      10: top(ok(system(R(),W(),Rw(),WIo(y),p))) ->= top(check(system(R(),WAo(W()),Rw(),y,p)))
      11: top(ok(system(r,W(),RIo(x),y,PR()))) ->= top(check(system(RAo(r),W(),x,y,PW())))
      12: top(ok(system(r,W(),RIn(x),y,PR()))) ->= top(check(system(RAn(r),W(),x,y,PW())))
      13: top(ok(system(R(),W(),x,WIo(y),PW()))) ->= top(check(system(R(),WAo(W()),x,y,PR())))
      14: top(ok(system(R(),W(),x,WIn(y),PW()))) ->= top(check(system(R(),WAn(W()),x,y,PR())))
      15: check(RIo(x)) ->= ok(RIo(x))
      16: check(RAo(x)) ->= RAo(check(x))
      17: check(RAn(x)) ->= RAn(check(x))
      18: check(WAo(x)) ->= WAo(check(x))
      19: check(WAn(x)) ->= WAn(check(x))
      20: check(RIo(x)) ->= RIo(check(x))
      21: check(RIn(x)) ->= RIn(check(x))
      22: check(WIo(x)) ->= WIo(check(x))
      23: check(WIn(x)) ->= WIn(check(x))
      24: check(system(v1,v2,v3,v4,v5)) ->= system(check(v1),v2,v3,v4,v5)
      25: check(system(v1,v2,v3,v4,v5)) ->= system(v1,check(v2),v3,v4,v5)
      26: check(system(v1,v2,v3,v4,v5)) ->= system(v1,v2,check(v3),v4,v5)
      27: check(system(v1,v2,v3,v4,v5)) ->= system(v1,v2,v3,check(v4),v5)
      28: check(system(v1,v2,v3,v4,v5)) ->= system(v1,v2,v3,v4,check(v5))
      29: RAo(ok(x)) ->= ok(RAo(x))
      30: RAn(ok(x)) ->= ok(RAn(x))
      31: WAo(ok(x)) ->= ok(WAo(x))
      32: WAn(ok(x)) ->= ok(WAn(x))
      33: RIo(ok(x)) ->= ok(RIo(x))
      34: RIn(ok(x)) ->= ok(RIn(x))
      35: WIo(ok(x)) ->= ok(WIo(x))
      36: WIn(ok(x)) ->= ok(WIn(x))
      37: system(ok(v1),v2,v3,v4,v5) ->= ok(system(v1,v2,v3,v4,v5))
      38: system(v1,ok(v2),v3,v4,v5) ->= ok(system(v1,v2,v3,v4,v5))
      39: system(v1,v2,ok(v3),v4,v5) ->= ok(system(v1,v2,v3,v4,v5))
      40: system(v1,v2,v3,ok(v4),v5) ->= ok(system(v1,v2,v3,v4,v5))
      41: system(v1,v2,v3,v4,ok(v5)) ->= ok(system(v1,v2,v3,v4,v5))
Number of strict rules: 4
Direct POLO(bPol) ... removes: 1 3 10 7 11 13
      RAo	w: x1 + 1
      RIo	w: x1 + 18750
      W 	w: 3566
      WIn	w: x1
      top	w: x1
      PR	w: 6812
      WAo	w: 2 * x1 + 120
      system	w: 2 * x1 + x2 + x3 + x4 + x5 + 19621
      Ww	w: 12063
      WIo	w: x1 + 3687
      check	w: x1
      ok	w: x1
      Rw	w: 25192
      R 	w: 26051
      WAn	w: x1
      RIn	w: x1
      PW	w: 6812
      RAn	w: x1
Number of strict rules: 2
Direct POLO(bPol) ... failed.
Uncurrying ... failed.
