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

    ↳32 TRUE

  ↳33 QDP

    ↳34 QDPOrderProof (⇔)

    ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 QDPOrderProof (⇔)

    ↳41 QDP

    ↳42 PisEmptyProof (⇔)

    ↳43 TRUE

  ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 PisEmptyProof (⇔)

    ↳50 TRUE

  ↳51 QDP

    ↳52 QDPOrderProof (⇔)

    ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 PisEmptyProof (⇔)

    ↳57 TRUE

  ↳58 QDP

    ↳59 QDPOrderProof (⇔)

    ↳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 PisEmptyProof (⇔)

    ↳85 TRUE

  ↳86 QDP

    ↳87 QDPOrderProof (⇔)

    ↳88 QDP

    ↳89 QDPOrderProof (⇔)

    ↳90 QDP

    ↳91 PisEmptyProof (⇔)

    ↳92 TRUE

  ↳93 QDP

    ↳94 QDPOrderProof (⇔)

    ↳95 QDP

    ↳96 QDPOrderProof (⇔)

    ↳97 QDP

    ↳98 PisEmptyProof (⇔)

    ↳99 TRUE

  ↳100 QDP

    ↳101 QDPOrderProof (⇔)

    ↳102 QDP

    ↳103 QDPOrderProof (⇔)

    ↳104 QDP

    ↳105 PisEmptyProof (⇔)

    ↳106 TRUE

  ↳107 QDP

    ↳108 QDPOrderProof (⇔)

    ↳109 QDP

    ↳110 QDPOrderProof (⇔)

    ↳111 QDP

    ↳112 PisEmptyProof (⇔)

    ↳113 TRUE

  ↳114 QDP

    ↳115 QDPOrderProof (⇔)

    ↳116 QDP

    ↳117 QDPOrderProof (⇔)

    ↳118 QDP

    ↳119 QDPOrderProof (⇔)

    ↳120 QDP

    ↳121 QDPOrderProof (⇔)

    ↳122 QDP

    ↳123 QDPOrderProof (⇔)

    ↳124 QDP

    ↳125 QDPOrderProof (⇔)

    ↳126 QDP

    ↳127 QDPOrderProof (⇔)

    ↳128 QDP

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(active(X)) → LENGTH(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[LENGTH₁, active₁]

Status:

active₁: [1]
LENGTH₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

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

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[S₁, active₁]

Status:

active₁: [1]
S₁: [1]

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

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

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(active(X)) → ISNAT(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[ISNAT₁, active₁]

Status:

active₁: [1]
ISNAT₁: [1]

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

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

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(mark(X)) → ISNAT(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

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

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

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

(24) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE

(26) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U62^1₂, mark₁]

Status:

mark₁: [1]
U62^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(29) 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)
U62¹(X1, active(X2)) → U62¹(X1, X2)

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

[U62^1₂, active₁]

Status:

active₁: [1]
U62^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(30) 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.

(31) PisEmptyProof (EQUIVALENT transformation)

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

(32) TRUE

(33) 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.

(34) 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, 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(x2, x3)
mark(x1) = mark(x1)
active(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > U61^1₂

Status:

U61^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(35) Obligation:

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

U61¹(mark(X1), X2, 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.

(36) 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)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U61^1₃, mark₁]

Status:

U61^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > U61^1₁

Status:

active₁: [1]
U61^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

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

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

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

[U61^1₃, active₁]

Status:

active₁: [1]
U61^1₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(41) 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.

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) 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.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(active(X)) → ISNATLIST(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[ISNATLIST₁, active₁]

Status:

active₁: [1]
ISNATLIST₁: [1]

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

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

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

(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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(48) 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.

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) 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.

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X)) → U52¹(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U52^1₁, active₁]

Status:

U52^1₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U52¹(mark(X)) → U52¹(X)

The TRS R consists of the following rules:

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

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

(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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(55) 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.

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) 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.

