Input TRS:
    1: a__zeros() -> cons(0(),zeros())
    2: a__U11(tt(),V1) -> a__U12(a__isNatList(V1))
    3: a__U12(tt()) -> tt()
    4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
    5: a__U22(tt()) -> tt()
    6: a__U31(tt(),V) -> a__U32(a__isNatList(V))
    7: a__U32(tt()) -> tt()
    8: a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2)
    9: a__U42(tt(),V2) -> a__U43(a__isNatIList(V2))
    10: a__U43(tt()) -> tt()
    11: a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2)
    12: a__U52(tt(),V2) -> a__U53(a__isNatList(V2))
    13: a__U53(tt()) -> tt()
    14: a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2)
    15: a__U62(tt(),V2) -> a__U63(a__isNatIList(V2))
    16: a__U63(tt()) -> tt()
    17: a__U71(tt(),L) -> s(a__length(mark(L)))
    18: a__U81(tt()) -> nil()
    19: a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL))
    20: a__and(tt(),X) -> mark(X)
    21: a__isNat(0()) -> tt()
    22: a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
    23: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
    24: a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
    25: a__isNatIList(zeros()) -> tt()
    26: a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
    27: a__isNatIListKind(nil()) -> tt()
    28: a__isNatIListKind(zeros()) -> tt()
    29: a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
    30: a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2))
    31: a__isNatKind(0()) -> tt()
    32: a__isNatKind(length(V1)) -> a__isNatIListKind(V1)
    33: a__isNatKind(s(V1)) -> a__isNatKind(V1)
    34: a__isNatList(nil()) -> tt()
    35: a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
    36: a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
    37: a__length(nil()) -> 0()
    38: a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
    39: a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL)))
    40: a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))),IL,M,N)
    41: mark(zeros()) -> a__zeros()
    42: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
    43: mark(U12(X)) -> a__U12(mark(X))
    44: mark(isNatList(X)) -> a__isNatList(X)
    45: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
    46: mark(U22(X)) -> a__U22(mark(X))
    47: mark(isNat(X)) -> a__isNat(X)
    48: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
    49: mark(U32(X)) -> a__U32(mark(X))
    50: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
    51: mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
    52: mark(U43(X)) -> a__U43(mark(X))
    53: mark(isNatIList(X)) -> a__isNatIList(X)
    54: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
    55: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
    56: mark(U53(X)) -> a__U53(mark(X))
    57: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
    58: mark(U62(X1,X2)) -> a__U62(mark(X1),X2)
    59: mark(U63(X)) -> a__U63(mark(X))
    60: mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
    61: mark(length(X)) -> a__length(mark(X))
    62: mark(U81(X)) -> a__U81(mark(X))
    63: mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4)
    64: mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
    65: mark(and(X1,X2)) -> a__and(mark(X1),X2)
