Termination w.r.t. Q proof of /tmp/tmpC4KHZj/LengthOfFiniteLists_nokinds_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 PisEmptyProof (⇔)

    ↳25 TRUE

  ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 QDPOrderProof (⇔)

    ↳34 QDP

    ↳35 PisEmptyProof (⇔)

    ↳36 TRUE

  ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 QDPOrderProof (⇔)

    ↳41 QDP

    ↳42 QDPOrderProof (⇔)

    ↳43 QDP

    ↳44 QDPOrderProof (⇔)

    ↳45 QDP

    ↳46 PisEmptyProof (⇔)

    ↳47 TRUE

  ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 QDPOrderProof (⇔)

    ↳52 QDP

    ↳53 PisEmptyProof (⇔)

    ↳54 TRUE

  ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 QDPOrderProof (⇔)

    ↳59 QDP

    ↳60 PisEmptyProof (⇔)

    ↳61 TRUE

  ↳62 QDP

    ↳63 QDPOrderProof (⇔)

    ↳64 QDP

    ↳65 QDPOrderProof (⇔)

    ↳66 QDP

    ↳67 QDPOrderProof (⇔)

    ↳68 QDP

    ↳69 QDPOrderProof (⇔)

    ↳70 QDP

    ↳71 PisEmptyProof (⇔)

    ↳72 TRUE

  ↳73 QDP

    ↳74 QDPOrderProof (⇔)

    ↳75 QDP

    ↳76 QDPOrderProof (⇔)

    ↳77 QDP

    ↳78 PisEmptyProof (⇔)

    ↳79 TRUE

  ↳80 QDP

    ↳81 QDPOrderProof (⇔)

    ↳82 QDP

    ↳83 QDPOrderProof (⇔)

    ↳84 QDP

    ↳85 PisEmptyProof (⇔)

    ↳86 TRUE

  ↳87 QDP

    ↳88 QDPOrderProof (⇔)

    ↳89 QDP

    ↳90 QDPOrderProof (⇔)

    ↳91 QDP

    ↳92 QDPOrderProof (⇔)

    ↳93 QDP

    ↳94 QDPOrderProof (⇔)

    ↳95 QDP

    ↳96 PisEmptyProof (⇔)

    ↳97 TRUE

  ↳98 QDP

    ↳99 QDPOrderProof (⇔)

    ↳100 QDP

    ↳101 QDPOrderProof (⇔)

    ↳102 QDP

    ↳103 PisEmptyProof (⇔)

    ↳104 TRUE

  ↳105 QDP

    ↳106 QDPOrderProof (⇔)

    ↳107 QDP

    ↳108 QDPOrderProof (⇔)

    ↳109 QDP

    ↳110 PisEmptyProof (⇔)

    ↳111 TRUE

  ↳112 QDP

    ↳113 QDPOrderProof (⇔)

    ↳114 QDP

    ↳115 QDPOrderProof (⇔)

    ↳116 QDP

    ↳117 PisEmptyProof (⇔)

    ↳118 TRUE

  ↳119 QDP

    ↳120 QDPOrderProof (⇔)

    ↳121 QDP

    ↳122 QDPOrderProof (⇔)

    ↳123 QDP

    ↳124 QDPOrderProof (⇔)

    ↳125 QDP

    ↳126 QDPOrderProof (⇔)

    ↳127 QDP

    ↳128 PisEmptyProof (⇔)

    ↳129 TRUE

  ↳130 QDP

    ↳131 QDPOrderProof (⇔)

    ↳132 QDP

    ↳133 QDPOrderProof (⇔)

