Termination w.r.t. Q proof of /tmp/tmpcW_YBs/OvConsOS_nosorts_noand

Termination w.r.t. Q of the given QTRS could be disproven:

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 DependencyPairsProof (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 QDP

↳7 MRRProof (⇔)

↳8 QDP

↳9 DependencyGraphProof (⇔)

↳10 AND

  ↳11 QDP

    ↳12 MRRProof (⇔)

    ↳13 QDP

    ↳14 UsableRulesProof (⇔)

    ↳15 QDP

    ↳16 QDPSizeChangeProof (⇔)

    ↳17 TRUE

  ↳18 QDP

    ↳19 Narrowing (⇔)

    ↳20 QDP

    ↳21 DependencyGraphProof (⇔)

    ↳22 QDP

    ↳23 Narrowing (⇔)

    ↳24 QDP

    ↳25 DependencyGraphProof (⇔)

    ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 QDPOrderProof (⇔)

    ↳34 QDP

    ↳35 QDPOrderProof (⇔)

    ↳36 QDP

    ↳37 QDPOrderProof (⇔)

    ↳38 QDP

    ↳39 QDPOrderProof (⇔)

    ↳40 QDP

    ↳41 NonTerminationProof (⇔)

    ↳42 FALSE

(0) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(tt, L)
a__take(0, IL) → nil
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U21(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U22(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U23(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U21(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U22(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U23(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 1
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__length(nil) → 0
a__take(0, IL) → nil

(2) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

Q is empty.

(3) DependencyPairsProof (EQUIVALENT transformation)

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

(4) Obligation:

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

A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__U12(tt, L) → MARK(L)
A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(6) Obligation:

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

A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → MARK(L)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(7) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

A__U12(tt, L) → MARK(L)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__LENGTH(x₁)) = 1 + x₁
POL(A__TAKE(x₁, x₂)) = x₁ + x₂
POL(A__U11(x₁, x₂)) = 1 + x₁ + x₂
POL(A__U12(x₁, x₂)) = 1 + x₁ + x₂
POL(A__U21(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(A__U22(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(A__U23(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 2 + x₁ + x₂
POL(U12(x₁, x₂)) = 2 + x₁ + x₂
POL(U21(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U22(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U23(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U12(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U21(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U22(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U23(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__length(x₁)) = 2 + x₁
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(length(x₁)) = 2 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

(8) Obligation:

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

A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(10) Complex Obligation (AND)

(11) Obligation:

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

A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(12) MRRProof (EQUIVALENT transformation)

A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(A__TAKE(x₁, x₂)) = 4 + x₁ + x₂
POL(A__U21(x₁, x₂, x₃, x₄)) = 3 + x₁ + x₂ + x₃ + x₄
POL(A__U22(x₁, x₂, x₃, x₄)) = 2 + x₁ + x₂ + x₃ + x₄
POL(A__U23(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U21(x₁, x₂, x₃, x₄)) = 5 + x₁ + x₂ + x₃ + x₄
POL(U22(x₁, x₂, x₃, x₄)) = 5 + x₁ + x₂ + x₃ + x₄
POL(U23(x₁, x₂, x₃, x₄)) = 5 + x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U21(x₁, x₂, x₃, x₄)) = 5 + x₁ + x₂ + x₃ + x₄
POL(a__U22(x₁, x₂, x₃, x₄)) = 5 + x₁ + x₂ + x₃ + x₄
POL(a__U23(x₁, x₂, x₃, x₄)) = 5 + x₁ + x₂ + x₃ + x₄
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 5 + x₁ + x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 5 + x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

(13) Obligation:

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

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

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(14) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(15) Obligation:

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

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

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

(16) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MARK(cons(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1

(17) TRUE

(18) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(19) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A__U12(tt, L) → A__LENGTH(mark(L)) at position [0] we obtained the following new rules [LPAR04]:

A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, 0) → A__LENGTH(0)
A__U12(tt, tt) → A__LENGTH(tt)
A__U12(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U12(tt, nil) → A__LENGTH(nil)

(20) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, 0) → A__LENGTH(0)
A__U12(tt, tt) → A__LENGTH(tt)
A__U12(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U12(tt, nil) → A__LENGTH(nil)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(22) Obligation:

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

A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(23) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A__U12(tt, zeros) → A__LENGTH(a__zeros) at position [0] we obtained the following new rules [LPAR04]:

A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U12(tt, zeros) → A__LENGTH(zeros)

(24) Obligation:

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

A__U11(tt, L) → A__U12(tt, L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U12(tt, zeros) → A__LENGTH(zeros)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(26) Obligation:

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

A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__U12(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(tt) =
/ 0 \

\ 0 /

POL(U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(A__LENGTH(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(a__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(mark(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__length(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(zeros) =
/ 0 \

\ 0 /

POL(0) =
/ 0 \

\ 0 /

POL(a__zeros) =
/ 0 \

\ 1 /

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(nil) =
/ 0 \

\ 1 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(28) Obligation:

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

A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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.