    66: mark(isNatIListKind(X)) -> a__isNatIListKind(X)
    67: mark(isNatKind(X)) -> a__isNatKind(X)
    68: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    69: mark(0()) -> 0()
    70: mark(tt()) -> tt()
    71: mark(s(X)) -> s(mark(X))
    72: mark(nil()) -> nil()
    73: a__zeros() -> zeros()
    74: a__U11(X1,X2) -> U11(X1,X2)
    75: a__U12(X) -> U12(X)
    76: a__isNatList(X) -> isNatList(X)
    77: a__U21(X1,X2) -> U21(X1,X2)
    78: a__U22(X) -> U22(X)
    79: a__isNat(X) -> isNat(X)
    80: a__U31(X1,X2) -> U31(X1,X2)
    81: a__U32(X) -> U32(X)
    82: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
    83: a__U42(X1,X2) -> U42(X1,X2)
    84: a__U43(X) -> U43(X)
    85: a__isNatIList(X) -> isNatIList(X)
    86: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
    87: a__U52(X1,X2) -> U52(X1,X2)
    88: a__U53(X) -> U53(X)
    89: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
    90: a__U62(X1,X2) -> U62(X1,X2)
    91: a__U63(X) -> U63(X)
    92: a__U71(X1,X2) -> U71(X1,X2)
    93: a__length(X) -> length(X)
    94: a__U81(X) -> U81(X)
    95: a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4)
    96: a__take(X1,X2) -> take(X1,X2)
    97: a__and(X1,X2) -> and(X1,X2)
    98: a__isNatIListKind(X) -> isNatIListKind(X)
    99: a__isNatKind(X) -> isNatKind(X)
Number of strict rules: 99
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U11(tt(),V1) -> #a__U12(a__isNatList(V1))
   #2: #a__U11(tt(),V1) -> #a__isNatList(V1)
   #3: #mark(U12(X)) -> #a__U12(mark(X))
   #4: #mark(U12(X)) -> #mark(X)
   #5: #a__isNatIListKind(cons(V1,V2)) -> #a__and(a__isNatKind(V1),isNatIListKind(V2))
   #6: #a__isNatIListKind(cons(V1,V2)) -> #a__isNatKind(V1)
   #7: #a__isNatList(cons(V1,V2)) -> #a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
   #8: #a__isNatList(cons(V1,V2)) -> #a__and(a__isNatKind(V1),isNatIListKind(V2))
   #9: #a__isNatList(cons(V1,V2)) -> #a__isNatKind(V1)
   #10: #mark(isNatIListKind(X)) -> #a__isNatIListKind(X)
   #11: #mark(U22(X)) -> #a__U22(mark(X))
   #12: #mark(U22(X)) -> #mark(X)
   #13: #mark(U11(X1,X2)) -> #a__U11(mark(X1),X2)
   #14: #mark(U11(X1,X2)) -> #mark(X1)
   #15: #mark(zeros()) -> #a__zeros()
   #16: #mark(isNat(X)) -> #a__isNat(X)
   #17: #mark(isNatIList(X)) -> #a__isNatIList(X)
   #18: #mark(s(X)) -> #mark(X)
   #19: #mark(U31(X1,X2)) -> #a__U31(mark(X1),X2)
   #20: #mark(U31(X1,X2)) -> #mark(X1)
   #21: #mark(U62(X1,X2)) -> #a__U62(mark(X1),X2)
   #22: #mark(U62(X1,X2)) -> #mark(X1)
   #23: #mark(length(X)) -> #a__length(mark(X))
   #24: #mark(length(X)) -> #mark(X)
   #25: #a__length(cons(N,L)) -> #a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)
   #26: #a__length(cons(N,L)) -> #a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N)))
   #27: #a__length(cons(N,L)) -> #a__and(a__isNatList(L),isNatIListKind(L))
   #28: #a__length(cons(N,L)) -> #a__isNatList(L)
   #29: #a__U31(tt(),V) -> #a__U32(a__isNatList(V))
   #30: #a__U31(tt(),V) -> #a__isNatList(V)
   #31: #mark(U63(X)) -> #a__U63(mark(X))
   #32: #mark(U63(X)) -> #mark(X)
   #33: #mark(U52(X1,X2)) -> #a__U52(mark(X1),X2)
   #34: #mark(U52(X1,X2)) -> #mark(X1)
   #35: #mark(isNatKind(X)) -> #a__isNatKind(X)
   #36: #a__take(s(M),cons(N,IL)) -> #a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))),IL,M,N)