(59) 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)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U51^1₂, mark₁]

Status:

U51^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

[U51^1₂, active₁]

Status:

active₁: [1]
U51^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(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(active(X)) → ISNATILIST(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[ISNATILIST₁, active₁]

Status:

active₁: [1]
ISNATILIST₁: [1]

The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

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

ISNATILIST(mark(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

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

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

(68) 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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(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(active(X)) → U42¹(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U42^1₁, active₁]

Status:

active₁: [1]
U42^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

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

U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

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

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

(75) 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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U41^1₂, mark₁]

Status:

U41^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(82) 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)
U41¹(X1, active(X2)) → U41¹(X1, X2)

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

[U41^1₂, active₁]

Status:

active₁: [1]
U41^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(83) 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.

(84) PisEmptyProof (EQUIVALENT transformation)

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

(85) TRUE

(86) 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.

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X)) → U31¹(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U31^1₁, active₁]

Status:

active₁: [1]
U31^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

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

U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

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

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

(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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(90) 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.

(91) PisEmptyProof (EQUIVALENT transformation)

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

(92) TRUE

(93) 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.

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X)) → U21¹(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U21^1₁, active₁]

Status:

active₁: [1]
U21^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

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

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

(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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(97) 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.

(98) PisEmptyProof (EQUIVALENT transformation)

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

(99) TRUE

(100) 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.

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X)) → U11¹(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U11^1₁, active₁]

Status:

active₁: [1]
U11^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(102) Obligation:

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

U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

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

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

(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) = x1
mark(x1) = mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(104) 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.

(105) PisEmptyProof (EQUIVALENT transformation)

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

(106) TRUE

(107) 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.

(108) 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(mark(X1), X2) → CONS(X1, X2)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[CONS₂, mark₁]

Status:

CONS₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(110) 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)
CONS(X1, active(X2)) → CONS(X1, X2)

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

[CONS₂, active₁]

Status:

active₁: [1]
CONS₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(111) 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.

(112) PisEmptyProof (EQUIVALENT transformation)

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

(113) TRUE

(114) 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.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U21(X)) → ACTIVE(U21(mark(X)))

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 with status [LPO].
Quasi-Precedence:

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

Status:

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

The following usable rules [FROCOS05] were oriented:

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

(116) Obligation:

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

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

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U11(X)) → ACTIVE(U11(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
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1) = U31
s(x1) = s
length(x1) = length
active(x1) = active
nil = nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

U21₁ > [MARK, cons, U41, U42, isNatIList, zeros, 0, U51, U52, isNatList, U61, U62, isNat, U31, s, length, active] > U11 > [mark, tt]
nil > [MARK, cons, U41, U42, isNatIList, zeros, 0, U51, U52, isNatList, U61, U62, isNat, U31, s, length, active] > U11 > [mark, tt]

Status:

active: []
MARK: []
U51: []
U62: []
isNatList: []
U31: []
U42: []
U52: []
U11: []
isNatIList: []
s: []
length: []
0: []
isNat: []
U61: []
cons: []
tt: []
zeros: []
U41: []
mark: []
nil: []
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(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))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, L, N)) → MARK(U62(isNat(N), L))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → ACTIVE(U31(mark(X)))
ACTIVE(U62(tt, L)) → MARK(s(length(L)))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U41(X1, X2)) → MARK(X1)
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(U31(X)) → ACTIVE(U31(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) = x1
U41(x1, x2) = U41
tt = tt
U42(x1) = U42
isNatIList(x1) = isNatIList
zeros = zeros
0 = 0
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U11(x1) = U11
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
U21(x1) = U21(x1)
U31(x1) = U31
s(x1) = s
length(x1) = length
active(x1) = active
nil = nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

[0, nil] > tt > active > U31
U21₁ > [MARK, cons, U41, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length] > tt > active > U31
U21₁ > [MARK, cons, U41, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length] > U11 > active > U31

Status:

active: []
MARK: []
U51: []
U62: []
isNatList: []
U31: []
U42: []
U52: []
U11: []
isNatIList: []
s: []
length: []
0: []
isNat: []
U61: []
cons: []
tt: []
zeros: []
U41: []
nil: []
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(120) 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))
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)
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.

