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

    ↳11 QDP

    ↳12 QDPOrderProof (⇔)

    ↳13 QDP

    ↳14 PisEmptyProof (⇔)

    ↳15 TRUE

  ↳16 QDP

    ↳17 QDPOrderProof (⇔)

    ↳18 QDP

    ↳19 QDPOrderProof (⇔)

    ↳20 QDP

    ↳21 PisEmptyProof (⇔)

    ↳22 TRUE

  ↳23 QDP

    ↳24 QDPOrderProof (⇔)

    ↳25 QDP

    ↳26 QDPOrderProof (⇔)

    ↳27 QDP

    ↳28 QDPOrderProof (⇔)

    ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 PisEmptyProof (⇔)

    ↳33 TRUE

  ↳34 QDP

    ↳35 QDPOrderProof (⇔)

    ↳36 QDP

    ↳37 QDPOrderProof (⇔)

    ↳38 QDP

    ↳39 QDPOrderProof (⇔)

    ↳40 QDP

    ↳41 QDPOrderProof (⇔)

    ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 PisEmptyProof (⇔)

    ↳46 TRUE

  ↳47 QDP

    ↳48 QDPOrderProof (⇔)

    ↳49 QDP

    ↳50 QDPOrderProof (⇔)

    ↳51 QDP

    ↳52 PisEmptyProof (⇔)

    ↳53 TRUE

  ↳54 QDP

    ↳55 QDPOrderProof (⇔)

    ↳56 QDP

    ↳57 QDPOrderProof (⇔)

    ↳58 QDP

    ↳59 PisEmptyProof (⇔)

    ↳60 TRUE

  ↳61 QDP

    ↳62 QDPOrderProof (⇔)

    ↳63 QDP

    ↳64 QDPOrderProof (⇔)

    ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 QDPOrderProof (⇔)

    ↳69 QDP

    ↳70 PisEmptyProof (⇔)

    ↳71 TRUE

  ↳72 QDP

    ↳73 QDPOrderProof (⇔)

    ↳74 QDP

    ↳75 QDPOrderProof (⇔)

    ↳76 QDP

    ↳77 QDPOrderProof (⇔)

    ↳78 QDP

    ↳79 PisEmptyProof (⇔)

    ↳80 TRUE

  ↳81 QDP

    ↳82 QDPOrderProof (⇔)

    ↳83 QDP

    ↳84 QDPOrderProof (⇔)

    ↳85 QDP

    ↳86 QDPOrderProof (⇔)

    ↳87 QDP

    ↳88 PisEmptyProof (⇔)

    ↳89 TRUE

  ↳90 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(mark(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) = x1
mark(x1) = mark(x1)
active(x1) = 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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > s > cons
length > 0 > mark₁ > s > cons
length > 0 > nil > cons
length > U11 > U12 > mark₁ > s > cons
length > tt > U12 > mark₁ > s > cons
take > U21 > U22 > tt > U12 > mark₁ > s > cons
take > U21 > U22 > U23 > mark₁ > s > cons
take > nil > cons

Status:

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

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(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.

(8) 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)

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) = 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) = x1
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

TAKE₂ > cons
zeros > 0 > mark₁ > cons
take > U21 > tt > length > 0 > mark₁ > cons
take > U21 > tt > length > U11 > U12 > mark₁ > cons
take > U21 > tt > length > U11 > U12 > s > cons
take > U21 > tt > U23 > mark₁ > cons
take > nil > 0 > mark₁ > cons

Status:

U21: []
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
TAKE₂: [1,2]
tt: []
cons: []
zeros: []
U23: []
nil: []

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:
The TRS P consists of the following rules:

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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22(x1)
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > 0 > mark₁ > U11₁ > active₁ > cons > TAKE₁
zeros > 0 > mark₁ > U22₁ > active₁ > cons > TAKE₁
length > 0 > mark₁ > U11₁ > active₁ > cons > TAKE₁
length > 0 > mark₁ > U22₁ > active₁ > cons > TAKE₁
length > tt > U12 > mark₁ > U11₁ > active₁ > cons > TAKE₁
length > tt > U12 > mark₁ > U22₁ > active₁ > cons > TAKE₁
length > tt > U12 > s > active₁ > cons > TAKE₁
length > tt > U23 > mark₁ > U11₁ > active₁ > cons > TAKE₁
length > tt > U23 > mark₁ > U22₁ > active₁ > cons > TAKE₁
take > U21 > tt > U12 > mark₁ > U11₁ > active₁ > cons > TAKE₁
take > U21 > tt > U12 > mark₁ > U22₁ > active₁ > cons > TAKE₁
take > U21 > tt > U12 > s > active₁ > cons > TAKE₁
take > U21 > tt > U23 > mark₁ > U11₁ > active₁ > cons > TAKE₁
take > U21 > tt > U23 > mark₁ > U22₁ > active₁ > cons > TAKE₁
take > nil > mark₁ > U11₁ > active₁ > cons > TAKE₁
take > nil > mark₁ > U22₁ > active₁ > cons > TAKE₁

