Input TRS:
    1: active(eq(0(),0())) -> mark(true())
    2: active(eq(s(X),s(Y))) -> mark(eq(X,Y))
    3: active(eq(X,Y)) -> mark(false())
    4: active(inf(X)) -> mark(cons(X,inf(s(X))))
    5: active(take(0(),X)) -> mark(nil())
    6: active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L)))
    7: active(length(nil())) -> mark(0())
    8: active(length(cons(X,L))) -> mark(s(length(L)))
    9: active(inf(X)) -> inf(active(X))
    10: active(take(X1,X2)) -> take(active(X1),X2)
    11: active(take(X1,X2)) -> take(X1,active(X2))
    12: active(length(X)) -> length(active(X))
    13: inf(mark(X)) -> mark(inf(X))
    14: take(mark(X1),X2) -> mark(take(X1,X2))
    15: take(X1,mark(X2)) -> mark(take(X1,X2))
    16: length(mark(X)) -> mark(length(X))
    17: proper(eq(X1,X2)) -> eq(proper(X1),proper(X2))
    18: proper(0()) -> ok(0())
    19: proper(true()) -> ok(true())
    20: proper(s(X)) -> s(proper(X))
    21: proper(false()) -> ok(false())
    22: proper(inf(X)) -> inf(proper(X))
    23: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    24: proper(take(X1,X2)) -> take(proper(X1),proper(X2))
    25: proper(nil()) -> ok(nil())
    26: proper(length(X)) -> length(proper(X))
    27: eq(ok(X1),ok(X2)) -> ok(eq(X1,X2))
    28: s(ok(X)) -> ok(s(X))
    29: inf(ok(X)) -> ok(inf(X))
    30: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    31: take(ok(X1),ok(X2)) -> ok(take(X1,X2))
    32: length(ok(X)) -> ok(length(X))
    33: top(mark(X)) -> top(proper(X))
    34: top(ok(X)) -> top(active(X))
Number of strict rules: 34
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #active(eq(s(X),s(Y))) -> #eq(X,Y)
   #2: #inf(ok(X)) -> #inf(X)
   #3: #active(take(s(X),cons(Y,L))) -> #cons(Y,take(X,L))
   #4: #active(take(s(X),cons(Y,L))) -> #take(X,L)
   #5: #inf(mark(X)) -> #inf(X)
   #6: #active(inf(X)) -> #inf(active(X))
   #7: #active(inf(X)) -> #active(X)
   #8: #active(take(X1,X2)) -> #take(X1,active(X2))
   #9: #active(take(X1,X2)) -> #active(X2)
   #10: #proper(take(X1,X2)) -> #take(proper(X1),proper(X2))
   #11: #proper(take(X1,X2)) -> #proper(X1)
   #12: #proper(take(X1,X2)) -> #proper(X2)
   #13: #proper(cons(X1,X2)) -> #cons(proper(X1),proper(X2))
   #14: #proper(cons(X1,X2)) -> #proper(X1)
   #15: #proper(cons(X1,X2)) -> #proper(X2)
   #16: #active(length(X)) -> #length(active(X))
   #17: #active(length(X)) -> #active(X)
   #18: #take(ok(X1),ok(X2)) -> #take(X1,X2)
   #19: #take(mark(X1),X2) -> #take(X1,X2)
   #20: #cons(ok(X1),ok(X2)) -> #cons(X1,X2)
   #21: #proper(s(X)) -> #s(proper(X))
   #22: #proper(s(X)) -> #proper(X)
   #23: #active(take(X1,X2)) -> #take(active(X1),X2)
   #24: #active(take(X1,X2)) -> #active(X1)
   #25: #top(mark(X)) -> #top(proper(X))
   #26: #top(mark(X)) -> #proper(X)
   #27: #s(ok(X)) -> #s(X)
   #28: #proper(inf(X)) -> #inf(proper(X))
   #29: #proper(inf(X)) -> #proper(X)
   #30: #top(ok(X)) -> #top(active(X))
   #31: #top(ok(X)) -> #active(X)
   #32: #eq(ok(X1),ok(X2)) -> #eq(X1,X2)
   #33: #proper(eq(X1,X2)) -> #eq(proper(X1),proper(X2))
   #34: #proper(eq(X1,X2)) -> #proper(X1)
   #35: #proper(eq(X1,X2)) -> #proper(X2)
   #36: #length(ok(X)) -> #length(X)
   #37: #proper(length(X)) -> #length(proper(X))
   #38: #proper(length(X)) -> #proper(X)
   #39: #length(mark(X)) -> #length(X)
   #40: #active(length(cons(X,L))) -> #s(length(L))
   #41: #active(length(cons(X,L))) -> #length(L)
   #42: #take(X1,mark(X2)) -> #take(X1,X2)
   #43: #active(inf(X)) -> #cons(X,inf(s(X)))
   #44: #active(inf(X)) -> #inf(s(X))
   #45: #active(inf(X)) -> #s(X)
Number of SCCs: 9, DPs: 25, edges: 121
	SCC { #27 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: 0
   top(x1)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: x1
   nil()	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   inf(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #27
Number of SCCs: 8, DPs: 24, edges: 120
	SCC { #20 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: x2
     s(x1)	weight: 0
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: 0
   top(x1)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   inf(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #20
Number of SCCs: 7, DPs: 23, edges: 119
	SCC { #32 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: 0
   top(x1)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: x2
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   inf(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #32
Number of SCCs: 6, DPs: 22, edges: 118
	SCC { #2 #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: 0
   top(x1)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 #inf(x1)	weight: x1
active(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   inf(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #2 #5
Number of SCCs: 5, DPs: 20, edges: 114
	SCC { #36 #39 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: 0
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: 0
   top(x1)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: 0
  cons(x1,x2)	weight: 0
#active(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: x1
   inf(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #36 #39
Number of SCCs: 4, DPs: 18, edges: 110
	SCC { #25 #30 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... succeeded.
