Termination w.r.t. Q proof of /tmp/tmpywEfEw/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 PisEmptyProof (⇔)

    ↳45 TRUE

  ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 PisEmptyProof (⇔)

    ↳52 TRUE

  ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 PisEmptyProof (⇔)

    ↳59 TRUE

  ↳60 QDP

    ↳61 QDPOrderProof (⇔)

    ↳62 QDP

    ↳63 PisEmptyProof (⇔)

    ↳64 TRUE

  ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 QDPOrderProof (⇔)

    ↳69 QDP

    ↳70 PisEmptyProof (⇔)

    ↳71 TRUE

  ↳72 QDP

    ↳73 QDPOrderProof (⇔)

    ↳74 QDP

    ↳75 QDPOrderProof (⇔)

    ↳76 QDP

    ↳77 PisEmptyProof (⇔)

    ↳78 TRUE

  ↳79 QDP

    ↳80 QDPOrderProof (⇔)

    ↳81 QDP

    ↳82 QDPOrderProof (⇔)

    ↳83 QDP

    ↳84 QDPOrderProof (⇔)

    ↳85 QDP

    ↳86 PisEmptyProof (⇔)

    ↳87 TRUE

  ↳88 QDP

    ↳89 QDPOrderProof (⇔)

    ↳90 QDP

    ↳91 QDPOrderProof (⇔)

    ↳92 QDP

    ↳93 PisEmptyProof (⇔)

    ↳94 TRUE

  ↳95 QDP

    ↳96 QDPOrderProof (⇔)

    ↳97 QDP

    ↳98 QDPOrderProof (⇔)

    ↳99 QDP

    ↳100 PisEmptyProof (⇔)

    ↳101 TRUE

  ↳102 QDP

    ↳103 QDPOrderProof (⇔)

    ↳104 QDP

    ↳105 QDPOrderProof (⇔)

    ↳106 QDP

    ↳107 PisEmptyProof (⇔)

    ↳108 TRUE

  ↳109 QDP

    ↳110 QDPOrderProof (⇔)

    ↳111 QDP

    ↳112 QDPOrderProof (⇔)

    ↳113 QDP

    ↳114 QDPOrderProof (⇔)

    ↳115 QDP

    ↳116 PisEmptyProof (⇔)

    ↳117 TRUE

  ↳118 QDP

    ↳119 QDPOrderProof (⇔)

    ↳120 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(mark(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) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [LENGTH₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [LENGTH₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [LENGTH₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [LENGTH₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [LENGTH₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [LENGTH₁, mark₁, 0, U11₁, tt, s]
nil > [LENGTH₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(7) Obligation:

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

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.

(8) 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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > LENGTH₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > LENGTH₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > LENGTH₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > LENGTH₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > LENGTH₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > LENGTH₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > LENGTH₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > LENGTH₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(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(mark(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) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [S₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [S₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [S₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [S₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [S₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [S₁, mark₁, 0, U11₁, tt, s]
nil > [S₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(14) Obligation:

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

S(active(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(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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > S₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > S₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > S₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > S₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > S₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > S₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > S₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > S₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > S₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > S₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > S₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > S₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > S₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > S₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(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(mark(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) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [ISNAT₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [ISNAT₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [ISNAT₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [ISNAT₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [ISNAT₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [ISNAT₁, mark₁, 0, U11₁, tt, s]
nil > [ISNAT₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(21) Obligation:

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

ISNAT(active(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(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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNAT₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > ISNAT₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > ISNAT₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > ISNAT₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > ISNAT₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > ISNAT₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNAT₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > ISNAT₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(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) = x2
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31(x1)
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = x1
isNat(x1) = isNat
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > [cons, U61] > U41 > U42 > tt > s > [mark₁, U11₁, U31₁, length₁]
zeros > [cons, U61] > isNat > U21 > tt > s > [mark₁, U11₁, U31₁, length₁]
zeros > [cons, U61] > isNat > isNatList > U51 > U52 > tt > s > [mark₁, U11₁, U31₁, length₁]
zeros > 0 > tt > s > [mark₁, U11₁, U31₁, length₁]
isNatIList > U41 > U42 > tt > s > [mark₁, U11₁, U31₁, length₁]
isNatIList > isNatList > U51 > U52 > tt > s > [mark₁, U11₁, U31₁, length₁]
nil > 0 > tt > s > [mark₁, U11₁, U31₁, length₁]