Status:

TAKE₁: [1]
U21: []
U12: []
mark₁: [1]
s: []
length: []
0: []
take: []
active₁: [1]
U22₁: [1]
tt: []
cons: []
zeros: []
U11₁: [1]
U23: []
nil: []

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)

(11) Obligation:

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

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.

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length(x1)
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

TAKE₂ > U11₁
zeros > cons > tt > U12 > s > mark₁ > active₁ > length₁ > U11₁
zeros > 0 > mark₁ > active₁ > length₁ > U11₁
take > U21 > U22 > U23 > cons > tt > U12 > s > mark₁ > active₁ > length₁ > U11₁
take > nil > 0 > mark₁ > active₁ > length₁ > U11₁

Status:

U21: []
U22: []
U12: []
mark₁: [1]
s: []
0: []
take: []
active₁: [1]
TAKE₂: [2,1]
tt: []
cons: []
zeros: []
U11₁: [1]
length₁: [1]
U23: []
nil: []

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)

(13) 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.

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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.

(17) 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) = U23¹(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11(x1)
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(x1)
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

U23^1₁ > nil
zeros > cons > mark₁ > U11₁ > nil
zeros > cons > mark₁ > U23₁ > active₁ > 0 > nil
length > tt > cons > mark₁ > U11₁ > nil
length > tt > cons > mark₁ > U23₁ > active₁ > 0 > nil
length > tt > U12 > mark₁ > U11₁ > nil
length > tt > U12 > mark₁ > U23₁ > active₁ > 0 > nil
length > tt > U12 > s > active₁ > 0 > nil
take > U21 > U22 > tt > cons > mark₁ > U11₁ > nil
take > U21 > U22 > tt > cons > mark₁ > U23₁ > active₁ > 0 > nil
take > U21 > U22 > tt > U12 > mark₁ > U11₁ > nil
take > U21 > U22 > tt > U12 > mark₁ > U23₁ > active₁ > 0 > nil
take > U21 > U22 > tt > U12 > s > active₁ > 0 > nil

Status:

U21: []
U22: []
U12: []
mark₁: [1]
s: []
U23^1₁: [1]
length: []
0: []
take: []
U23₁: [1]
active₁: [1]
tt: []
cons: []
zeros: []
U11₁: [1]
nil: []

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:
The TRS P consists of the following rules:

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, 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.

(19) 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(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, 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(x1, x2, x3, x4)
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21(x1)
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > cons > U11₁ > active₁ > U23^1₄
zeros > mark₁ > cons > U21₁ > active₁ > U23^1₄
zeros > mark₁ > s > U21₁ > active₁ > U23^1₄
length > 0 > nil > mark₁ > cons > U11₁ > active₁ > U23^1₄
length > 0 > nil > mark₁ > cons > U21₁ > active₁ > U23^1₄
length > 0 > nil > mark₁ > s > U21₁ > active₁ > U23^1₄
length > tt > U12 > mark₁ > cons > U11₁ > active₁ > U23^1₄
length > tt > U12 > mark₁ > cons > U21₁ > active₁ > U23^1₄
length > tt > U12 > mark₁ > s > U21₁ > active₁ > U23^1₄
length > tt > U22 > U23 > mark₁ > cons > U11₁ > active₁ > U23^1₄
length > tt > U22 > U23 > mark₁ > cons > U21₁ > active₁ > U23^1₄
length > tt > U22 > U23 > mark₁ > s > U21₁ > active₁ > U23^1₄
take > tt > U12 > mark₁ > cons > U11₁ > active₁ > U23^1₄
take > tt > U12 > mark₁ > cons > U21₁ > active₁ > U23^1₄
take > tt > U12 > mark₁ > s > U21₁ > active₁ > U23^1₄
take > tt > U22 > U23 > mark₁ > cons > U11₁ > active₁ > U23^1₄
take > tt > U22 > U23 > mark₁ > cons > U21₁ > active₁ > U23^1₄
take > tt > U22 > U23 > mark₁ > s > U21₁ > active₁ > U23^1₄
take > nil > mark₁ > cons > U11₁ > active₁ > U23^1₄
take > nil > mark₁ > cons > U21₁ > active₁ > U23^1₄
take > nil > mark₁ > s > U21₁ > active₁ > U23^1₄

Status:

U22: []
U12: []
mark₁: [1]
s: []
length: []
0: []
take: []
active₁: [1]
tt: []
cons: []
zeros: []
U23^1₄: [4,1,2,3]
U11₁: [1]
U23: []
nil: []
U21₁: [1]

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)

(20) 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.

