Termination w.r.t. Q proof of /tmp/tmp2DmtJB/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 PisEmptyProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 QDPOrderProof (⇔)

    ↳18 QDP

    ↳19 PisEmptyProof (⇔)

    ↳20 TRUE

  ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 QDPOrderProof (⇔)

    ↳25 QDP

    ↳26 PisEmptyProof (⇔)

    ↳27 TRUE

  ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 PisEmptyProof (⇔)

    ↳34 TRUE

  ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 PisEmptyProof (⇔)

    ↳41 TRUE

  ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 PisEmptyProof (⇔)

    ↳48 TRUE

  ↳49 QDP

    ↳50 QDPOrderProof (⇔)

    ↳51 QDP

    ↳52 QDPOrderProof (⇔)

    ↳53 QDP

    ↳54 PisEmptyProof (⇔)

    ↳55 TRUE

  ↳56 QDP

    ↳57 QDPOrderProof (⇔)

    ↳58 QDP

    ↳59 QDPOrderProof (⇔)

    ↳60 QDP

    ↳61 PisEmptyProof (⇔)

    ↳62 TRUE

  ↳63 QDP

    ↳64 QDPOrderProof (⇔)

    ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 PisEmptyProof (⇔)

    ↳69 TRUE

  ↳70 QDP

    ↳71 QDPOrderProof (⇔)

    ↳72 QDP

    ↳73 QDPOrderProof (⇔)

    ↳74 QDP

    ↳75 QDPOrderProof (⇔)

    ↳76 QDP

    ↳77 PisEmptyProof (⇔)

    ↳78 TRUE

  ↳79 QDP

    ↳80 QDPOrderProof (⇔)

    ↳81 QDP

    ↳82 QDPOrderProof (⇔)

    ↳83 QDP

    ↳84 PisEmptyProof (⇔)

    ↳85 TRUE

  ↳86 QDP

(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 12 SCCs with 31 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(ok(X1), ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)

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

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

[active₁, zeros, proper₁] > length₁ > [TAKE₁, ok₁, cons₁, 0, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]
tt > length₁ > [TAKE₁, ok₁, cons₁, 0, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]

Status:

TAKE₁: [1]
ok₁: [1]
active₁: [1]
zeros: []
cons₁: [1]
0: []
tt: []
U12₁: [1]
s₁: [1]
length₁: [1]
U21₁: [1]
U22₁: [1]
U23₁: [1]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) 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)

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

TAKE₁ > [0, nil, ok, top]
active₁ > zeros > [0, nil, ok, top]
active₁ > [cons₂, U11₂, tt, U12₂, s₁, length₁, U21₄, U22₄, U23₄, take₂] > mark₁ > [0, nil, ok, top]

Status:

TAKE₁: [1]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [1,2]
tt: []
U12₂: [1,2]
s₁: [1]
length₁: [1]
U21₄: [1,2,3,4]
U22₄: [1,2,3,4]
U23₄: [1,2,4,3]
take₂: [1,2]
nil: []
ok: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(9) Obligation:

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

TAKE(X1, mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

zeros > [active₁, cons₂, 0, U12₂, U21₄, U22₄, take₂, proper₁] > U11₂ > [mark₁, tt, nil, ok₁, top]
zeros > [active₁, cons₂, 0, U12₂, U21₄, U22₄, take₂, proper₁] > U23₄ > [mark₁, tt, nil, ok₁, top]

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
U21₄: [4,2,3,1]
U22₄: [4,3,2,1]
U23₄: [1,4,3,2]
take₂: [2,1]
nil: []
proper₁: [1]
ok₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(11) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

U23¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U23¹(X1, X2, X3, X4)
U23¹(mark(X1), X2, X3, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(ok(X1), ok(X2), ok(X3), ok(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(x4)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x2)
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12(x2)
s(x1) = s(x1)
length(x1) = length(x1)
U21(x1, x2, x3, x4) = U21(x1)
U22(x1, x2, x3, x4) = U22(x1)
U23(x1, x2, x3, x4) = U23(x1)
take(x1, x2) = x2
nil = nil
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, zeros, proper₁] > length₁ > [U23^1₁, ok₁, mark, cons₁, 0, U11₁, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]
tt > length₁ > [U23^1₁, ok₁, mark, cons₁, 0, U11₁, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]