Dependency Pairs:
     #1: #top(ok(system(R(),W(),Rw(),WIn(y),p))) ->? #top(check(system(R(),WAn(W()),Rw(),y,p)))
     #2: #top(ok(system(R(),W(),Rw(),WIn(y),p))) ->? #check(system(R(),WAn(W()),Rw(),y,p))
     #3: #top(ok(system(R(),W(),Rw(),WIn(y),p))) ->? #WAn(W())
     #4: #check(system(v1,v2,v3,v4,v5)) ->? #check(v1)
     #5: #check(WIn(x)) ->? #check(x)
     #6: #top(ok(system(r,W(),RIn(x),y,PR()))) ->? #top(check(system(RAn(r),W(),x,y,PW())))
     #7: #top(ok(system(r,W(),RIn(x),y,PR()))) ->? #check(system(RAn(r),W(),x,y,PW()))
     #8: #top(ok(system(r,W(),RIn(x),y,PR()))) ->? #RAn(r)
     #9: #top(ok(system(R(),W(),x,WIn(y),PW()))) ->? #top(check(system(R(),WAn(W()),x,y,PR())))
     #10: #top(ok(system(R(),W(),x,WIn(y),PW()))) ->? #check(system(R(),WAn(W()),x,y,PR()))
     #11: #top(ok(system(R(),W(),x,WIn(y),PW()))) ->? #WAn(W())
     #12: #RAn(ok(x)) ->? #RAn(x)
     #13: #check(system(v1,v2,v3,v4,v5)) ->? #check(v2)
     #14: #check(RIo(x)) ->? #check(x)
     #15: #check(system(v1,v2,v3,v4,v5)) ->? #check(v5)
     #16: #check(WIo(x)) ->? #check(x)
     #17: #check(system(v1,v2,v3,v4,v5)) ->? #check(v4)
     #18: #check(RAn(x)) ->? #RAn(check(x))
     #19: #check(RAn(x)) ->? #check(x)
     #20: #check(WAn(x)) ->? #WAn(check(x))
     #21: #check(WAn(x)) ->? #check(x)
     #22: #WAn(ok(x)) ->? #WAn(x)
     #23: #check(system(v1,v2,v3,v4,v5)) ->? #check(v3)
     #24: #check(RIn(x)) ->? #check(x)
     #25: #check(RAo(x)) ->? #check(x)
     #26: #top(ok(system(r,W(),RIn(x),Ww(),p))) ->? #top(check(system(RAn(r),W(),x,Ww(),p)))
     #27: #top(ok(system(r,W(),RIn(x),Ww(),p))) ->? #check(system(RAn(r),W(),x,Ww(),p))
     #28: #top(ok(system(r,W(),RIn(x),Ww(),p))) ->? #RAn(r)
     #29: #check(WAo(x)) ->? #check(x)
Number of SCCs: 4, DPs: 19
  SCC { #12 }
POLO(Sum)... succeeded.
      RAo	w: x1 + 1
      RIo	w: x1 + 50736
      W 	w: 1
      #RAn	w: x1
      WIn	w: x1
      top	w: x1
      PR	w: 1
      #check	w: 0
      #top	w: 0
      WAo	w: x1 + 11049
      system	w: x1 + x2 + x3 + x4 + x5 + 1
      #WAn	w: 0
      Ww	w: 31328
      WIo	w: x1 + 1
      check	w: x1 + 1
      ok	w: x1 + 1
      Rw	w: 2831
      R 	w: 0
      WAn	w: x1
      RIn	w: x1
      PW	w: 1
      RAn	w: x1
    Removed DPs: #12
Number of SCCs: 3, DPs: 18
  SCC { #22 }
POLO(Sum)... succeeded.
      RAo	w: x1 + 1
      RIo	w: x1 + 35495
      W 	w: 1
      #RAn	w: 0
      WIn	w: x1
      top	w: x1
      PR	w: 1
      #check	w: 0
      #top	w: 0
      WAo	w: x1 + 11049
      system	w: x1 + x2 + x3 + x4 + x5 + 1
      #WAn	w: x1
      Ww	w: 1
      WIo	w: x1 + 1
      check	w: x1 + 1
      ok	w: x1 + 1
      Rw	w: 1
      R 	w: 0
      WAn	w: x1
      RIn	w: x1
      PW	w: 1
      RAn	w: x1
    Removed DPs: #22
Number of SCCs: 2, DPs: 17
  SCC { #1 #6 #9 #26 }
POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... QWPOpS(mSum)... Mat2b... failed.
Finding a loop...  failed.