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(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(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.

(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(mark(X1), X2, X3, 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)

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(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

Lexicographic path order with status [LPO].
Precedence:

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

Status:

U21: []
U22: []
U22^1₂: [1,2]
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
active₁: [1]
tt: []
cons: []
zeros: []
U23: []
nil: []

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, X3, mark(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) = 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(x1)
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > active₁ > cons
zeros > mark₁ > active₁ > 0
zeros > mark₁ > active₁ > s
zeros > mark₁ > active₁ > nil
zeros > mark₁ > U21₁
take > tt > length > U11 > U12 > mark₁ > active₁ > cons
take > tt > length > U11 > U12 > mark₁ > active₁ > 0
take > tt > length > U11 > U12 > mark₁ > active₁ > s
take > tt > length > U11 > U12 > mark₁ > active₁ > nil
take > tt > length > U11 > U12 > mark₁ > U21₁
take > tt > U22 > U23 > mark₁ > active₁ > cons
take > tt > U22 > U23 > mark₁ > active₁ > 0
take > tt > U22 > U23 > mark₁ > active₁ > s
take > tt > U22 > U23 > mark₁ > active₁ > nil
take > tt > U22 > U23 > mark₁ > U21₁

Status:

U22: []
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
active₁: [1]
tt: []
cons: []
zeros: []
U23: []
nil: []
U21₁: [1]

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:
The TRS P consists of the following rules:

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

(28) 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)

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(x3)
mark(x1) = mark(x1)
active(x1) = 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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > nil
zeros > cons
length > 0 > mark₁ > nil
length > U11 > U12 > s > mark₁ > nil
length > U11 > U12 > s > tt
take > U21 > U22 > tt
take > U21 > U22 > U23 > mark₁ > nil
take > U21 > U22 > U23 > cons

Status:

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

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)

(29) Obligation:

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

U22¹(X1, X2, active(X3), 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.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, X2, active(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) = U22¹(x3)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
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

Lexicographic path order with status [LPO].
Precedence:

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

Status:

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

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)

(31) 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.

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) 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.

(35) 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(mark(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) = U21¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = x1
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = x1
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > cons > tt > U12 > mark₁ > s > U21^1₂
zeros > 0 > mark₁ > s > U21^1₂
take > U21 > U22 > U23 > cons > tt > U12 > mark₁ > s > U21^1₂
take > nil > 0 > mark₁ > s > U21^1₂

Status:

U21: []
U22: []
U12: []
U21^1₂: [1,2]
mark₁: [1]
s: []
0: []
take: []
tt: []
cons: []
zeros: []
U23: []
nil: []

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:
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(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.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, X2, X3, mark(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) = x4
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = x1
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22(x1)
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > cons > mark₁ > 0 > U22₁
length > tt > U12 > mark₁ > 0 > U22₁
length > tt > U12 > s > U22₁
length > tt > U23 > cons > mark₁ > 0 > U22₁
take > U21 > tt > U12 > mark₁ > 0 > U22₁
take > U21 > tt > U12 > s > U22₁
take > U21 > tt > U23 > cons > mark₁ > 0 > U22₁
take > nil > 0 > U22₁

Status:

U21: []
U12: []
mark₁: [1]
s: []
length: []
0: []
take: []
U22₁: [1]
tt: []
cons: []
zeros: []
U23: []
nil: []

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)

(38) Obligation:

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

U21¹(X1, X2, mark(X3), 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.