The following usable rules [FROCOS05] were oriented:

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)

(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(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) = U62¹(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

U62^1₁ > [0, tt, U31, isNatList₁, U61, U62, s]
[zeros, cons] > [U11, isNat] > [mark₁, active₁, U21, isNatIList₁, length] > [0, tt, U31, isNatList₁, U61, U62, s]
[zeros, cons] > U41 > U42 > [mark₁, active₁, U21, isNatIList₁, length] > [0, tt, U31, isNatList₁, U61, U62, s]
[zeros, cons] > U51 > U52 > [mark₁, active₁, U21, isNatIList₁, length] > [0, tt, U31, isNatList₁, U61, U62, s]
nil > [mark₁, active₁, U21, isNatIList₁, length] > [0, tt, U31, isNatList₁, U61, U62, s]

The following usable rules [FROCOS05] were oriented:

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)

(30) 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)

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(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) = x1
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = x1
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31(x1)
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = x1
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > [mark₁, U31₁] > tt
isNatIList₁ > U41 > U42 > [mark₁, U31₁] > tt
isNatIList₁ > isNatList₁ > U51 > U52 > [mark₁, U31₁] > tt
length > 0 > [mark₁, U31₁] > tt
length > U11 > [mark₁, U31₁] > tt
length > isNatList₁ > U51 > U52 > [mark₁, U31₁] > tt
length > U61 > U62 > s > U21 > [mark₁, U31₁] > tt
nil > 0 > [mark₁, U31₁] > tt

The following usable rules [FROCOS05] were oriented:

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)

(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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62(x1)
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
zeros > cons > U51 > U52 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
zeros > cons > U61 > isNat > U11 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
zeros > cons > U61 > isNat > U21 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
isNatIList > U31 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
isNatIList > U41 > U42 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
isNatIList > isNat > U11 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
isNatIList > isNat > U21 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
length > U61 > isNat > U11 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
length > U61 > isNat > U21 > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]
nil > tt > [mark₁, 0, s] > [U62^1₁, active₁, isNatList₁, U62₁]

The following usable rules [FROCOS05] were oriented:

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)

(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(mark(X1), X2, X3) → U61¹(X1, X2, X3)
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) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = x2
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41(x2)
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

isNatIList₁ > U31 > [mark₁, active₁, U21, U52] > zeros > 0
isNatIList₁ > U31 > [mark₁, active₁, U21, U52] > [tt, U41₁, U42, nil] > length > 0
isNatIList₁ > U31 > [mark₁, active₁, U21, U52] > [tt, U41₁, U42, nil] > length > [U61, U62, s]
isNatIList₁ > [U51, isNatList] > [U11, isNat] > [mark₁, active₁, U21, U52] > zeros > 0
isNatIList₁ > [U51, isNatList] > [U11, isNat] > [mark₁, active₁, U21, U52] > [tt, U41₁, U42, nil] > length > 0
isNatIList₁ > [U51, isNatList] > [U11, isNat] > [mark₁, active₁, U21, U52] > [tt, U41₁, U42, nil] > length > [U61, U62, s]

The following usable rules [FROCOS05] were oriented:

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)

(39) Obligation:

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

U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(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, 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(x3)
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = cons(x1)
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31(x1)
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

U61^1₁ > [mark₁, cons₁, U31₁, isNat₁]
zeros > 0 > [tt, nil] > [mark₁, cons₁, U31₁, isNat₁]
[U11, length] > 0 > [tt, nil] > [mark₁, cons₁, U31₁, isNat₁]
[U11, length] > U61 > U62 > s > U21 > [tt, nil] > [mark₁, cons₁, U31₁, isNat₁]
isNatIList > U41 > U42 > [tt, nil] > [mark₁, cons₁, U31₁, isNat₁]
isNatIList > isNatList > U51 > U52 > [tt, nil] > [mark₁, cons₁, U31₁, isNat₁]

The following usable rules [FROCOS05] were oriented:

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)

(41) Obligation:

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

U61¹(X1, mark(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.

