Termination w.r.t. Q proof of /tmp/tmpwaRvx7/OvConsOS_nosorts_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 QDPOrderProof (⇔)

    ↳18 QDP

    ↳19 PisEmptyProof (⇔)

    ↳20 TRUE

  ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 QDPOrderProof (⇔)

    ↳25 QDP

    ↳26 QDPOrderProof (⇔)

    ↳27 QDP

    ↳28 PisEmptyProof (⇔)

    ↳29 TRUE

  ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 QDPOrderProof (⇔)

    ↳34 QDP

    ↳35 QDPOrderProof (⇔)

    ↳36 QDP

    ↳37 PisEmptyProof (⇔)

    ↳38 TRUE

  ↳39 QDP

    ↳40 QDPOrderProof (⇔)

    ↳41 QDP

    ↳42 PisEmptyProof (⇔)

    ↳43 TRUE

  ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 PisEmptyProof (⇔)

    ↳48 TRUE

  ↳49 QDP

    ↳50 QDPOrderProof (⇔)

    ↳51 QDP

    ↳52 QDPOrderProof (⇔)

    ↳53 QDP

    ↳54 PisEmptyProof (⇔)

    ↳55 TRUE

  ↳56 QDP

    ↳57 QDPOrderProof (⇔)

    ↳58 QDP

    ↳59 QDPOrderProof (⇔)

    ↳60 QDP

    ↳61 PisEmptyProof (⇔)

    ↳62 TRUE

  ↳63 QDP

    ↳64 QDPOrderProof (⇔)

    ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 PisEmptyProof (⇔)

    ↳69 TRUE

  ↳70 QDP

    ↳71 QDPOrderProof (⇔)

    ↳72 QDP

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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, L)) → MARK(U12(tt, L))
ACTIVE(U11(tt, L)) → U12¹(tt, L)
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(U12(tt, L)) → S(length(L))
ACTIVE(U12(tt, L)) → LENGTH(L)
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(U21(tt, IL, M, N)) → U22¹(tt, IL, M, N)
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
ACTIVE(U22(tt, IL, M, N)) → U23¹(tt, IL, M, N)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U23(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(U23(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
ACTIVE(length(cons(N, L))) → U11¹(tt, L)
ACTIVE(take(0, IL)) → MARK(nil)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → U21¹(tt, IL, M, N)
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(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(U12(X1, X2)) → U12¹(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
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(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U21(X1, X2, X3, X4)) → U21¹(mark(X1), X2, X3, X4)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U22(X1, X2, X3, X4)) → U22¹(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → U23¹(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
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(X1), X2) → U11¹(X1, X2)
U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
U21¹(mark(X1), X2, X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, mark(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, mark(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, mark(X4)) → U21¹(X1, X2, X3, X4)
U21¹(active(X1), X2, X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, active(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, active(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, active(X4)) → U21¹(X1, X2, X3, X4)
U22¹(mark(X1), X2, X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, mark(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, mark(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, mark(X4)) → U22¹(X1, X2, X3, X4)
U22¹(active(X1), X2, X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, active(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, active(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, active(X4)) → U22¹(X1, X2, X3, X4)
U23¹(mark(X1), X2, X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, mark(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, mark(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, mark(X4)) → U23¹(X1, X2, X3, X4)
U23¹(active(X1), X2, X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, active(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, active(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, active(X4)) → U23¹(X1, X2, X3, X4)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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 10 SCCs with 24 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2) = TAKE(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [TAKE₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [TAKE₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

TAKE₁: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(7) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2) = TAKE(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [TAKE₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [TAKE₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

TAKE₂: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(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, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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:

U23¹(X1, mark(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(mark(X1), X2, X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, mark(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, mark(X4)) → U23¹(X1, X2, X3, X4)
U23¹(active(X1), X2, X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, active(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, active(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, active(X4)) → U23¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

U23¹(mark(X1), X2, X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(active(X1), X2, X3, X4) → U23¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1, x2, x3, x4) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(14) Obligation:

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

U23¹(X1, mark(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, mark(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, mark(X4)) → U23¹(X1, X2, X3, X4)
U23¹(X1, active(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, active(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, active(X4)) → U23¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

U23¹(X1, mark(X2), X3, X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, active(X2), X3, X4) → U23¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1, x2, x3, x4) = x2
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(16) Obligation:

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

U23¹(X1, X2, mark(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, mark(X4)) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, active(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, active(X4)) → U23¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(X1, X2, mark(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, mark(X4)) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, active(X3), X4) → U23¹(X1, X2, X3, X4)
U23¹(X1, X2, X3, active(X4)) → U23¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1, x2, x3, x4) = U23¹(x2, x3, x4)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U23^1₃, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U23^1₃, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U23^1₃: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(18) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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