#cons(x1,x2)	weight: 0; 0
     s(x1)	weight: (/ 1 8) + x1_1; x1_1
#take(x1,x2)	weight: 0; 0
  take(x1,x2)	weight: (/ 1 8); (/ 1 4) + x2_2 + x1_2
   top(x1)	weight: 0; 0
    eq(x1,x2)	weight: (/ 1 8) + x2_1 + x1_1; (/ 1 4) + x2_1 + x1_1
 false()	weight: (/ 1 8); 0
 #top(x1)	weight: x1_2; 0
  true()	weight: (/ 1 8); (/ 1 8)
  #eq(x1,x2)	weight: 0; 0
proper(x1)	weight: x1_1; x1_2
    ok(x1)	weight: x1_1; x1_2
     0()	weight: 0; 0
   #s(x1)	weight: 0; 0
   nil()	weight: (/ 1 8); (/ 1 8)
  mark(x1)	weight: (/ 1 8); (/ 1 8) + x1_2
#proper(x1)	weight: 0; 0
 #inf(x1)	weight: 0; 0
active(x1)	weight: (/ 1 8) + x1_2; x1_2
  cons(x1,x2)	weight: (/ 1 8) + x1_2; 0
#active(x1)	weight: 0; 0
length(x1)	weight: (/ 1 8); (/ 1 4) + x1_2
#length(x1)	weight: 0; 0
   inf(x1)	weight: (/ 1 8); (/ 1 8) + x1_2
    Usable rules: { 1..32 }
    Removed DPs: #25
Number of SCCs: 4, DPs: 17, edges: 107
	SCC { #30 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: (/ 1 4) + x1
   top(x1)	weight: 0
    eq(x1,x2)	weight: (/ 1 4) + x2
 false()	weight: 0
 #top(x1)	weight: x1
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2) + x1
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 1 4)
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 4) + x1
#active(x1)	weight: 0
length(x1)	weight: (/ 1 4) + x1
#length(x1)	weight: 0
   inf(x1)	weight: (/ 1 4) + x1
    Usable rules: { 1..16 27..32 }
    Removed DPs: #30
Number of SCCs: 3, DPs: 16, edges: 106
	SCC { #18 #19 #42 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 8) + x1
#take(x1,x2)	weight: x1 + x2
  take(x1,x2)	weight: (/ 1 8) + x1
   top(x1)	weight: 0
    eq(x1,x2)	weight: (/ 1 8) + x2
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 4) + x1
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 1 8) + x1
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: (/ 1 8) + x1
  cons(x1,x2)	weight: (/ 1 8) + x1
#active(x1)	weight: 0
length(x1)	weight: (/ 1 8) + x1
#length(x1)	weight: 0
   inf(x1)	weight: (/ 1 8) + x1
    Usable rules: { }
    Removed DPs: #18 #19 #42
Number of SCCs: 3, DPs: 13, edges: 97
	SCC { #7 #9 #17 #24 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: (/ 1 4) + x1 + x2
   top(x1)	weight: 0
    eq(x1,x2)	weight: (/ 1 4) + x2
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 4) + x1
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 1 4) + x1
#proper(x1)	weight: 0
 #inf(x1)	weight: 0
active(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 4) + x1
#active(x1)	weight: x1
length(x1)	weight: (/ 1 4) + x1
#length(x1)	weight: 0
   inf(x1)	weight: (/ 1 4) + x1
    Usable rules: { }
    Removed DPs: #7 #9 #17 #24
Number of SCCs: 2, DPs: 9, edges: 81
	SCC { #11 #12 #14 #15 #22 #29 #34 #35 #38 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#cons(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 8) + x1
#take(x1,x2)	weight: 0
  take(x1,x2)	weight: (/ 1 8) + x1 + x2
   top(x1)	weight: 0
    eq(x1,x2)	weight: (/ 1 8) + x1 + x2
 false()	weight: 0
 #top(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 8) + x1
    ok(x1)	weight: (/ 1 8) + x1
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 1 8) + x1
#proper(x1)	weight: x1
 #inf(x1)	weight: 0
active(x1)	weight: (/ 3 8) + x1
  cons(x1,x2)	weight: (/ 1 8) + x1 + x2
#active(x1)	weight: 0
length(x1)	weight: (/ 1 8) + x1
#length(x1)	weight: 0
   inf(x1)	weight: (/ 1 8) + x1
    Usable rules: { }
    Removed DPs: #11 #12 #14 #15 #22 #29 #34 #35 #38
Number of SCCs: 1, DPs: 0, edges: 0
YES