Input TRS:
    1: active(nats()) -> mark(cons(0(),incr(nats())))
    2: active(pairs()) -> mark(cons(0(),incr(odds())))
    3: active(odds()) -> mark(incr(pairs()))
    4: active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
    5: active(head(cons(X,XS))) -> mark(X)
    6: active(tail(cons(X,XS))) -> mark(XS)
    7: active(cons(X1,X2)) -> cons(active(X1),X2)
    8: active(incr(X)) -> incr(active(X))
    9: active(s(X)) -> s(active(X))
    10: active(head(X)) -> head(active(X))
    11: active(tail(X)) -> tail(active(X))
    12: cons(mark(X1),X2) -> mark(cons(X1,X2))
    13: incr(mark(X)) -> mark(incr(X))
    14: s(mark(X)) -> mark(s(X))
    15: head(mark(X)) -> mark(head(X))
    16: tail(mark(X)) -> mark(tail(X))
    17: proper(nats()) -> ok(nats())
    18: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    19: proper(0()) -> ok(0())
    20: proper(incr(X)) -> incr(proper(X))
    21: proper(pairs()) -> ok(pairs())
    22: proper(odds()) -> ok(odds())
    23: proper(s(X)) -> s(proper(X))
    24: proper(head(X)) -> head(proper(X))
    25: proper(tail(X)) -> tail(proper(X))
    26: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    27: incr(ok(X)) -> ok(incr(X))
    28: s(ok(X)) -> ok(s(X))
    29: head(ok(X)) -> ok(head(X))
    30: tail(ok(X)) -> ok(tail(X))
    31: top(mark(X)) -> top(proper(X))
    32: top(ok(X)) -> top(active(X))
Number of strict rules: 32
Direct Order(PosReal,>,Poly) ... removes: 5 6
  incr(x1)	weight: 2 * x1
     s(x1)	weight: 2 * x1
   top(x1)	weight: x1
 pairs()	weight: 0
  tail(x1)	weight: (/ 1 4) + x1
proper(x1)	weight: x1
    ok(x1)	weight: x1
     0()	weight: 0
  odds()	weight: 0
  mark(x1)	weight: x1
  nats()	weight: 0
active(x1)	weight: x1
  head(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: x1 + x2
Number of strict rules: 30
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #active(pairs()) -> #cons(0(),incr(odds()))
   #2: #active(pairs()) -> #incr(odds())
   #3: #head(ok(X)) -> #head(X)
   #4: #incr(mark(X)) -> #incr(X)
   #5: #active(s(X)) -> #s(active(X))
   #6: #active(s(X)) -> #active(X)
   #7: #active(tail(X)) -> #tail(active(X))
   #8: #active(tail(X)) -> #active(X)
   #9: #proper(head(X)) -> #head(proper(X))
   #10: #proper(head(X)) -> #proper(X)
   #11: #proper(s(X)) -> #s(proper(X))
   #12: #proper(s(X)) -> #proper(X)
   #13: #cons(mark(X1),X2) -> #cons(X1,X2)
   #14: #top(mark(X)) -> #top(proper(X))
   #15: #top(mark(X)) -> #proper(X)
   #16: #s(mark(X)) -> #s(X)
   #17: #tail(ok(X)) -> #tail(X)
   #18: #proper(tail(X)) -> #tail(proper(X))
   #19: #proper(tail(X)) -> #proper(X)
   #20: #proper(incr(X)) -> #incr(proper(X))
   #21: #proper(incr(X)) -> #proper(X)
   #22: #active(cons(X1,X2)) -> #cons(active(X1),X2)
   #23: #active(cons(X1,X2)) -> #active(X1)
   #24: #active(head(X)) -> #head(active(X))
   #25: #active(head(X)) -> #active(X)
   #26: #s(ok(X)) -> #s(X)
   #27: #incr(ok(X)) -> #incr(X)
   #28: #top(ok(X)) -> #top(active(X))
   #29: #top(ok(X)) -> #active(X)
   #30: #cons(ok(X1),ok(X2)) -> #cons(X1,X2)
   #31: #tail(mark(X)) -> #tail(X)
   #32: #active(odds()) -> #incr(pairs())
   #33: #active(nats()) -> #cons(0(),incr(nats()))
   #34: #active(nats()) -> #incr(nats())
   #35: #active(incr(X)) -> #incr(active(X))
   #36: #active(incr(X)) -> #active(X)
   #37: #head(mark(X)) -> #head(X)
   #38: #active(incr(cons(X,XS))) -> #cons(s(X),incr(XS))
   #39: #active(incr(cons(X,XS))) -> #s(X)
   #40: #active(incr(cons(X,XS))) -> #incr(XS)
   #41: #proper(cons(X1,X2)) -> #cons(proper(X1),proper(X2))
   #42: #proper(cons(X1,X2)) -> #proper(X1)
   #43: #proper(cons(X1,X2)) -> #proper(X2)
Number of SCCs: 8, DPs: 23, edges: 85
	SCC { #4 #27 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: 0
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#incr(x1)	weight: x1
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: 0
  head(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #4 #27