(42) 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(X1, active(X2), X3) → U61¹(X1, X2, X3)
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) = U61¹(x2, x3)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > [cons, U61] > U41 > U42 > [mark₁, tt] > [active₁, 0]
zeros > [cons, U61] > U41 > U42 > [mark₁, tt] > nil
zeros > [cons, U61] > isNatList > U51 > U52 > [mark₁, tt] > [active₁, 0]
zeros > [cons, U61] > isNatList > U51 > U52 > [mark₁, tt] > nil
zeros > [cons, U61] > U62 > s > U21 > [mark₁, tt] > [active₁, 0]
zeros > [cons, U61] > U62 > s > U21 > [mark₁, tt] > nil
isNatIList > U31 > [mark₁, tt] > [active₁, 0]
isNatIList > U31 > [mark₁, tt] > nil
isNatIList > U41 > U42 > [mark₁, tt] > [active₁, 0]
isNatIList > U41 > U42 > [mark₁, tt] > nil
isNatIList > isNat > U11 > [mark₁, tt] > [active₁, 0]
isNatIList > isNat > U11 > [mark₁, tt] > nil
isNatIList > isNat > U21 > [mark₁, tt] > [active₁, 0]
isNatIList > isNat > U21 > [mark₁, tt] > nil
isNatIList > isNat > isNatList > U51 > U52 > [mark₁, tt] > [active₁, 0]
isNatIList > isNat > isNatList > U51 > U52 > [mark₁, tt] > nil
length > [cons, U61] > U41 > U42 > [mark₁, tt] > [active₁, 0]
length > [cons, U61] > U41 > U42 > [mark₁, tt] > nil
length > [cons, U61] > isNatList > U51 > U52 > [mark₁, tt] > [active₁, 0]
length > [cons, U61] > isNatList > U51 > U52 > [mark₁, tt] > nil
length > [cons, U61] > U62 > s > U21 > [mark₁, tt] > [active₁, 0]
length > [cons, U61] > U62 > s > U21 > [mark₁, tt] > nil
length > U11 > [mark₁, tt] > [active₁, 0]
length > U11 > [mark₁, tt] > nil

The following usable rules [FROCOS05] were oriented:

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)

(43) 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.

(44) PisEmptyProof (EQUIVALENT transformation)

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

(45) TRUE

(46) 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.

(47) 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: Combined order from the following AFS and order.
ISNATLIST(x1) = ISNATLIST(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]
nil > [ISNATLIST₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(48) Obligation:

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

ISNATLIST(active(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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNATLIST₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > ISNATLIST₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > ISNATLIST₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > ISNATLIST₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > ISNATLIST₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > ISNATLIST₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNATLIST₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > ISNATLIST₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(50) 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.

(51) PisEmptyProof (EQUIVALENT transformation)

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

(52) TRUE

(53) 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.

(54) 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: Combined order from the following AFS and order.
U52¹(x1) = U52¹(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [U52^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [U52^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [U52^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [U52^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [U52^1₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [U52^1₁, mark₁, 0, U11₁, tt, s]
nil > [U52^1₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(55) Obligation:

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

U52¹(active(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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > U52^1₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > U52^1₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U52^1₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U52^1₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > U52^1₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > U52^1₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U52^1₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > U52^1₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > U52^1₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > U52^1₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > U52^1₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U52^1₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U52^1₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > U52^1₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(57) 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.

(58) PisEmptyProof (EQUIVALENT transformation)

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

(59) TRUE

(60) 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.

