Input TRS:
    1: a__zeros() -> cons(0(),zeros())
    2: a__U11(tt(),L) -> s(a__length(mark(L)))
    3: a__U21(tt()) -> nil()
    4: a__U31(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
    5: a__and(tt(),X) -> mark(X)
    6: a__isNat(0()) -> tt()
    7: a__isNat(length(V1)) -> a__isNatList(V1)
    8: a__isNat(s(V1)) -> a__isNat(V1)
    9: a__isNatIList(V) -> a__isNatList(V)
    10: a__isNatIList(zeros()) -> tt()
    11: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
    12: a__isNatList(nil()) -> tt()
    13: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
    14: a__isNatList(take(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
    15: a__length(nil()) -> 0()
    16: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
    17: a__take(0(),IL) -> a__U21(a__isNatIList(IL))
    18: a__take(s(M),cons(N,IL)) -> a__U31(a__and(a__isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)
    19: mark(zeros()) -> a__zeros()
    20: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
    21: mark(length(X)) -> a__length(mark(X))
    22: mark(U21(X)) -> a__U21(mark(X))
    23: mark(U31(X1,X2,X3,X4)) -> a__U31(mark(X1),X2,X3,X4)
    24: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
    25: mark(and(X1,X2)) -> a__and(mark(X1),X2)
    26: mark(isNat(X)) -> a__isNat(X)
    27: mark(isNatList(X)) -> a__isNatList(X)
    28: mark(isNatIList(X)) -> a__isNatIList(X)
    29: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    30: mark(0()) -> 0()
    31: mark(tt()) -> tt()
    32: mark(s(X)) -> s(mark(X))
    33: mark(nil()) -> nil()
    34: a__zeros() -> zeros()
    35: a__U11(X1,X2) -> U11(X1,X2)
    36: a__length(X) -> length(X)
    37: a__U21(X) -> U21(X)
    38: a__U31(X1,X2,X3,X4) -> U31(X1,X2,X3,X4)
    39: a__take(X1,X2) -> take(X1,X2)
    40: a__and(X1,X2) -> and(X1,X2)
    41: a__isNat(X) -> isNat(X)
    42: a__isNatList(X) -> isNatList(X)
    43: a__isNatIList(X) -> isNatIList(X)
Number of strict rules: 43
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U11(tt(),L) -> #a__length(mark(L))
   #2: #a__U11(tt(),L) -> #mark(L)
   #3: #mark(cons(X1,X2)) -> #mark(X1)
   #4: #a__isNatList(cons(V1,V2)) -> #a__and(a__isNat(V1),isNatList(V2))
   #5: #a__isNatList(cons(V1,V2)) -> #a__isNat(V1)
   #6: #a__isNatIList(V) -> #a__isNatList(V)
   #7: #a__isNatIList(cons(V1,V2)) -> #a__and(a__isNat(V1),isNatIList(V2))
   #8: #a__isNatIList(cons(V1,V2)) -> #a__isNat(V1)
   #9: #mark(take(X1,X2)) -> #a__take(mark(X1),mark(X2))
   #10: #mark(take(X1,X2)) -> #mark(X1)
   #11: #mark(take(X1,X2)) -> #mark(X2)
   #12: #mark(U31(X1,X2,X3,X4)) -> #a__U31(mark(X1),X2,X3,X4)
   #13: #mark(U31(X1,X2,X3,X4)) -> #mark(X1)
   #14: #a__isNatList(take(V1,V2)) -> #a__and(a__isNat(V1),isNatIList(V2))
   #15: #a__isNatList(take(V1,V2)) -> #a__isNat(V1)
   #16: #mark(and(X1,X2)) -> #a__and(mark(X1),X2)
   #17: #mark(and(X1,X2)) -> #mark(X1)
   #18: #mark(U11(X1,X2)) -> #a__U11(mark(X1),X2)
   #19: #mark(U11(X1,X2)) -> #mark(X1)
   #20: #a__isNat(length(V1)) -> #a__isNatList(V1)
   #21: #a__and(tt(),X) -> #mark(X)
   #22: #mark(isNatIList(X)) -> #a__isNatIList(X)
   #23: #mark(U21(X)) -> #a__U21(mark(X))
   #24: #mark(U21(X)) -> #mark(X)
   #25: #mark(isNatList(X)) -> #a__isNatList(X)
   #26: #a__take(0(),IL) -> #a__U21(a__isNatIList(IL))
   #27: #a__take(0(),IL) -> #a__isNatIList(IL)
   #28: #mark(s(X)) -> #mark(X)
   #29: #mark(zeros()) -> #a__zeros()