   #37: #a__take(s(M),cons(N,IL)) -> #a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))))
   #38: #a__take(s(M),cons(N,IL)) -> #a__and(a__isNatIList(IL),isNatIListKind(IL))
   #39: #a__take(s(M),cons(N,IL)) -> #a__isNatIList(IL)
   #40: #mark(U42(X1,X2)) -> #a__U42(mark(X1),X2)
   #41: #mark(U42(X1,X2)) -> #mark(X1)
   #42: #a__U42(tt(),V2) -> #a__U43(a__isNatIList(V2))
   #43: #a__U42(tt(),V2) -> #a__isNatIList(V2)
   #44: #a__U51(tt(),V1,V2) -> #a__U52(a__isNat(V1),V2)
   #45: #a__U51(tt(),V1,V2) -> #a__isNat(V1)
   #46: #mark(U61(X1,X2,X3)) -> #a__U61(mark(X1),X2,X3)
   #47: #mark(U61(X1,X2,X3)) -> #mark(X1)
   #48: #a__isNatIList(V) -> #a__U31(a__isNatIListKind(V),V)
   #49: #a__isNatIList(V) -> #a__isNatIListKind(V)
   #50: #a__isNat(s(V1)) -> #a__U21(a__isNatKind(V1),V1)
   #51: #a__isNat(s(V1)) -> #a__isNatKind(V1)
   #52: #mark(U21(X1,X2)) -> #a__U21(mark(X1),X2)
   #53: #mark(U21(X1,X2)) -> #mark(X1)
   #54: #a__U52(tt(),V2) -> #a__U53(a__isNatList(V2))
   #55: #a__U52(tt(),V2) -> #a__isNatList(V2)
   #56: #mark(U53(X)) -> #a__U53(mark(X))
   #57: #mark(U53(X)) -> #mark(X)
   #58: #a__U61(tt(),V1,V2) -> #a__U62(a__isNat(V1),V2)
   #59: #a__U61(tt(),V1,V2) -> #a__isNat(V1)
   #60: #mark(U81(X)) -> #a__U81(mark(X))
   #61: #mark(U81(X)) -> #mark(X)
   #62: #a__isNatIListKind(take(V1,V2)) -> #a__and(a__isNatKind(V1),isNatIListKind(V2))
   #63: #a__isNatIListKind(take(V1,V2)) -> #a__isNatKind(V1)
   #64: #mark(U43(X)) -> #a__U43(mark(X))
   #65: #mark(U43(X)) -> #mark(X)
   #66: #mark(U32(X)) -> #a__U32(mark(X))
   #67: #mark(U32(X)) -> #mark(X)
   #68: #a__and(tt(),X) -> #mark(X)
   #69: #a__take(0(),IL) -> #a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL)))
   #70: #a__take(0(),IL) -> #a__and(a__isNatIList(IL),isNatIListKind(IL))
   #71: #a__take(0(),IL) -> #a__isNatIList(IL)
   #72: #mark(take(X1,X2)) -> #a__take(mark(X1),mark(X2))
   #73: #mark(take(X1,X2)) -> #mark(X1)
   #74: #mark(take(X1,X2)) -> #mark(X2)
   #75: #a__isNatKind(s(V1)) -> #a__isNatKind(V1)
   #76: #mark(isNatList(X)) -> #a__isNatList(X)
   #77: #mark(and(X1,X2)) -> #a__and(mark(X1),X2)
   #78: #mark(and(X1,X2)) -> #mark(X1)
   #79: #a__isNat(length(V1)) -> #a__U11(a__isNatIListKind(V1),V1)
   #80: #a__isNat(length(V1)) -> #a__isNatIListKind(V1)
   #81: #mark(U71(X1,X2)) -> #a__U71(mark(X1),X2)
   #82: #mark(U71(X1,X2)) -> #mark(X1)
   #83: #a__U71(tt(),L) -> #a__length(mark(L))
   #84: #a__U71(tt(),L) -> #mark(L)
   #85: #a__isNatKind(length(V1)) -> #a__isNatIListKind(V1)
   #86: #a__U91(tt(),IL,M,N) -> #mark(N)
   #87: #mark(U91(X1,X2,X3,X4)) -> #a__U91(mark(X1),X2,X3,X4)
   #88: #mark(U91(X1,X2,X3,X4)) -> #mark(X1)
   #89: #a__isNatIList(cons(V1,V2)) -> #a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
   #90: #a__isNatIList(cons(V1,V2)) -> #a__and(a__isNatKind(V1),isNatIListKind(V2))
   #91: #a__isNatIList(cons(V1,V2)) -> #a__isNatKind(V1)
   #92: #mark(cons(X1,X2)) -> #mark(X1)
   #93: #a__isNatList(take(V1,V2)) -> #a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2)
   #94: #a__isNatList(take(V1,V2)) -> #a__and(a__isNatKind(V1),isNatIListKind(V2))
   #95: #a__isNatList(take(V1,V2)) -> #a__isNatKind(V1)
   #96: #mark(U51(X1,X2,X3)) -> #a__U51(mark(X1),X2,X3)