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X1), 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) = U21¹(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(x1)
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take(x1, x2)
nil = nil

Lexicographic path order with status [LPO].
Precedence:

U21^1₂ > active₁
zeros > mark₁ > cons > U21₁ > active₁
zeros > mark₁ > take₂ > U21₁ > active₁
zeros > mark₁ > take₂ > nil > active₁
length > 0 > mark₁ > cons > U21₁ > active₁
length > 0 > mark₁ > take₂ > U21₁ > active₁
length > 0 > mark₁ > take₂ > nil > active₁
length > U11 > U12 > s > tt > U22 > U23 > mark₁ > cons > U21₁ > active₁
length > U11 > U12 > s > tt > U22 > U23 > mark₁ > take₂ > U21₁ > active₁
length > U11 > U12 > s > tt > U22 > U23 > mark₁ > take₂ > nil > active₁

Status:

U22: []
U12: []
U21^1₂: [2,1]
U11: []
mark₁: [1]
s: []
length: []
take₂: [1,2]
0: []
active₁: [1]
tt: []
cons: []
zeros: []
U23: []
nil: []
U21₁: [1]

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)

(40) 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, 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.

(41) 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)

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(x1, x2, x3, x4)
mark(x1) = mark(x1)
active(x1) = x1
zeros = zeros
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = x1
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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > U21^1₄
zeros > mark₁ > cons
zeros > mark₁ > s
length > 0 > mark₁ > U21^1₄
length > 0 > mark₁ > cons
length > 0 > mark₁ > s
length > tt > U12 > mark₁ > U21^1₄
length > tt > U12 > mark₁ > cons
length > tt > U12 > mark₁ > s
length > tt > U23 > mark₁ > U21^1₄
length > tt > U23 > mark₁ > cons
length > tt > U23 > mark₁ > s
take > U21 > tt > U12 > mark₁ > U21^1₄
take > U21 > tt > U12 > mark₁ > cons
take > U21 > tt > U12 > mark₁ > s
take > U21 > tt > U23 > mark₁ > U21^1₄
take > U21 > tt > U23 > mark₁ > cons
take > U21 > tt > U23 > mark₁ > s
take > U21 > U22 > U23 > mark₁ > U21^1₄
take > U21 > U22 > U23 > mark₁ > cons
take > U21 > U22 > U23 > mark₁ > s
take > nil > mark₁ > U21^1₄
take > nil > mark₁ > cons
take > nil > mark₁ > s

Status:

U21: []
U22: []
U12: []
mark₁: [1]
U21^1₄: [2,1,3,4]
s: []
length: []
0: []
take: []
tt: []
cons: []
zeros: []
U23: []
nil: []

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)

(42) Obligation:

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

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.

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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(x1, x2, x3, x4)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12(x1)
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23(x1)
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

U21^1₄ > nil
zeros > cons > mark₁ > active₁ > U23₁ > nil
zeros > cons > mark₁ > U12₁ > nil
zeros > 0 > active₁ > U23₁ > nil
length > 0 > active₁ > U23₁ > nil
length > U11 > tt > cons > mark₁ > active₁ > U23₁ > nil
length > U11 > tt > cons > mark₁ > U12₁ > nil
length > U11 > tt > s > active₁ > U23₁ > nil
take > U21 > U22 > tt > cons > mark₁ > active₁ > U23₁ > nil
take > U21 > U22 > tt > cons > mark₁ > U12₁ > nil
take > U21 > U22 > tt > s > active₁ > U23₁ > nil

Status:

U21: []
U22: []
U11: []
mark₁: [1]
U21^1₄: [4,1,3,2]
s: []
length: []
0: []
take: []
U23₁: [1]
active₁: [1]
tt: []
cons: []
zeros: []
U12₁: [1]
nil: []

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)

(44) 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.

(45) PisEmptyProof (EQUIVALENT transformation)

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

(46) TRUE

(47) 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.

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁
zeros > cons
length > 0 > mark₁
length > 0 > nil
length > U11 > tt > s > mark₁
length > U11 > tt > U23 > mark₁
length > U11 > tt > U23 > cons
take > U21 > tt > s > mark₁
take > U21 > tt > U23 > mark₁
take > U21 > tt > U23 > cons
take > nil

Status:

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

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)

(49) Obligation:

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

LENGTH(active(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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.

LENGTH(active(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1) = LENGTH(x1)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, 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

Lexicographic path order with status [LPO].
Precedence:

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

Status:

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

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:
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.

(52) PisEmptyProof (EQUIVALENT transformation)

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

(53) TRUE

(54) 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.

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁
zeros > cons
length > 0 > mark₁
length > 0 > nil
length > U11 > tt > s > mark₁
length > U11 > tt > U23 > mark₁
length > U11 > tt > U23 > cons
take > U21 > tt > s > mark₁
take > U21 > tt > U23 > mark₁
take > U21 > tt > U23 > cons
take > nil

Status:

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

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)

(56) Obligation:

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

S(active(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, 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

Lexicographic path order with status [LPO].
Precedence:

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

Status:

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

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:
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.