   #30: #mark(isNat(X)) -> #a__isNat(X)
   #31: #mark(length(X)) -> #a__length(mark(X))
   #32: #mark(length(X)) -> #mark(X)
   #33: #a__length(cons(N,L)) -> #a__U11(a__and(a__isNatList(L),isNat(N)),L)
   #34: #a__length(cons(N,L)) -> #a__and(a__isNatList(L),isNat(N))
   #35: #a__length(cons(N,L)) -> #a__isNatList(L)
   #36: #a__isNat(s(V1)) -> #a__isNat(V1)
   #37: #a__U31(tt(),IL,M,N) -> #mark(N)
   #38: #a__take(s(M),cons(N,IL)) -> #a__U31(a__and(a__isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)
   #39: #a__take(s(M),cons(N,IL)) -> #a__and(a__isNatIList(IL),and(isNat(M),isNat(N)))
   #40: #a__take(s(M),cons(N,IL)) -> #a__isNatIList(IL)
Number of SCCs: 1, DPs: 37, edges: 261
	SCC { #1..22 #24 #25 #27 #28 #30..40 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   U21(x1)	weight: (/ 1 4) + x1
isNatList(x1)	weight: 0
   U11(x1,x2)	weight: x1 + x2
     s(x1)	weight: x1
#a__U31(x1,x2,x3,x4)	weight: (/ 1 4) + x2 + x4
a__isNatIList(x1)	weight: 0
#a__isNat(x1)	weight: 0
  take(x1,x2)	weight: (/ 3 4) + x1 + x2
   and(x1,x2)	weight: x1 + x2
isNatIList(x1)	weight: 0
#a__U11(x1,x2)	weight: x2
 zeros()	weight: 0
a__U31(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x2 + x3 + x4
#a__take(x1,x2)	weight: (/ 1 2) + x2
a__isNatList(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__isNatIList(x1)	weight: 0
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: x2
#a__zeros()	weight: 0
a__U21(x1)	weight: (/ 1 4) + x1
   nil()	weight: 0
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: x1 + x2
#a__length(x1)	weight: x1
 isNat(x1)	weight: 0
   U31(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x2 + x3 + x4
#a__isNatList(x1)	weight: 0
  cons(x1,x2)	weight: x1 + x2
a__take(x1,x2)	weight: (/ 3 4) + x1 + x2
    tt()	weight: 0
a__isNat(x1)	weight: 0
a__and(x1,x2)	weight: x1 + x2
length(x1)	weight: x1
a__zeros()	weight: 0
a__length(x1)	weight: x1
    Usable rules: { 1..43 }
    Removed DPs: #9..13 #24 #27 #37..40
Number of SCCs: 1, DPs: 26, edges: 121
	SCC { #1..8 #14..22 #25 #28 #30..36 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   U21(x1)	weight: (/ 1 4) + x1
isNatList(x1)	weight: 0
   U11(x1,x2)	weight: (/ 1 2) + x1 + x2
     s(x1)	weight: x1
#a__U31(x1,x2,x3,x4)	weight: (/ 1 4)
a__isNatIList(x1)	weight: 0
#a__isNat(x1)	weight: 0
  take(x1,x2)	weight: (/ 3 4) + x1 + x2
   and(x1,x2)	weight: x1 + x2
isNatIList(x1)	weight: 0
#a__U11(x1,x2)	weight: (/ 1 4) + x2
 zeros()	weight: 0
a__U31(x1,x2,x3,x4)	weight: (/ 3 4) + x2 + x3 + x4
#a__take(x1,x2)	weight: (/ 1 2) + x2
a__isNatList(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__isNatIList(x1)	weight: 0
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: x2
#a__zeros()	weight: 0
a__U21(x1)	weight: (/ 1 4) + x1
   nil()	weight: 0
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: (/ 1 2) + x1 + x2
#a__length(x1)	weight: (/ 1 4) + x1
 isNat(x1)	weight: 0
   U31(x1,x2,x3,x4)	weight: (/ 3 4) + x2 + x3 + x4
#a__isNatList(x1)	weight: 0
  cons(x1,x2)	weight: x1 + x2
a__take(x1,x2)	weight: (/ 3 4) + x1 + x2
    tt()	weight: 0
a__isNat(x1)	weight: 0
a__and(x1,x2)	weight: x1 + x2
length(x1)	weight: (/ 1 2) + x1
a__zeros()	weight: 0
a__length(x1)	weight: (/ 1 2) + x1
    Usable rules: { 1..43 }
    Removed DPs: #2 #18 #19 #31 #32 #34 #35
Number of SCCs: 2, DPs: 19, edges: 59
	SCC { #1 #33 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... succeeded.
