Input TRS:
    1: a__zeros() -> cons(0(),zeros())
    2: a__U11(tt(),L) -> s(a__length(mark(L)))
    3: a__and(tt(),X) -> mark(X)
    4: a__isNat(0()) -> tt()
    5: a__isNat(length(V1)) -> a__isNatList(V1)
    6: a__isNat(s(V1)) -> a__isNat(V1)
    7: a__isNatIList(V) -> a__isNatList(V)
    8: a__isNatIList(zeros()) -> tt()
    9: a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2))
    10: a__isNatList(nil()) -> tt()
    11: a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2))
    12: a__length(nil()) -> 0()
    13: a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L)
    14: mark(zeros()) -> a__zeros()
    15: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
    16: mark(length(X)) -> a__length(mark(X))
    17: mark(and(X1,X2)) -> a__and(mark(X1),X2)
    18: mark(isNat(X)) -> a__isNat(X)
    19: mark(isNatList(X)) -> a__isNatList(X)
    20: mark(isNatIList(X)) -> a__isNatIList(X)
    21: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    22: mark(0()) -> 0()
    23: mark(tt()) -> tt()
    24: mark(s(X)) -> s(mark(X))
    25: mark(nil()) -> nil()
    26: a__zeros() -> zeros()
    27: a__U11(X1,X2) -> U11(X1,X2)
    28: a__length(X) -> length(X)
    29: a__and(X1,X2) -> and(X1,X2)
    30: a__isNat(X) -> isNat(X)
    31: a__isNatList(X) -> isNatList(X)
    32: a__isNatIList(X) -> isNatIList(X)
Number of strict rules: 32
Direct Order(PosReal,>,Poly) ... removes: 8 5 7 12
isNatList(x1)	weight: x1
   U11(x1,x2)	weight: (/ 1 4) + x1 + x2
     s(x1)	weight: x1
a__isNatIList(x1)	weight: (/ 1 4) + 2 * x1
   and(x1,x2)	weight: x1 + x2
isNatIList(x1)	weight: (/ 1 4) + 2 * x1
 zeros()	weight: 0
a__isNatList(x1)	weight: x1
     0()	weight: 0
   nil()	weight: 0
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: (/ 1 4) + x1 + x2
 isNat(x1)	weight: x1
  cons(x1,x2)	weight: x1 + 2 * x2
    tt()	weight: 0
a__isNat(x1)	weight: x1
a__and(x1,x2)	weight: x1 + x2
length(x1)	weight: (/ 1 4) + x1
a__zeros()	weight: 0
a__length(x1)	weight: (/ 1 4) + x1
Number of strict rules: 28
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: #a__isNat(s(V1)) -> #a__isNat(V1)
   #4: #a__length(cons(N,L)) -> #a__U11(a__and(a__isNatList(L),isNat(N)),L)
   #5: #a__length(cons(N,L)) -> #a__and(a__isNatList(L),isNat(N))
   #6: #a__length(cons(N,L)) -> #a__isNatList(L)
   #7: #a__isNatIList(cons(V1,V2)) -> #a__and(a__isNat(V1),isNatIList(V2))
   #8: #a__isNatIList(cons(V1,V2)) -> #a__isNat(V1)
   #9: #a__isNatList(cons(V1,V2)) -> #a__and(a__isNat(V1),isNatList(V2))
   #10: #a__isNatList(cons(V1,V2)) -> #a__isNat(V1)
   #11: #mark(s(X)) -> #mark(X)
   #12: #mark(zeros()) -> #a__zeros()
   #13: #mark(isNatIList(X)) -> #a__isNatIList(X)
   #14: #mark(and(X1,X2)) -> #a__and(mark(X1),X2)
   #15: #mark(and(X1,X2)) -> #mark(X1)
   #16: #mark(isNatList(X)) -> #a__isNatList(X)
   #17: #mark(cons(X1,X2)) -> #mark(X1)
   #18: #mark(length(X)) -> #a__length(mark(X))
   #19: #mark(length(X)) -> #mark(X)
   #20: #a__and(tt(),X) -> #mark(X)
   #21: #mark(U11(X1,X2)) -> #a__U11(mark(X1),X2)
   #22: #mark(U11(X1,X2)) -> #mark(X1)
   #23: #mark(isNat(X)) -> #a__isNat(X)
Number of SCCs: 2, DPs: 19, edges: 88
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatList(x1)	weight: 0
   U11(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
a__isNatIList(x1)	weight: 0
#a__isNat(x1)	weight: x1
   and(x1,x2)	weight: 0
isNatIList(x1)	weight: 0
#a__U11(x1,x2)	weight: 0
 zeros()	weight: 0
a__isNatList(x1)	weight: 0
#a__isNatIList(x1)	weight: 0
#mark(x1)	weight: 0
     0()	weight: 0
#a__and(x1,x2)	weight: 0
#a__zeros()	weight: 0
   nil()	weight: 0
  mark(x1)	weight: 0
a__U11(x1,x2)	weight: 0
#a__length(x1)	weight: 0
 isNat(x1)	weight: 0
#a__isNatList(x1)	weight: 0
  cons(x1,x2)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: 0
a__and(x1,x2)	weight: 0
length(x1)	weight: 0
a__zeros()	weight: 0
a__length(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #3
Number of SCCs: 1, DPs: 18, edges: 87
	SCC { #1 #2 #4..7 #9 #11 #13..22 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatList(x1)	weight: 0
   U11(x1,x2)	weight: (/ 1 2) + x1 + x2
     s(x1)	weight: x1
a__isNatIList(x1)	weight: (/ 1 4)
#a__isNat(x1)	weight: 0
   and(x1,x2)	weight: x1 + x2
isNatIList(x1)	weight: (/ 1 4)
#a__U11(x1,x2)	weight: (/ 1 4) + x2
 zeros()	weight: 0
a__isNatList(x1)	weight: 0
#a__isNatIList(x1)	weight: (/ 1 4)
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: x2
#a__zeros()	weight: 0
   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
#a__isNatList(x1)	weight: 0
  cons(x1,x2)	weight: 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..4 6 9..11 13..32 }
    Removed DPs: #2 #5 #6 #18 #19 #21 #22
Number of SCCs: 2, DPs: 11, edges: 31
	SCC { #1 #4 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... succeeded.