(59) PisEmptyProof (EQUIVALENT transformation)

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

(60) TRUE

(61) 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.

(62) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = 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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > s > cons
length > 0 > mark₁ > s > cons
length > 0 > nil > cons
length > U11 > U12 > mark₁ > s > cons
length > tt > U12 > mark₁ > s > cons
take > U21 > U22 > tt > U12 > mark₁ > s > cons
take > U21 > U22 > U23 > mark₁ > s > cons
take > nil > cons

Status:

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

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)

(63) Obligation:

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

U12¹(X1, mark(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.

(64) 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)

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) = 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) = x1
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

U12^1₂ > cons
zeros > 0 > mark₁ > cons
take > U21 > tt > length > 0 > mark₁ > cons
take > U21 > tt > length > U11 > U12 > mark₁ > cons
take > U21 > tt > length > U11 > U12 > s > cons
take > U21 > tt > U23 > mark₁ > cons
take > nil > 0 > mark₁ > cons

Status:

U21: []
U12^1₂: [1,2]
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
tt: []
cons: []
zeros: []
U23: []
nil: []

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:

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.

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22(x1)
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > 0 > mark₁ > U11₁ > active₁ > cons > U12^1₁
zeros > 0 > mark₁ > U22₁ > active₁ > cons > U12^1₁
length > 0 > mark₁ > U11₁ > active₁ > cons > U12^1₁
length > 0 > mark₁ > U22₁ > active₁ > cons > U12^1₁
length > tt > U12 > mark₁ > U11₁ > active₁ > cons > U12^1₁
length > tt > U12 > mark₁ > U22₁ > active₁ > cons > U12^1₁
length > tt > U12 > s > active₁ > cons > U12^1₁
length > tt > U23 > mark₁ > U11₁ > active₁ > cons > U12^1₁
length > tt > U23 > mark₁ > U22₁ > active₁ > cons > U12^1₁
take > U21 > tt > U12 > mark₁ > U11₁ > active₁ > cons > U12^1₁
take > U21 > tt > U12 > mark₁ > U22₁ > active₁ > cons > U12^1₁
take > U21 > tt > U12 > s > active₁ > cons > U12^1₁
take > U21 > tt > U23 > mark₁ > U11₁ > active₁ > cons > U12^1₁
take > U21 > tt > U23 > mark₁ > U22₁ > active₁ > cons > U12^1₁
take > nil > mark₁ > U11₁ > active₁ > cons > U12^1₁
take > nil > mark₁ > U22₁ > active₁ > cons > U12^1₁

Status:

U21: []
U12: []
mark₁: [1]
s: []
length: []
0: []
take: []
active₁: [1]
U22₁: [1]
tt: []
cons: []
U12^1₁: [1]
zeros: []
U11₁: [1]
U23: []
nil: []

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:
The TRS P consists of the following rules:

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.

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12
s(x1) = s
length(x1) = length(x1)
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

U12^1₂ > U11₁
zeros > cons > tt > U12 > s > mark₁ > active₁ > length₁ > U11₁
zeros > 0 > mark₁ > active₁ > length₁ > U11₁
take > U21 > U22 > U23 > cons > tt > U12 > s > mark₁ > active₁ > length₁ > U11₁
take > nil > 0 > mark₁ > active₁ > length₁ > U11₁

Status:

U21: []
U12^1₂: [2,1]
U22: []
U12: []
mark₁: [1]
s: []
0: []
take: []
active₁: [1]
tt: []
cons: []
zeros: []
U11₁: [1]
length₁: [1]
U23: []
nil: []

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)

(69) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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.

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) Obligation:

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

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.

(73) 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)

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) = 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(x1)
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

U11^1₁ > U21₁
zeros > cons > mark₁ > U21₁
length > 0 > mark₁ > U21₁
length > U11 > U12 > s > mark₁ > U21₁
length > tt > U12 > s > mark₁ > U21₁
length > tt > U22 > U23 > cons > mark₁ > U21₁
take > tt > U12 > s > mark₁ > U21₁
take > tt > U22 > U23 > cons > mark₁ > U21₁
take > nil > 0 > mark₁ > U21₁

Status:

U22: []
U11^1₁: [1]
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
tt: []
cons: []
zeros: []
U23: []
nil: []
U21₁: [1]

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)

(74) 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)
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.

(75) 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) = 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

Lexicographic path order with status [LPO].
Precedence:

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

Status:

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

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)

(76) Obligation:

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

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.