   U21(x1)	weight: max{0, (- (/ 1 8)) + x1_1}; 0
isNatList(x1)	weight: (max (/ 1 4) 0); x1_2
   U11(x1,x2)	weight: max{0, (/ 1 2) + x2_1 + x1_2}; x1_2
     s(x1)	weight: max{0, (/ 1 4) + x1_1}; 0
#a__U31(x1,x2,x3,x4)	weight: 0; 0
a__isNatIList(x1)	weight: (max (/ 1 4) 0); 0
#a__isNat(x1)	weight: 0; 0
  take(x1,x2)	weight: max{0, (/ 1 8) + x1_1}; 0
   and(x1,x2)	weight: max{0, x2_1}; x2_2 + x1_2
isNatIList(x1)	weight: (max (/ 1 4) 0); 0
#a__U11(x1,x2)	weight: max{0, (/ 1 4) + x2_1 + x1_2}; 0
 zeros()	weight: (/ 1 8); (- (/ 1 4))
a__U31(x1,x2,x3,x4)	weight: max{0, (/ 3 8) + x3_1 + x1_2}; 0
#a__take(x1,x2)	weight: 0; 0
a__isNatList(x1)	weight: (max (/ 1 4) 0); x1_2
#a__U21(x1)	weight: 0; 0
#a__isNatIList(x1)	weight: 0; 0
#mark(x1)	weight: 0; 0
     0()	weight: 0; (- (/ 1 8))
#a__and(x1,x2)	weight: 0; 0
#a__zeros()	weight: 0; 0
a__U21(x1)	weight: max{0, (- (/ 1 8)) + x1_1}; 0
   nil()	weight: 0; 0
  mark(x1)	weight: max{0, x1_1}; x1_2
a__U11(x1,x2)	weight: max{0, (/ 1 2) + x2_1 + x1_2}; x1_2
#a__length(x1)	weight: max{0, (/ 1 8) + x1_1}; 0
 isNat(x1)	weight: max{0, x1_1}; 0
   U31(x1,x2,x3,x4)	weight: max{0, (/ 3 8) + x3_1 + x1_2}; 0
#a__isNatList(x1)	weight: 0; 0
  cons(x1,x2)	weight: max{0, (/ 1 4) + x2_1 + x2_2}; x2_2
a__take(x1,x2)	weight: max{0, (/ 1 8) + x1_1}; 0
    tt()	weight: 0; 0
a__isNat(x1)	weight: max{0, x1_1}; 0
a__and(x1,x2)	weight: max{0, x2_1}; x2_2 + x1_2
length(x1)	weight: max{0, (/ 1 4) + x1_1}; 0
a__zeros()	weight: (/ 1 8); (- (/ 1 4))
a__length(x1)	weight: max{0, (/ 1 4) + x1_1}; 0
    Usable rules: { 1..43 }
    Removed DPs: #1 #33
Number of SCCs: 1, DPs: 17, edges: 57
	SCC { #3..8 #14..17 #20..22 #25 #28 #30 #36 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   U21(x1)	weight: (/ 1 8) + x1
isNatList(x1)	weight: (/ 5 8) + x1
   U11(x1,x2)	weight: (/ 1 8)
     s(x1)	weight: (/ 1 16) + x1
#a__U31(x1,x2,x3,x4)	weight: (/ 1 16)
a__isNatIList(x1)	weight: (/ 1 8)
#a__isNat(x1)	weight: (/ 7 16) + x1
  take(x1,x2)	weight: (/ 1 4) + x1 + x2
   and(x1,x2)	weight: (/ 1 8) + x1 + x2
isNatIList(x1)	weight: (/ 11 16) + x1
#a__U11(x1,x2)	weight: (/ 1 16)
 zeros()	weight: 0
a__U31(x1,x2,x3,x4)	weight: (/ 7 16)
#a__take(x1,x2)	weight: (/ 1 8)
a__isNatList(x1)	weight: (/ 3 8) + x1
#a__U21(x1)	weight: 0
#a__isNatIList(x1)	weight: (/ 5 8) + x1
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: (/ 1 16) + x2
#a__zeros()	weight: 0
a__U21(x1)	weight: (/ 1 16) + x1
   nil()	weight: 0
  mark(x1)	weight: (/ 1 16)
a__U11(x1,x2)	weight: (/ 1 16) + x1 + x2
#a__length(x1)	weight: 0
 isNat(x1)	weight: (/ 1 2) + x1
   U31(x1,x2,x3,x4)	weight: (/ 1 2) + x2 + x3 + x4
#a__isNatList(x1)	weight: (/ 9 16) + x1
  cons(x1,x2)	weight: (/ 3 16) + x1 + x2
a__take(x1,x2)	weight: (/ 1 8) + x1 + x2
    tt()	weight: 0
a__isNat(x1)	weight: (/ 1 8) + x1
a__and(x1,x2)	weight: x2
length(x1)	weight: (/ 3 16) + x1
a__zeros()	weight: 0
a__length(x1)	weight: (/ 1 8)
    Usable rules: { }
    Removed DPs: #3..8 #14..17 #20..22 #25 #28 #30 #36
Number of SCCs: 0, DPs: 0, edges: 0
YES