(121) 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))

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
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList
U11(x1) = U11
U61(x1, x2, x3) = U61
U62(x1, x2) = U62
isNat(x1) = isNat
U21(x1) = U21(x1)
s(x1) = s
length(x1) = length
U31(x1) = U31(x1)
active(x1) = x1
nil = nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

0 > mark > [MARK, U41, tt, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length, U31₁] > cons
0 > mark > [MARK, U41, tt, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length, U31₁] > U11
0 > mark > [MARK, U41, tt, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length, U31₁] > U21₁
nil > mark > [MARK, U41, tt, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length, U31₁] > cons
nil > mark > [MARK, U41, tt, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length, U31₁] > U11
nil > mark > [MARK, U41, tt, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, s, length, U31₁] > U21₁

Status:

MARK: []
U51: []
U62: []
isNatList: []
U42: []
U52: []
U11: []
isNatIList: []
s: []
length: []
0: []
isNat: []
U61: []
cons: []
tt: []
U31₁: [1]
zeros: []
U41: []
mark: []
nil: []
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(122) 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)
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.

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

0 > tt > [mark, active] > cons₂ > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, length] > [U11₁, s]
0 > tt > [mark, active] > U21₁ > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, length] > [U11₁, s]
0 > tt > [mark, active] > U31₁ > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, length] > [U11₁, s]
nil > tt > [mark, active] > cons₂ > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, length] > [U11₁, s]
nil > tt > [mark, active] > U21₁ > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, length] > [U11₁, s]
nil > tt > [mark, active] > U31₁ > [U41, MARK, U42, isNatIList, zeros, U51, U52, isNatList, U61, U62, isNat, length] > [U11₁, s]

Status:

active: []
MARK: []
U51: []
U62: []
isNatList: []
U42: []
U52: []
isNatIList: []
s: []
length: []
0: []
isNat: []
cons₂: [2,1]
U61: []
tt: []
U31₁: [1]
zeros: []
U41: []
mark: []
U11₁: [1]
nil: []
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(124) 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)
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)) → 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.

(125) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U52(X)) → ACTIVE(U52(mark(X)))

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[0, nil] > [U11, U21, mark, active] > [U41, MARK, U42, isNatIList, cons₁, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > tt > U31
s > [U11, U21, mark, active] > [U41, MARK, U42, isNatIList, cons₁, zeros, U51, isNatList, U61, U62, isNat, length] > U52 > tt > U31

Status:

active: []
MARK: []
U51: []
U21: []
cons₁: [1]
U62: []
isNatList: []
U31: []
U42: []
U52: []
U11: []
isNatIList: []
s: []
length: []
0: []
isNat: []
U61: []
tt: []
zeros: []
U41: []
mark: []
nil: []

The following usable rules [FROCOS05] were oriented:

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

(126) 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)
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)
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.

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U42(X)) → ACTIVE(U42(mark(X)))

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

U52 > [mark, active] > U11₁ > [U41, tt, MARK, isNatIList, zeros, U51, isNatList, U61, U62, isNat, s₁, length] > U42
U52 > [mark, active] > U11₁ > [U41, tt, MARK, isNatIList, zeros, U51, isNatList, U61, U62, isNat, s₁, length] > cons₂
U52 > [mark, active] > U11₁ > [U41, tt, MARK, isNatIList, zeros, U51, isNatList, U61, U62, isNat, s₁, length] > [0, nil]
U52 > [mark, active] > U11₁ > [U41, tt, MARK, isNatIList, zeros, U51, isNatList, U61, U62, isNat, s₁, length] > U21₁

Status:

active: []
MARK: []
U51: []
U62: []
isNatList: []
U42: []
U52: []
isNatIList: []
length: []
0: []
isNat: []
cons₂: [2,1]
U61: []
tt: []
zeros: []
U41: []
mark: []
U11₁: [1]
s₁: [1]
nil: []
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

U62(mark(X1), X2) → U62(X1, X2)
U62(active(X1), X2) → U62(X1, X2)
U62(X1, active(X2)) → U62(X1, X2)
U62(X1, mark(X2)) → U62(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)

(128) 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.