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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) = x2
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
cons(x1, x2) = cons
0 = 0
U11(x1, x2) = U11
tt = tt
U12(x1, x2) = U12(x1)
s(x1) = s
length(x1) = length
U21(x1, x2, x3, x4) = U21
U22(x1, x2, x3, x4) = U22
U23(x1, x2, x3, x4) = U23(x1)
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

zeros > cons > mark₁ > U12₁ > active₁ > nil > 0
zeros > cons > mark₁ > s > active₁ > nil > 0
zeros > cons > mark₁ > U23₁ > active₁ > nil > 0
length > U11 > tt > cons > mark₁ > U12₁ > active₁ > nil > 0
length > U11 > tt > cons > mark₁ > s > active₁ > nil > 0
length > U11 > tt > cons > mark₁ > U23₁ > active₁ > nil > 0
take > U21 > U22 > tt > cons > mark₁ > U12₁ > active₁ > nil > 0
take > U21 > U22 > tt > cons > mark₁ > s > active₁ > nil > 0
take > U21 > U22 > tt > cons > mark₁ > U23₁ > active₁ > nil > 0

Status:

U21: []
U22: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
U23₁: [1]
active₁: [1]
tt: []
cons: []
zeros: []
U12₁: [1]
nil: []

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)

(78) 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.

(79) PisEmptyProof (EQUIVALENT transformation)

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

(80) TRUE

(81) 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.

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

zeros > mark₁ > nil
zeros > cons > nil
length > 0 > mark₁ > nil
length > U11 > U12 > s > mark₁ > nil
length > U11 > U12 > s > tt > nil
take > U21 > U22 > tt > nil
take > U21 > U22 > U23 > mark₁ > nil
take > U21 > U22 > U23 > cons > nil

Status:

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

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)

(83) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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.

(84) 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(x1)
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

CONS₁ > U23₁
zeros > mark₁ > active₁ > 0 > U23₁
zeros > mark₁ > cons > U23₁
zeros > mark₁ > nil > 0 > U23₁
length > U11 > U12 > s > U21 > U22 > tt > mark₁ > active₁ > 0 > U23₁
length > U11 > U12 > s > U21 > U22 > tt > mark₁ > cons > U23₁
length > U11 > U12 > s > U21 > U22 > tt > mark₁ > nil > 0 > U23₁
take > U21 > U22 > tt > mark₁ > active₁ > 0 > U23₁
take > U21 > U22 > tt > mark₁ > cons > U23₁
take > U21 > U22 > tt > mark₁ > nil > 0 > U23₁

Status:

U21: []
U22: []
CONS₁: [1]
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
U23₁: [1]
active₁: [1]
tt: []
cons: []
zeros: []
nil: []

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)

(85) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1, x2)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark(x1)
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(x1)
U23(x1, x2, x3, x4) = U23
take(x1, x2) = take
nil = nil

Lexicographic path order with status [LPO].
Precedence:

CONS₂ > cons
zeros > mark₁ > U22₁ > active₁ > 0 > cons
zeros > mark₁ > nil > active₁ > 0 > cons
length > U11 > U12 > mark₁ > U22₁ > active₁ > 0 > cons
length > U11 > U12 > mark₁ > nil > active₁ > 0 > cons
length > U11 > U12 > s > active₁ > 0 > cons
length > tt > U12 > mark₁ > U22₁ > active₁ > 0 > cons
length > tt > U12 > mark₁ > nil > active₁ > 0 > cons
length > tt > U12 > s > active₁ > 0 > cons
length > tt > U23 > mark₁ > U22₁ > active₁ > 0 > cons
length > tt > U23 > mark₁ > nil > active₁ > 0 > cons
take > U21 > tt > U12 > mark₁ > U22₁ > active₁ > 0 > cons
take > U21 > tt > U12 > mark₁ > nil > active₁ > 0 > cons
take > U21 > tt > U12 > s > active₁ > 0 > cons
take > U21 > tt > U23 > mark₁ > U22₁ > active₁ > 0 > cons
take > U21 > tt > U23 > mark₁ > nil > active₁ > 0 > cons

Status:

U21: []
U12: []
U11: []
mark₁: [1]
s: []
length: []
0: []
take: []
active₁: [1]
U22₁: [1]
tt: []
cons: []
zeros: []
CONS₂: [2,1]
U23: []
nil: []

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)

(87) 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.

(88) PisEmptyProof (EQUIVALENT transformation)

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

(89) TRUE

(90) 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.