(61) 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)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
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) = U51¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > [mark₁, cons] > [U51^1₂, active₁, nil] > 0 > s
zeros > [mark₁, cons] > [U51^1₂, active₁, nil] > tt > s
[U11, isNatList, isNat₁, length] > U21 > [mark₁, cons] > [U51^1₂, active₁, nil] > 0 > s
[U11, isNatList, isNat₁, length] > U21 > [mark₁, cons] > [U51^1₂, active₁, nil] > tt > s
[U11, isNatList, isNat₁, length] > U51 > U52 > [mark₁, cons] > [U51^1₂, active₁, nil] > 0 > s
[U11, isNatList, isNat₁, length] > U51 > U52 > [mark₁, cons] > [U51^1₂, active₁, nil] > tt > s
[U11, isNatList, isNat₁, length] > U61 > U62 > [mark₁, cons] > [U51^1₂, active₁, nil] > 0 > s
[U11, isNatList, isNat₁, length] > U61 > U62 > [mark₁, cons] > [U51^1₂, active₁, nil] > tt > s
[U41, isNatIList₁] > U31 > [mark₁, cons] > [U51^1₂, active₁, nil] > 0 > s
[U41, isNatIList₁] > U31 > [mark₁, cons] > [U51^1₂, active₁, nil] > tt > s
[U41, isNatIList₁] > U42 > [mark₁, cons] > [U51^1₂, active₁, nil] > 0 > s
[U41, isNatIList₁] > U42 > [mark₁, cons] > [U51^1₂, active₁, nil] > tt > s

The following usable rules [FROCOS05] were oriented:

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)

(62) 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.

(63) PisEmptyProof (EQUIVALENT transformation)

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

(64) TRUE

(65) 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.

(66) 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: Combined order from the following AFS and order.
ISNATILIST(x1) = ISNATILIST(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]
nil > [ISNATILIST₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(67) Obligation:

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

ISNATILIST(active(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.

(68) 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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNATILIST₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > ISNATILIST₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > ISNATILIST₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > ISNATILIST₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > ISNATILIST₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > ISNATILIST₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > ISNATILIST₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > ISNATILIST₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(69) 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.

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) 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.

(73) 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: Combined order from the following AFS and order.
U42¹(x1) = U42¹(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [U42^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [U42^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [U42^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [U42^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [U42^1₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [U42^1₁, mark₁, 0, U11₁, tt, s]
nil > [U42^1₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(74) Obligation:

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

U42¹(active(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.

(75) 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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > U42^1₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > U42^1₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U42^1₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U42^1₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > U42^1₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > U42^1₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U42^1₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > U42^1₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > U42^1₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > U42^1₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > U42^1₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U42^1₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U42^1₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > U42^1₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(76) 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.

(77) PisEmptyProof (EQUIVALENT transformation)

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

(78) TRUE

(79) 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.

(80) 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) = x1
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = x1
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > [mark₁, 0]
[isNatIList, U51, isNatList₁] > U31 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > U41 > U42 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > U52 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > [U61, isNat] > U11 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > [U61, isNat] > U21 > [tt, nil] > U62 > s > [mark₁, 0]
length > [U61, isNat] > U11 > [tt, nil] > U62 > s > [mark₁, 0]
length > [U61, isNat] > U21 > [tt, nil] > U62 > s > [mark₁, 0]

The following usable rules [FROCOS05] were oriented:

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)

(81) Obligation:

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

U41¹(X1, mark(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.

(82) 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)
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
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = x2
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51(x2)
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > zeros
[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > U11 > [tt, U52]
[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > [U51₁, isNatList₁] > [tt, U52]
[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > nil
isNatIList > [U41, U42] > [mark₁, active₁, U31] > zeros
isNatIList > [U41, U42] > [mark₁, active₁, U31] > U11 > [tt, U52]
isNatIList > [U41, U42] > [mark₁, active₁, U31] > [U51₁, isNatList₁] > [tt, U52]
isNatIList > [U41, U42] > [mark₁, active₁, U31] > nil
isNat₁ > [U21, s] > [mark₁, active₁, U31] > zeros
isNat₁ > [U21, s] > [mark₁, active₁, U31] > U11 > [tt, U52]
isNat₁ > [U21, s] > [mark₁, active₁, U31] > [U51₁, isNatList₁] > [tt, U52]
isNat₁ > [U21, s] > [mark₁, active₁, U31] > nil

The following usable rules [FROCOS05] were oriented:

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)

(83) 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.