Status:

U23^1₁: [1]
ok₁: [1]
mark: []
active₁: [1]
zeros: []
cons₁: [1]
0: []
U11₁: [1]
tt: []
U12₁: [1]
s₁: [1]
length₁: [1]
U21₁: [1]
U22₁: [1]
U23₁: [1]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(16) Obligation:

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

U23¹(mark(X1), X2, X3, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

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

U23^1₄ > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > zeros > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > U11₂ > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > [U21₄, U22₄, U23₄] > [mark₁, 0, tt, nil, top]

Status:

U23^1₄: [1,4,3,2]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
U21₄: [4,1,3,2]
U22₄: [4,3,2,1]
U23₄: [1,4,3,2]
take₂: [1,2]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(18) Obligation:

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

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

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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

U22¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U22¹(X1, X2, X3, X4)
U22¹(mark(X1), X2, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(ok(X1), ok(X2), ok(X3), ok(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(x4)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x2)
0 = 0
U11(x1, x2) = U11(x1)
tt = tt
U12(x1, x2) = U12(x2)
s(x1) = s(x1)
length(x1) = length(x1)
U21(x1, x2, x3, x4) = U21(x1)
U22(x1, x2, x3, x4) = U22(x1)
U23(x1, x2, x3, x4) = U23(x1)
take(x1, x2) = x2
nil = nil
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, zeros, proper₁] > length₁ > [U22^1₁, ok₁, mark, cons₁, 0, U11₁, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]
tt > length₁ > [U22^1₁, ok₁, mark, cons₁, 0, U11₁, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]

Status:

U22^1₁: [1]
ok₁: [1]
mark: []
active₁: [1]
zeros: []
cons₁: [1]
0: []
U11₁: [1]
tt: []
U12₁: [1]
s₁: [1]
length₁: [1]
U21₁: [1]
U22₁: [1]
U23₁: [1]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(23) Obligation:

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

U22¹(mark(X1), X2, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

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

U22^1₄ > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > zeros > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > U11₂ > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > [U21₄, U22₄, U23₄] > [mark₁, 0, tt, nil, top]

Status:

U22^1₄: [1,4,3,2]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
U21₄: [4,1,3,2]
U22₄: [4,3,2,1]
U23₄: [1,4,3,2]
take₂: [1,2]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(25) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(26) PisEmptyProof (EQUIVALENT transformation)

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

(27) TRUE

(28) Obligation:

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

U21¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U21¹(X1, X2, X3, X4)
U21¹(mark(X1), X2, X3, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

[active₁, zeros, proper₁] > length₁ > [U21^1₁, ok₁, mark, cons₁, 0, U11₁, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]
tt > length₁ > [U21^1₁, ok₁, mark, cons₁, 0, U11₁, U12₁, s₁, U21₁, U22₁, U23₁, nil, top]

Status:

U21^1₁: [1]
ok₁: [1]
mark: []
active₁: [1]
zeros: []
cons₁: [1]
0: []
U11₁: [1]
tt: []
U12₁: [1]
s₁: [1]
length₁: [1]
U21₁: [1]
U22₁: [1]
U23₁: [1]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(30) Obligation:

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

U21¹(mark(X1), X2, X3, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

U21^1₄ > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > zeros > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > U11₂ > [mark₁, 0, tt, nil, top]
[active₁, cons₂, U12₂, take₂, proper₁] > [U21₄, U22₄, U23₄] > [mark₁, 0, tt, nil, top]

Status:

U21^1₄: [1,4,3,2]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
U21₄: [4,1,3,2]
U22₄: [4,3,2,1]
U23₄: [1,4,3,2]
take₂: [1,2]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(32) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(33) PisEmptyProof (EQUIVALENT transformation)