    ↳134 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U11(tt)) → MARK(tt)
ACTIVE(U21(tt)) → MARK(tt)
ACTIVE(U31(tt)) → MARK(tt)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U41(tt, V2)) → U42¹(isNatIList(V2))
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
ACTIVE(U42(tt)) → MARK(tt)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U51(tt, V2)) → U52¹(isNatList(V2))
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
ACTIVE(U52(tt)) → MARK(tt)
ACTIVE(U61(tt, L, N)) → MARK(U62(isNat(N), L))
ACTIVE(U61(tt, L, N)) → U62¹(isNat(N), L)
ACTIVE(U61(tt, L, N)) → ISNAT(N)
ACTIVE(U62(tt, L)) → MARK(s(length(L)))
ACTIVE(U62(tt, L)) → S(length(L))
ACTIVE(U62(tt, L)) → LENGTH(L)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(isNat(length(V1))) → U11¹(isNatList(V1))
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(isNat(s(V1))) → U21¹(isNat(V1))
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
ACTIVE(isNatIList(V)) → U31¹(isNatList(V))
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(isNat(V1), V2)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNatList(cons(V1, V2))) → U51¹(isNat(V1), V2)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(N, L))) → MARK(U61(isNatList(L), L, N))
ACTIVE(length(cons(N, L))) → U61¹(isNatList(L), L, N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(0) → ACTIVE(0)
MARK(U11(X)) → ACTIVE(U11(mark(X)))
MARK(U11(X)) → U11¹(mark(X))
MARK(U11(X)) → MARK(X)
MARK(tt) → ACTIVE(tt)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
MARK(U21(X)) → U21¹(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → ACTIVE(U31(mark(X)))
MARK(U31(X)) → U31¹(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U41(X1, X2)) → U41¹(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U42(X)) → U42¹(mark(X))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U51(X1, X2)) → U51¹(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
MARK(U52(X)) → U52¹(mark(X))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U61(X1, X2, X3)) → U61¹(mark(X1), X2, X3)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2)) → ACTIVE(U62(mark(X1), X2))
MARK(U62(X1, X2)) → U62¹(mark(X1), X2)
MARK(U62(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(nil) → ACTIVE(nil)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
U11¹(mark(X)) → U11¹(X)
U11¹(active(X)) → U11¹(X)
U21¹(mark(X)) → U21¹(X)
U21¹(active(X)) → U21¹(X)
U31¹(mark(X)) → U31¹(X)
U31¹(active(X)) → U31¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)
U42¹(mark(X)) → U42¹(X)
U42¹(active(X)) → U42¹(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)
U52¹(mark(X)) → U52¹(X)
U52¹(active(X)) → U52¹(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)
U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(X1, mark(X2)) → U62¹(X1, X2)
U62¹(active(X1), X2) → U62¹(X1, X2)
U62¹(X1, active(X2)) → U62¹(X1, X2)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 45 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(active(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1) = LENGTH(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[LENGTH₁, active₁]

Status:

active₁: multiset
LENGTH₁: multiset

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > LENGTH₁

Status:

LENGTH₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[S₁, active₁]

Status:

active₁: multiset
S₁: multiset

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > S₁

Status:

mark₁: multiset
S₁: multiset

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) TRUE

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(active(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = ISNAT(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[ISNAT₁, active₁]

Status:

active₁: multiset
ISNAT₁: multiset

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(mark(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > ISNAT₁

Status:

mark₁: multiset
ISNAT₁: multiset

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(25) TRUE

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(X1, mark(X2)) → U62¹(X1, X2)
U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(active(X1), X2) → U62¹(X1, X2)
U62¹(X1, active(X2)) → U62¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, mark(X2)) → U62¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2) = U62¹(x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U62^1₁, mark₁]

Status:

mark₁: multiset
U62^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(active(X1), X2) → U62¹(X1, X2)
U62¹(X1, active(X2)) → U62¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X1), X2) → U62¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2) = U62¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U62^1₂

Status:

mark₁: [1]
U62^1₂: multiset

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(active(X1), X2) → U62¹(X1, X2)
U62¹(X1, active(X2)) → U62¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, active(X2)) → U62¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2) = x2
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(active(X1), X2) → U62¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(active(X1), X2) → U62¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2) = U62¹(x1)
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U62^1₁, active₁]

Status:

active₁: [1]
U62^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(36) TRUE

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3) = U61¹(x1, x2, x3)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U61^1₃: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3) = U61¹(x2)
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U61^1₁, active₁]

Status:

active₁: [1]
U61^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3) = x3
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3) = U61¹(x1)
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U61^1₁, active₁]

Status:

active₁: multiset
U61^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(46) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(47) TRUE

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(active(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1) = ISNATLIST(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[ISNATLIST₁, active₁]

Status:

active₁: multiset
ISNATLIST₁: multiset

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(mark(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > ISNATLIST₁

Status:

ISNATLIST₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(53) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(54) TRUE

(55) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(active(X)) → U52¹(X)
U52¹(mark(X)) → U52¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1) = U52¹(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U52^1₁, active₁]

Status:

U52^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(mark(X)) → U52¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U52^1₁

Status:

U52^1₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(60) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(61) TRUE

(62) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, mark(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2) = U51¹(x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U51^1₁, mark₁]

Status:

mark₁: multiset
U51^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2) = U51¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U51^1₂

Status:

U51^1₂: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, active(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2) = x2
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(active(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(active(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2) = U51¹(x1)
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U51^1₁, active₁]

Status:

active₁: [1]
U51^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(71) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(72) TRUE

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(active(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1) = ISNATILIST(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[ISNATILIST₁, active₁]

Status:

active₁: multiset
ISNATILIST₁: multiset

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(mark(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(mark(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > ISNATILIST₁

Status:

ISNATILIST₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(78) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(79) TRUE

(80) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(active(X)) → U42¹(X)
U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(active(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1) = U42¹(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U42^1₁, active₁]

Status:

active₁: multiset
U42^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U42^1₁

Status:

mark₁: multiset
U42^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(84) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(85) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(86) TRUE

(87) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, mark(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = U41¹(x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U41^1₁, mark₁]

Status:

U41^1₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = U41¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U41^1₂

Status:

U41^1₂: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, active(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = x2
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(93) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(active(X1), X2) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(active(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = U41¹(x1)
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U41^1₁, active₁]

Status:

active₁: [1]
U41^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(96) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(97) TRUE

(98) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(active(X)) → U31¹(X)
U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X)) → U31¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1) = U31¹(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U31^1₁, active₁]

Status:

active₁: multiset
U31^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(100) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X)) → U31¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U31^1₁

Status:

U31^1₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(102) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(103) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(104) TRUE

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(active(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X)) → U21¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1) = U21¹(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U21^1₁, active₁]

Status:

active₁: multiset
U21^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X)) → U21¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U21^1₁

Status:

mark₁: multiset
U21^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(110) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(111) TRUE

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(active(X)) → U11¹(X)
U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1) = U11¹(x1)
active(x1) = active(x1)
mark(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U11^1₁, active₁]

Status:

active₁: multiset
U11^1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U11^1₁

Status:

U11^1₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(117) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(118) TRUE

(119) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[CONS₁, mark₁]

Status:

CONS₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(121) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > CONS₂

Status:

CONS₂: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = x2
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(125) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1)
active(x1) = active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[CONS₁, active₁]

Status:

active₁: [1]
CONS₁: multiset

The following usable rules [FROCOS05] were oriented: none

(127) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(128) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(129) TRUE

(130) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
ACTIVE(U61(tt, L, N)) → MARK(U62(isNat(N), L))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → ACTIVE(U31(mark(X)))
ACTIVE(U62(tt, L)) → MARK(s(length(L)))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U61(isNatList(L), L, N))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2)) → ACTIVE(U62(mark(X1), X2))
MARK(U62(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U21(X)) → ACTIVE(U21(mark(X)))
MARK(U42(X)) → ACTIVE(U42(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK
cons(x1, x2) = cons
ACTIVE(x1) = x1
mark(x1) = mark
U41(x1, x2) = U41
tt = tt
U42(x1) = U42
isNatIList(x1) = isNatIList
zeros = zeros
0 = 0
U11(x1) = U11
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U21(x1) = U21
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
U31(x1) = U31
s(x1) = s
length(x1) = length
active(x1) = active(x1)
nil = nil

Recursive path order with status [RPO].
Quasi-Precedence:

[mark, nil] > [MARK, cons, U41, isNatIList, zeros, 0, U11, U51, U52, isNatList, U61, U62, isNat, U31, s, length] > U42 > [tt, U21] > active₁

Status:

MARK: multiset
U51: multiset
U21: multiset
U62: multiset
isNatList: multiset
U31: multiset
U42: multiset
U52: multiset
U11: multiset
isNatIList: multiset
s: multiset
length: multiset
0: multiset
isNat: multiset
active₁: multiset
U61: multiset
cons: multiset
tt: multiset
zeros: multiset
U41: multiset
mark: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U62(mark(X1), X2) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)

(132) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, L, N)) → MARK(U62(isNat(N), L))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → ACTIVE(U31(mark(X)))
ACTIVE(U62(tt, L)) → MARK(s(length(L)))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U41(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U61(isNatList(L), L, N))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2)) → ACTIVE(U62(mark(X1), X2))
MARK(U62(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(133) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
MARK(U31(X)) → ACTIVE(U31(mark(X)))
MARK(U52(X)) → ACTIVE(U52(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK
cons(x1, x2) = cons
ACTIVE(x1) = x1
mark(x1) = mark(x1)
U41(x1, x2) = U41
tt = tt
U42(x1) = U42(x1)
isNatIList(x1) = isNatIList
zeros = zeros
0 = 0
U11(x1) = U11
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
U21(x1) = U21
U31(x1) = U31
s(x1) = s
length(x1) = length
active(x1) = active(x1)
nil = nil

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > cons > active₁ > U31
[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > cons > active₁ > s > U21
[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > 0 > active₁ > U31
[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > 0 > active₁ > s > U21
[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > U11 > active₁ > U31
[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > U11 > active₁ > s > U21
[MARK, U41, isNatIList, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > [mark₁, tt, U42₁] > nil

Status:

MARK: []
U51: []
U21: []
U62: []
isNatList: []
U31: multiset
U52: multiset
mark₁: [1]
U11: []
isNatIList: []
s: []
length: []
0: multiset
isNat: []
active₁: [1]
U61: []
cons: []
tt: multiset
U42₁: multiset
zeros: multiset
U41: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U62(mark(X1), X2) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)

(134) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, L, N)) → MARK(U62(isNat(N), L))
MARK(U21(X)) → MARK(X)
ACTIVE(U62(tt, L)) → MARK(s(length(L)))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U41(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U61(isNatList(L), L, N))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2)) → ACTIVE(U62(mark(X1), X2))
MARK(U62(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2)) → active(U62(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.