Number of SCCs: 7, DPs: 21, edges: 81
	SCC { #17 #31 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: 0
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: x1
  mark(x1)	weight: (/ 1 2) + x1
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: 0
  head(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #17 #31
Number of SCCs: 6, DPs: 19, edges: 77
	SCC { #16 #26 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: 0
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: x1
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: 0
  head(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #16 #26
Number of SCCs: 5, DPs: 17, edges: 73
	SCC { #3 #37 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: 0
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: x1
  tail(x1)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: 0
  head(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #3 #37
Number of SCCs: 4, DPs: 15, edges: 69
	SCC { #14 #28 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
  incr(x1)	weight: (/ 3 8) + x1
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 8) + x1
   top(x1)	weight: 0
 pairs()	weight: (/ 1 2)
 #top(x1)	weight: x1
#head(x1)	weight: 0
  tail(x1)	weight: (/ 1 8) + x1
proper(x1)	weight: (/ 1 4) + x1
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: (/ 1 8)
   #s(x1)	weight: 0
  odds()	weight: (/ 9 8)
#tail(x1)	weight: 0
  mark(x1)	weight: (/ 3 8) + x1
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: (/ 1 2)
active(x1)	weight: (/ 1 8) + x1
  head(x1)	weight: (/ 1 8) + x1
  cons(x1,x2)	weight: max{0, (/ 1 8) + x1}
#active(x1)	weight: 0
    Usable rules: { 1..4 7..30 }
    Removed DPs: #14 #28
Number of SCCs: 3, DPs: 13, edges: 65
	SCC { #13 #30 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: (/ 1 4)
#cons(x1,x2)	weight: x2
     s(x1)	weight: (/ 1 4)
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: (/ 1 4)
proper(x1)	weight: (/ 1 8)
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: (/ 1 2)
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: (/ 1 8) + x1
  head(x1)	weight: (/ 1 4)
  cons(x1,x2)	weight: (/ 1 8) + x2
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #30
Number of SCCs: 3, DPs: 12, edges: 62
	SCC { #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: (/ 1 2)
#cons(x1,x2)	weight: x1
     s(x1)	weight: (/ 1 2)
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: (/ 1 2)
proper(x1)	weight: (/ 1 4)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: (/ 1 4) + x1
  head(x1)	weight: (/ 1 2)
  cons(x1,x2)	weight: x2
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #13
Number of SCCs: 2, DPs: 11, edges: 61
	SCC { #6 #8 #23 #25 #36 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: (/ 1 4) + x1
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: (/ 1 4) + x1
proper(x1)	weight: 0
    ok(x1)	weight: x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: x1
#incr(x1)	weight: 0
#proper(x1)	weight: 0
  nats()	weight: 0
active(x1)	weight: x1
  head(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 4) + x1
#active(x1)	weight: x1
    Usable rules: { }
    Removed DPs: #6 #8 #23 #25 #36
Number of SCCs: 2, DPs: 6, edges: 36
	SCC { #10 #12 #19 #21 #42 #43 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  incr(x1)	weight: (/ 1 4) + x1
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
   top(x1)	weight: 0
 pairs()	weight: 0
 #top(x1)	weight: 0
#head(x1)	weight: 0
  tail(x1)	weight: (/ 1 4) + x1
proper(x1)	weight: (/ 1 4)
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  odds()	weight: 0
#tail(x1)	weight: 0
  mark(x1)	weight: x1
#incr(x1)	weight: 0
#proper(x1)	weight: x1
  nats()	weight: 0
active(x1)	weight: x1
  head(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 4) + x1 + x2
#active(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #10 #12 #19 #21 #42 #43
Number of SCCs: 1, DPs: 0, edges: 0
YES