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

(34) TRUE

(35) Obligation:

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

LENGTH(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

LENGTH₁ > [0, nil]
top > active₁ > [tt, U12₂, s₁, length₁, take₁] > [ok₁, zeros] > [0, nil]
top > proper₁ > [tt, U12₂, s₁, length₁, take₁] > [ok₁, zeros] > [0, nil]

Status:

LENGTH₁: [1]
ok₁: [1]
active₁: [1]
zeros: []
0: []
tt: []
U12₂: [1,2]
s₁: [1]
length₁: [1]
take₁: [1]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(37) Obligation:

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

LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

LENGTH₁ > [mark₁, 0, tt, nil, ok₁, top]
[active₁, cons₂, U12₂, take₂, proper₁] > zeros > [mark₁, 0, tt, nil, ok₁, top]
[active₁, cons₂, U12₂, take₂, proper₁] > U11₂ > [mark₁, 0, tt, nil, ok₁, top]
[active₁, cons₂, U12₂, take₂, proper₁] > [U21₄, U22₄, U23₄] > [mark₁, 0, tt, nil, ok₁, top]

Status:

LENGTH₁: [1]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
U21₄: [4,1,3,2]
U22₄: [4,3,2,1]
U23₄: [1,4,3,2]
take₂: [2,1]
nil: []
proper₁: [1]
ok₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(39) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(40) PisEmptyProof (EQUIVALENT transformation)

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

(41) TRUE

(42) Obligation:

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

S(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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

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

S₁ > [0, nil]
top > active₁ > [tt, U12₂, s₁, length₁, take₁] > [ok₁, zeros] > [0, nil]
top > proper₁ > [tt, U12₂, s₁, length₁, take₁] > [ok₁, zeros] > [0, nil]

Status:

S₁: [1]
ok₁: [1]
active₁: [1]
zeros: []
0: []
tt: []
U12₂: [1,2]
s₁: [1]
length₁: [1]
take₁: [1]
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(44) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

S₁ > [mark₁, 0, tt, nil, ok₁, top]
[active₁, cons₂, U12₂, take₂, proper₁] > zeros > [mark₁, 0, tt, nil, ok₁, top]
[active₁, cons₂, U12₂, take₂, proper₁] > U11₂ > [mark₁, 0, tt, nil, ok₁, top]
[active₁, cons₂, U12₂, take₂, proper₁] > [U21₄, U22₄, U23₄] > [mark₁, 0, tt, nil, ok₁, top]

Status:

S₁: [1]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
U21₄: [4,1,3,2]
U22₄: [4,3,2,1]
U23₄: [1,4,3,2]
take₂: [2,1]
nil: []
proper₁: [1]
ok₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(46) Obligation:

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

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

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

(47) PisEmptyProof (EQUIVALENT transformation)

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

(48) TRUE

(49) Obligation:

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

U12¹(ok(X1), ok(X2)) → U12¹(X1, X2)
U12¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(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
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1, x2) = U12(x1, x2)
s(x1) = s(x1)
length(x1) = length(x1)
U21(x1, x2, x3, x4) = U21(x1, x2, x3, x4)
U22(x1, x2, x3, x4) = U22(x1, x2, x3, x4)
U23(x1, x2, x3, x4) = U23(x1, x2, x3, x4)
take(x1, x2) = take(x1, x2)
nil = nil
proper(x1) = x1
top(x1) = top

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

[active₁, U11₂, tt, length₁] > zeros > [mark₁, 0, s₁, nil, top]
[active₁, U11₂, tt, length₁] > [cons₂, U21₄, U22₄, U23₄, take₂] > [mark₁, 0, s₁, nil, top]
[active₁, U11₂, tt, length₁] > U12₂ > [mark₁, 0, s₁, nil, top]

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [2,1]
0: []
U11₂: [1,2]
tt: []
U12₂: [1,2]
s₁: [1]
length₁: [1]
U21₄: [1,4,3,2]
U22₄: [1,4,3,2]
U23₄: [1,4,3,2]
take₂: [1,2]
nil: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(51) Obligation:

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

U12¹(ok(X1), ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(ok(X1), ok(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) = x2
ok(x1) = ok(x1)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
U12(x1, x2) = x2
s(x1) = x1
length(x1) = x1
U21(x1, x2, x3, x4) = x1
U22(x1, x2, x3, x4) = x1
U23(x1, x2, x3, x4) = x1
take(x1, x2) = x2
nil = nil
proper(x1) = proper(x1)
top(x1) = top

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

[ok₁, active₁, proper₁] > [zeros, mark, 0, tt, nil, top]

Status:

ok₁: [1]
active₁: [1]
zeros: []
mark: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(53) Obligation:

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

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

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

(54) PisEmptyProof (EQUIVALENT transformation)

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

(55) TRUE

(56) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(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
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1, x2) = U12(x1, x2)
s(x1) = s(x1)
length(x1) = length(x1)
U21(x1, x2, x3, x4) = U21(x1, x2, x3, x4)
U22(x1, x2, x3, x4) = U22(x1, x2, x3, x4)
U23(x1, x2, x3, x4) = U23(x1, x2, x3, x4)
take(x1, x2) = take(x1, x2)
nil = nil
proper(x1) = x1
top(x1) = top

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

[active₁, U11₂, tt, length₁] > zeros > [mark₁, 0, s₁, nil, top]
[active₁, U11₂, tt, length₁] > [cons₂, U21₄, U22₄, U23₄, take₂] > [mark₁, 0, s₁, nil, top]
[active₁, U11₂, tt, length₁] > U12₂ > [mark₁, 0, s₁, nil, top]

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [2,1]
0: []
U11₂: [1,2]
tt: []
U12₂: [1,2]
s₁: [1]
length₁: [1]
U21₄: [1,4,3,2]
U22₄: [1,4,3,2]
U23₄: [1,4,3,2]
take₂: [1,2]
nil: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(58) Obligation:

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

U11¹(ok(X1), ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(ok(X1), ok(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
ok(x1) = ok(x1)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
U12(x1, x2) = x2
s(x1) = x1
length(x1) = x1
U21(x1, x2, x3, x4) = x1
U22(x1, x2, x3, x4) = x1
U23(x1, x2, x3, x4) = x1
take(x1, x2) = x2
nil = nil
proper(x1) = proper(x1)
top(x1) = top

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

[ok₁, active₁, proper₁] > [zeros, mark, 0, tt, nil, top]

Status:

ok₁: [1]
active₁: [1]
zeros: []
mark: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(60) Obligation:

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

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

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

(61) PisEmptyProof (EQUIVALENT transformation)

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

(62) TRUE

(63) Obligation:

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

CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

[active₁, U11₂, tt, length₁] > zeros > [mark₁, 0, s₁, nil, top]
[active₁, U11₂, tt, length₁] > [cons₂, U21₄, U22₄, U23₄, take₂] > [mark₁, 0, s₁, nil, top]
[active₁, U11₂, tt, length₁] > U12₂ > [mark₁, 0, s₁, nil, top]

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [2,1]
0: []
U11₂: [1,2]
tt: []
U12₂: [1,2]
s₁: [1]
length₁: [1]
U21₄: [1,4,3,2]
U22₄: [1,4,3,2]
U23₄: [1,4,3,2]
take₂: [1,2]
nil: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(65) Obligation:

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

CONS(ok(X1), ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

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

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

[ok₁, active₁, proper₁] > [zeros, mark, 0, tt, nil, top]

Status:

ok₁: [1]
active₁: [1]
zeros: []
mark: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(67) Obligation:

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

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

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

(68) PisEmptyProof (EQUIVALENT transformation)

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

(69) TRUE

(70) Obligation:

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

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U21(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U22(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U23(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)