(84) 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) = x1
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > [cons, U51] > U41 > U42 > mark₁ > [active₁, tt, s, nil]
zeros > [cons, U51] > U52 > mark₁ > [active₁, tt, s, nil]
zeros > [cons, U51] > isNatList₁ > mark₁ > [active₁, tt, s, nil]
zeros > [cons, U51] > U61 > U62 > mark₁ > [active₁, tt, s, nil]
zeros > 0 > mark₁ > [active₁, tt, s, nil]
isNatIList₁ > [U11, isNat₁] > U21 > mark₁ > [active₁, tt, s, nil]
isNatIList₁ > [U11, isNat₁] > isNatList₁ > mark₁ > [active₁, tt, s, nil]
isNatIList₁ > U31 > mark₁ > [active₁, tt, s, nil]
isNatIList₁ > U41 > U42 > mark₁ > [active₁, tt, s, nil]
length > 0 > mark₁ > [active₁, tt, s, nil]
length > U61 > U62 > mark₁ > [active₁, tt, s, nil]

The following usable rules [FROCOS05] were oriented:

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)

(85) 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.

(86) PisEmptyProof (EQUIVALENT transformation)

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

(87) TRUE

(88) 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.

(89) 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: Combined order from the following AFS and order.
U31¹(x1) = U31¹(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [U31^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [U31^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [U31^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [U31^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [U31^1₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [U31^1₁, mark₁, 0, U11₁, tt, s]
nil > [U31^1₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(90) Obligation:

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

U31¹(active(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.

(91) 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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > U31^1₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > U31^1₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U31^1₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U31^1₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > U31^1₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > U31^1₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U31^1₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > U31^1₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > U31^1₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > U31^1₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > U31^1₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U31^1₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U31^1₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > U31^1₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(92) 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.

(93) PisEmptyProof (EQUIVALENT transformation)

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

(94) TRUE

(95) 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.

(96) 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: Combined order from the following AFS and order.
U21¹(x1) = U21¹(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [U21^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [U21^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [U21^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [U21^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [U21^1₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [U21^1₁, mark₁, 0, U11₁, tt, s]
nil > [U21^1₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(97) Obligation:

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

U21¹(active(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.

(98) 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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > U21^1₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > U21^1₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U21^1₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U21^1₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > U21^1₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > U21^1₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U21^1₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > U21^1₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > U21^1₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > U21^1₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > U21^1₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U21^1₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U21^1₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > U21^1₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(99) 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.

(100) PisEmptyProof (EQUIVALENT transformation)

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

(101) TRUE

(102) 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.

(103) 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: Combined order from the following AFS and order.
U11¹(x1) = U11¹(x1)
active(x1) = x1
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1) = U11(x1)
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > [U11^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U31 > [U11^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > U41 > U42 > [U11^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > U21 > [U11^1₁, mark₁, 0, U11₁, tt, s]
isNatIList₁ > isNat > isNatList > U51 > U52 > [U11^1₁, mark₁, 0, U11₁, tt, s]
length > U61 > U62 > [U11^1₁, mark₁, 0, U11₁, tt, s]
nil > [U11^1₁, mark₁, 0, U11₁, tt, s]

The following usable rules [FROCOS05] were oriented:

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)

(104) Obligation:

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

U11¹(active(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.

(105) 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)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length(x1)
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁ > U11^1₁
zeros > cons > U41 > U42 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > U11^1₁
zeros > cons > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U11^1₁
zeros > cons > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U11^1₁
zeros > cons > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > U11^1₁
zeros > cons > U61 > U62 > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > U11^1₁
zeros > cons > U61 > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U11^1₁
zeros > cons > U61 > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
zeros > 0 > tt > mark₁ > active₁ > U11^1₁
zeros > 0 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > U11^1₁
isNatIList₁ > U31 > tt > mark₁ > active₁ > length₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > U11^1₁
isNatIList₁ > U41 > U42 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > U11^1₁
isNatIList₁ > isNatList > U51 > U52 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > U11^1₁
isNatIList₁ > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > length₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > U11^1₁
isNatIList₁ > isNatList > isNat₁ > [U21, s] > tt > mark₁ > active₁ > length₁
nil > 0 > tt > mark₁ > active₁ > U11^1₁
nil > 0 > tt > mark₁ > active₁ > length₁

The following usable rules [FROCOS05] were oriented:

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)

(106) 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.

(107) PisEmptyProof (EQUIVALENT transformation)

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

(108) TRUE

(109) 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.