isNatList(x1)	weight: (/ 1 8) + x1_2; (/ 1 8)
   U11(x1,x2)	weight: (/ 1 8); (/ 1 4)
     s(x1)	weight: x1_1; (/ 1 8) + x1_1
a__isNatIList(x1)	weight: (/ 1 4) + x1_2; (/ 1 4) + x1_2
#a__isNat(x1)	weight: 0; 0
   and(x1,x2)	weight: (/ 1 8) + x2_1 + x1_1; (/ 1 8)
isNatIList(x1)	weight: (/ 1 8) + x1_2; (/ 1 8)
#a__U11(x1,x2)	weight: x2_1 + x1_2; 0
 zeros()	weight: (/ 1 8); (/ 1 8)
a__isNatList(x1)	weight: (/ 1 4) + x1_2; (/ 7 8) + x1_2
#a__isNatIList(x1)	weight: 0; 0
#mark(x1)	weight: 0; 0
     0()	weight: (/ 1 2); (/ 1 8)
#a__and(x1,x2)	weight: 0; 0
#a__zeros()	weight: 0; 0
   nil()	weight: (/ 1 8); (/ 1 2)
  mark(x1)	weight: (/ 5 8) + x1_1; (/ 3 4) + x1_1
a__U11(x1,x2)	weight: (/ 1 8); (/ 1 4)
#a__length(x1)	weight: (/ 5 8) + x1_1; 0
 isNat(x1)	weight: (/ 1 8) + x1_1; (/ 1 8)
#a__isNatList(x1)	weight: 0; 0
  cons(x1,x2)	weight: x2_1 + x2_2 + x1_1; (/ 1 8) + x2_2 + x1_1
    tt()	weight: (/ 5 8); (/ 11 8)
a__isNat(x1)	weight: (/ 1 8) + x1_1; (/ 7 8) + x1_1
a__and(x1,x2)	weight: (/ 1 8) + x2_1 + x1_1; (/ 1 8) + x2_1 + x1_1
length(x1)	weight: (/ 1 8); (/ 1 8)
a__zeros()	weight: (/ 3 4); (/ 3 4)
a__length(x1)	weight: (/ 1 8); (/ 1 4)
    Usable rules: { 1..4 6 9..11 13..32 }
    Removed DPs: #1 #4
Number of SCCs: 1, DPs: 9, edges: 29
	SCC { #7 #9 #11 #13..17 #20 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatList(x1)	weight: (/ 1 4) + x1
   U11(x1,x2)	weight: (/ 3 8)
     s(x1)	weight: (/ 1 8) + x1
a__isNatIList(x1)	weight: 0
#a__isNat(x1)	weight: 0
   and(x1,x2)	weight: (/ 3 8) + x1 + x2
isNatIList(x1)	weight: (/ 1 8) + x1
#a__U11(x1,x2)	weight: (/ 1 8)
 zeros()	weight: 0
a__isNatList(x1)	weight: (/ 1 8)
#a__isNatIList(x1)	weight: x1
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: (/ 1 8) + x2
#a__zeros()	weight: 0
   nil()	weight: 0
  mark(x1)	weight: (/ 3 8) + x1
a__U11(x1,x2)	weight: (/ 1 2) + x1
#a__length(x1)	weight: 0
 isNat(x1)	weight: (/ 1 8)
#a__isNatList(x1)	weight: (/ 1 8) + x1
  cons(x1,x2)	weight: (/ 3 8) + x1 + x2
    tt()	weight: 0
a__isNat(x1)	weight: (/ 5 8) + x1
a__and(x1,x2)	weight: (/ 1 4)
length(x1)	weight: (/ 1 8) + x1
a__zeros()	weight: 0
a__length(x1)	weight: (/ 1 4) + x1
    Usable rules: { }
    Removed DPs: #7 #9 #11 #13..17 #20
Number of SCCs: 0, DPs: 0, edges: 0
YES