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

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

cons₂ > U11₂ > [U21₄, U22₄, mark, 0, tt, nil, ok, top]
U12₂ > [U21₄, U22₄, mark, 0, tt, nil, ok, top]
[U23₄, take₂] > [U21₄, U22₄, mark, 0, tt, nil, ok, top]
zeros > [U21₄, U22₄, mark, 0, tt, nil, ok, top]

Status:

cons₂: [1,2]
U11₂: [2,1]
U12₂: [2,1]
U21₄: [4,3,2,1]
U22₄: [4,3,1,2]
U23₄: [4,3,2,1]
take₂: [1,2]
zeros: []
mark: []
0: []
tt: []
nil: []
ok: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(72) Obligation:

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

PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(length(X)) → PROPER(X)

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

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

PROPER₁ > [cons₁, 0, U11₁, U12₁, U22₁, U23₁, take₁, nil, ok, top]
zeros > [cons₁, 0, U11₁, U12₁, U22₁, U23₁, take₁, nil, ok, top]
tt > [length₁, active₁, proper₁] > [cons₁, 0, U11₁, U12₁, U22₁, U23₁, take₁, nil, ok, top]

Status:

PROPER₁: [1]
length₁: [1]
active₁: [1]
zeros: []
cons₁: [1]
0: []
U11₁: [1]
tt: []
U12₁: [1]
U22₁: [1]
U23₁: [1]
take₁: [1]
nil: []
proper₁: [1]
ok: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(74) Obligation:

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

PROPER(s(X)) → PROPER(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

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

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

PROPER₁ > [s₁, mark, cons₁, 0, U11₂, U12₂, U21₁, U22₁, U23₁, take₁, nil, ok, top]
[active₁, zeros] > length₁ > [s₁, mark, cons₁, 0, U11₂, U12₂, U21₁, U22₁, U23₁, take₁, nil, ok, top]
tt > length₁ > [s₁, mark, cons₁, 0, U11₂, U12₂, U21₁, U22₁, U23₁, take₁, nil, ok, top]

Status:

PROPER₁: [1]
s₁: [1]
active₁: [1]
zeros: []
mark: []
cons₁: [1]
0: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
length₁: [1]
U21₁: [1]
U22₁: [1]
U23₁: [1]
take₁: [1]
nil: []
ok: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(76) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(77) PisEmptyProof (EQUIVALENT transformation)

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

(78) TRUE

(79) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U22(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U23(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U22(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U23(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)

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

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

ACTIVE₁ > [U11₂, U21₄, U22₄, 0, tt, nil, ok, top]
[cons₂, U12₂, s₁, U23₄, take₂, active₁, zeros] > [U11₂, U21₄, U22₄, 0, tt, nil, ok, top]

Status:

ACTIVE₁: [1]
U11₂: [1,2]
cons₂: [1,2]
U12₂: [1,2]
s₁: [1]
U21₄: [3,2,4,1]
U22₄: [3,2,4,1]
U23₄: [3,2,4,1]
take₂: [2,1]
active₁: [1]
zeros: []
0: []
tt: []
nil: []
ok: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(81) Obligation:

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

ACTIVE(length(X)) → ACTIVE(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(length(X)) → ACTIVE(X)

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

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

ACTIVE₁ > [length₁, mark, cons₁, 0, U11₂, U12₁, s₁, U21₁, U22₁, U23₁, take₁, nil, ok, top]
[active₁, zeros] > [length₁, mark, cons₁, 0, U11₂, U12₁, s₁, U21₁, U22₁, U23₁, take₁, nil, ok, top]
tt > [length₁, mark, cons₁, 0, U11₂, U12₁, s₁, U21₁, U22₁, U23₁, take₁, nil, ok, top]

Status:

ACTIVE₁: [1]
length₁: [1]
active₁: [1]
zeros: []
mark: []
cons₁: [1]
0: []
U11₂: [2,1]
tt: []
U12₁: [1]
s₁: [1]
U21₁: [1]
U22₁: [1]
U23₁: [1]
take₁: [1]
nil: []
ok: []
top: []

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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(83) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(84) PisEmptyProof (EQUIVALENT transformation)

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

(85) TRUE

(86) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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