(110) 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) = x1
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = x1
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

zeros > [mark₁, 0]
[isNatIList, U51, isNatList₁] > U31 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > U41 > U42 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > U52 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > [U61, isNat] > U11 > [tt, nil] > U62 > s > [mark₁, 0]
[isNatIList, U51, isNatList₁] > [U61, isNat] > U21 > [tt, nil] > U62 > s > [mark₁, 0]
length > [U61, isNat] > U11 > [tt, nil] > U62 > s > [mark₁, 0]
length > [U61, isNat] > U21 > [tt, nil] > U62 > s > [mark₁, 0]

The following usable rules [FROCOS05] were oriented:

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)

(111) Obligation:

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

CONS(X1, mark(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.

(112) 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)
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
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = x2
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51(x2)
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > zeros
[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > U11 > [tt, U52]
[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > [U51₁, isNatList₁] > [tt, U52]
[0, U61, U62, length] > [U21, s] > [mark₁, active₁, U31] > nil
isNatIList > [U41, U42] > [mark₁, active₁, U31] > zeros
isNatIList > [U41, U42] > [mark₁, active₁, U31] > U11 > [tt, U52]
isNatIList > [U41, U42] > [mark₁, active₁, U31] > [U51₁, isNatList₁] > [tt, U52]
isNatIList > [U41, U42] > [mark₁, active₁, U31] > nil
isNat₁ > [U21, s] > [mark₁, active₁, U31] > zeros
isNat₁ > [U21, s] > [mark₁, active₁, U31] > U11 > [tt, U52]
isNat₁ > [U21, s] > [mark₁, active₁, U31] > [U51₁, isNatList₁] > [tt, U52]
isNat₁ > [U21, s] > [mark₁, active₁, U31] > nil

The following usable rules [FROCOS05] were oriented:

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)

(113) 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.

(114) 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) = x1
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1) = U11
tt = tt
U21(x1) = U21
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = x1
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat(x1)
s(x1) = s
length(x1) = length
nil = nil

Lexicographic Path Order [LPO].
Precedence:

[zeros, cons] > [active₁, mark₁, U61, U62] > U11 > [0, tt, U31, U42, U52, s]
[zeros, cons] > [active₁, mark₁, U61, U62] > U21 > [0, tt, U31, U42, U52, s]
[zeros, cons] > [active₁, mark₁, U61, U62] > U41 > [0, tt, U31, U42, U52, s]
[zeros, cons] > [active₁, mark₁, U61, U62] > isNatList > U51 > [0, tt, U31, U42, U52, s]
isNat₁ > [active₁, mark₁, U61, U62] > U11 > [0, tt, U31, U42, U52, s]
isNat₁ > [active₁, mark₁, U61, U62] > U21 > [0, tt, U31, U42, U52, s]
isNat₁ > [active₁, mark₁, U61, U62] > U41 > [0, tt, U31, U42, U52, s]
isNat₁ > [active₁, mark₁, U61, U62] > isNatList > U51 > [0, tt, U31, U42, U52, s]
length > [active₁, mark₁, U61, U62] > U11 > [0, tt, U31, U42, U52, s]
length > [active₁, mark₁, U61, U62] > U21 > [0, tt, U31, U42, U52, s]
length > [active₁, mark₁, U61, U62] > U41 > [0, tt, U31, U42, U52, s]
length > [active₁, mark₁, U61, U62] > isNatList > U51 > [0, tt, U31, U42, U52, s]
nil > [active₁, mark₁, U61, U62] > U11 > [0, tt, U31, U42, U52, s]
nil > [active₁, mark₁, U61, U62] > U21 > [0, tt, U31, U42, U52, s]
nil > [active₁, mark₁, U61, U62] > U41 > [0, tt, U31, U42, U52, s]
nil > [active₁, mark₁, U61, U62] > isNatList > U51 > [0, tt, U31, U42, U52, s]

The following usable rules [FROCOS05] were oriented:

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)

(115) 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.

(116) PisEmptyProof (EQUIVALENT transformation)

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

(117) TRUE

(118) 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.

(119) 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(U21(X)) → ACTIVE(U21(mark(X)))
MARK(U31(X)) → ACTIVE(U31(mark(X)))
MARK(U42(X)) → ACTIVE(U42(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
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
nil = nil

Lexicographic Path Order [LPO].
Precedence:

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

The following usable rules [FROCOS05] were oriented:

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)

(120) 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.