U22¹(X1, mark(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(mark(X1), X2, X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, mark(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, mark(X4)) → U22¹(X1, X2, X3, X4)
U22¹(active(X1), X2, X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, active(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, active(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, active(X4)) → U22¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

U22¹(mark(X1), X2, X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(active(X1), X2, X3, X4) → U22¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2, x3, x4) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(23) Obligation:

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

U22¹(X1, mark(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, mark(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, mark(X4)) → U22¹(X1, X2, X3, X4)
U22¹(X1, active(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, active(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, active(X4)) → U22¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, mark(X2), X3, X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, active(X2), X3, X4) → U22¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2, x3, x4) = x2
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(25) Obligation:

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

U22¹(X1, X2, mark(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, mark(X4)) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, active(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, active(X4)) → U22¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, X2, mark(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, mark(X4)) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, active(X3), X4) → U22¹(X1, X2, X3, X4)
U22¹(X1, X2, X3, active(X4)) → U22¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2, x3, x4) = U22¹(x2, x3, x4)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U22^1₃, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U22^1₃, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U22^1₃: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(27) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

U21¹(X1, mark(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(mark(X1), X2, X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, mark(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, mark(X4)) → U21¹(X1, X2, X3, X4)
U21¹(active(X1), X2, X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, active(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, active(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, active(X4)) → U21¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2, X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(active(X1), X2, X3, X4) → U21¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2, x3, x4) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(32) Obligation:

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

U21¹(X1, mark(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, mark(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, mark(X4)) → U21¹(X1, X2, X3, X4)
U21¹(X1, active(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, active(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, active(X4)) → U21¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2), X3, X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, active(X2), X3, X4) → U21¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2, x3, x4) = x2
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(34) Obligation:

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

U21¹(X1, X2, mark(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, mark(X4)) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, active(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, active(X4)) → U21¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, X2, mark(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, mark(X4)) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, active(X3), X4) → U21¹(X1, X2, X3, X4)
U21¹(X1, X2, X3, active(X4)) → U21¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2, x3, x4) = U21¹(x2, x3, x4)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U21^1₃, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U21^1₃, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U21^1₃: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(36) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(37) PisEmptyProof (EQUIVALENT transformation)

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

(38) TRUE

(39) 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, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1) = LENGTH(x1)
active(x1) = active(x1)
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

[U21, U22, U23, take] > [active₁, mark₁] > zeros > [LENGTH₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

LENGTH₁: [1]
active₁: [1]
mark₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(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, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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:

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, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

S(active(X)) → S(X)
S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
active(x1) = active(x1)
mark(x1) = mark(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

[U21, U22, U23, take] > [active₁, mark₁] > zeros > [S₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

S₁: [1]
active₁: [1]
mark₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(46) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(47) PisEmptyProof (EQUIVALENT transformation)

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

(48) TRUE

(49) Obligation:

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

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2) = U12¹(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U12^1₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U12^1₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U12^1₁: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(51) Obligation:

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

U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2) = U12¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U12^1₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U12^1₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U12^1₂: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(53) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(54) PisEmptyProof (EQUIVALENT transformation)

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

(55) TRUE

(56) Obligation:

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

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U11^1₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U11^1₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U11^1₁: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(58) Obligation:

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

U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [U11^1₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [U11^1₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

U11^1₂: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(60) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(61) PisEmptyProof (EQUIVALENT transformation)

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

(62) TRUE

(63) 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, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [CONS₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [CONS₁, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

CONS₁: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(65) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

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

zeros > [CONS₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]
[U21, U22, U23, take] > [CONS₂, mark₁, active₁, cons, 0, U11, tt, U12, s, length, nil]

Status:

CONS₂: multiset
mark₁: [1]
active₁: [1]
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U12: []
s: []
length: []
U21: []
U22: []
U23: []
take: []
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(67) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(68) PisEmptyProof (EQUIVALENT transformation)

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

(69) TRUE

(70) Obligation:

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

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
MARK(U12(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(length(X)) → MARK(X)
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK
cons(x1, x2) = cons
ACTIVE(x1) = x1
mark(x1) = mark(x1)
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12
zeros = zeros
0 = 0
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
active(x1) = active(x1)
nil = nil

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

tt > [MARK, U11, U12, zeros, length, U21, U22, U23, take] > [mark₁, active₁] > cons > [0, nil]
tt > [MARK, U11, U12, zeros, length, U21, U22, U23, take] > [mark₁, active₁] > s > [0, nil]

Status:

MARK: multiset
cons: multiset
mark₁: multiset
U11: multiset
tt: multiset
U12: multiset
zeros: multiset
0: multiset
s: multiset
length: multiset
U21: multiset
U22: multiset
U23: multiset
take: multiset
active₁: multiset
nil: multiset

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(72) Obligation:

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

ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(length(X)) → MARK(X)
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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