   #97: #mark(U51(X1,X2,X3)) -> #mark(X1)
   #98: #a__U41(tt(),V1,V2) -> #a__U42(a__isNat(V1),V2)
   #99: #a__U41(tt(),V1,V2) -> #a__isNat(V1)
   #100: #a__U62(tt(),V2) -> #a__U63(a__isNatIList(V2))
   #101: #a__U62(tt(),V2) -> #a__isNatIList(V2)
   #102: #a__U21(tt(),V1) -> #a__U22(a__isNat(V1))
   #103: #a__U21(tt(),V1) -> #a__isNat(V1)
   #104: #mark(U41(X1,X2,X3)) -> #a__U41(mark(X1),X2,X3)
   #105: #mark(U41(X1,X2,X3)) -> #mark(X1)
Number of SCCs: 1, DPs: 90, edges: 1323
	SCC { #2 #4..10 #12..14 #16..28 #30 #32..41 #43..53 #55 #57..59 #61..63 #65 #67 #68 #70..99 #101 #103..105 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#a__isNatKind(x1)	weight: 0
isNatKind(x1)	weight: 0
a__U63(x1)	weight: x1
   U21(x1,x2)	weight: x1
isNatList(x1)	weight: 0
#a__U71(x1,x2)	weight: (/ 1 4) + x2
   U11(x1,x2)	weight: x1
     s(x1)	weight: x1
#a__U31(x1,x2)	weight: 0
a__isNatIList(x1)	weight: 0
#a__isNatIListKind(x1)	weight: 0
   U42(x1,x2)	weight: x1
   U91(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x2 + x3 + x4
#a__isNat(x1)	weight: 0
  take(x1,x2)	weight: (/ 3 4) + x1 + x2
   U71(x1,x2)	weight: (/ 1 2) + x1 + x2
a__U62(x1,x2)	weight: x1
   and(x1,x2)	weight: x1 + x2
isNatIList(x1)	weight: 0
#a__U43(x1)	weight: 0
   U43(x1)	weight: x1
#a__U51(x1,x2,x3)	weight: 0
   U63(x1)	weight: x1
a__U22(x1)	weight: x1
#a__U11(x1,x2)	weight: 0
 zeros()	weight: 0
a__U31(x1,x2)	weight: x1
a__U51(x1,x2,x3)	weight: x1
a__U81(x1)	weight: (/ 1 4) + x1
#a__U53(x1)	weight: 0
#a__take(x1,x2)	weight: (/ 1 2) + x2
   U12(x1)	weight: x1
a__isNatList(x1)	weight: 0
a__U43(x1)	weight: x1
#a__U62(x1,x2)	weight: 0
#a__U42(x1,x2)	weight: 0
a__U41(x1,x2,x3)	weight: x1
#a__U12(x1)	weight: 0
#a__U21(x1,x2)	weight: 0
#a__isNatIList(x1)	weight: 0
#a__U81(x1)	weight: 0
#a__U61(x1,x2,x3)	weight: 0
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: x2
#a__zeros()	weight: 0
a__U21(x1,x2)	weight: x1
a__U32(x1)	weight: x1
a__U91(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x2 + x3 + x4
   nil()	weight: 0
isNatIListKind(x1)	weight: 0
   U62(x1,x2)	weight: x1
#a__U52(x1,x2)	weight: 0
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: x1
   U32(x1)	weight: x1
a__U53(x1)	weight: x1
a__U42(x1,x2)	weight: x1
a__U52(x1,x2)	weight: x1
#a__length(x1)	weight: (/ 1 4) + x1
a__U12(x1)	weight: x1
a__isNatIListKind(x1)	weight: 0
#a__U63(x1)	weight: 0
 isNat(x1)	weight: 0
   U52(x1,x2)	weight: x1
   U61(x1,x2,x3)	weight: x1
#a__U22(x1)	weight: 0
   U31(x1,x2)	weight: x1
#a__isNatList(x1)	weight: 0
a__U71(x1,x2)	weight: (/ 1 2) + x1 + x2
  cons(x1,x2)	weight: x1 + x2
a__U61(x1,x2,x3)	weight: x1
a__take(x1,x2)	weight: (/ 3 4) + x1 + x2
   U81(x1)	weight: (/ 1 4) + x1
#a__U41(x1,x2,x3)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: 0
a__isNatKind(x1)	weight: 0
   U22(x1)	weight: x1
   U51(x1,x2,x3)	weight: x1
a__and(x1,x2)	weight: x1 + x2
   U53(x1)	weight: x1
length(x1)	weight: (/ 1 2) + x1
   U41(x1,x2,x3)	weight: x1
#a__U32(x1)	weight: 0
a__zeros()	weight: 0
#a__U91(x1,x2,x3,x4)	weight: (/ 1 4) + x4
a__length(x1)	weight: (/ 1 2) + x1
    Usable rules: { 1..99 }
    Removed DPs: #23 #24 #26..28 #36..39 #61 #70..74 #81 #82 #84 #86..88
Number of SCCs: 2, DPs: 69, edges: 750
	SCC { #25 #83 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)...