A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(tt) =
/ 0 \

\ 0 /

POL(U11(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 0 /

· x₂

POL(A__LENGTH(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(a__U11(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 1 /

· x₂

POL(mark(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 1 \

\ 1 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(a__length(x₁)) =
/ 1 \

\ 1 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(zeros) =
/ 0 \

\ 0 /

POL(0) =
/ 1 \

\ 0 /

POL(a__zeros) =
/ 1 \

\ 0 /

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U12(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(nil) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(30) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__LENGTH(x₁)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(tt) =
/ 0 \

\ 0 /

POL(A__U12(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__length(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(mark(x₁)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(U21(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__take(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(zeros) =
/ 0 \

\ 0 /

POL(0) =
/ 1 \

\ 0 /

POL(a__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__zeros) =
/ 0 \

\ 0 /

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(nil) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(32) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__LENGTH(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(tt) =
/ 0 \

\ 0 /

POL(A__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__length(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(mark(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 1 /

· x₁

POL(U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(zeros) =
/ 1 \

\ 1 /

POL(0) =
/ 1 \

\ 1 /

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U11(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__zeros) =
/ 0 \

\ 0 /

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U11(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(nil) =
/ 1 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(34) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__LENGTH(x₁)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(tt) =
/ 1 \

\ 0 /

POL(A__U12(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(mark(x₁)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__take(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(zeros) =
/ 0 \

\ 1 /

POL(0) =
/ 0 \

\ 1 /

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__length(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__U11(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__zeros) =
/ 0 \

\ 0 /

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U21(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U11(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(nil) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(36) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__LENGTH(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(tt) =
/ 0 \

\ 0 /

POL(A__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(U22(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(mark(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(U23(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__take(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(zeros) =
/ 0 \

\ 0 /

POL(0) =
/ 1 \

\ 0 /

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__length(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(a__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(a__zeros) =
/ 0 \

\ 0 /

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U21(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(nil) =
/ 1 \

\ 1 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(38) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__LENGTH(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂

POL(A__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(tt) =
/ 0 \

\ 0 /

POL(A__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(U22(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U22(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(mark(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 1 /

· x₁

POL(take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(a__take(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(zeros) =
/ 0 \

\ 1 /

POL(0) =
/ 1 \

\ 0 /

POL(a__U21(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__U23(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(a__length(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__zeros) =
/ 1 \

\ 1 /

POL(a__U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U23(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U21(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U12(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(length(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U11(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(nil) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
mark(s(X)) → s(mark(X))
mark(tt) → tt
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))

(40) Obligation:

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

A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(41) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = A__U11(tt, zeros) evaluates to t =A__U11(tt, zeros)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

A__U11(tt, zeros) → A__U12(tt, zeros)
with rule A__U11(tt, L) → A__U12(tt, L) at position [] and matcher [L / zeros]

A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
with rule A__U12(tt, zeros) → A__LENGTH(cons(0, zeros)) at position [] and matcher [ ]

A__LENGTH(cons(0, zeros)) → A__U11(tt, zeros)
with rule A__LENGTH(cons(N, L)) → A__U11(tt, L)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) MRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Complex Obligation (AND)

(11) Obligation:

(12) MRRProof (EQUIVALENT transformation)

(13) Obligation:

(14) UsableRulesProof (EQUIVALENT transformation)

(15) Obligation:

(16) QDPSizeChangeProof (EQUIVALENT transformation)

(17) TRUE

(18) Obligation:

(19) Narrowing (EQUIVALENT transformation)

(20) Obligation:

(21) DependencyGraphProof (EQUIVALENT transformation)

(22) Obligation:

(23) Narrowing (EQUIVALENT transformation)

(24) Obligation:

(25) DependencyGraphProof (EQUIVALENT transformation)

(26) Obligation:

(27) QDPOrderProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) QDPOrderProof (EQUIVALENT transformation)

(40) Obligation:

(41) NonTerminationProof (EQUIVALENT transformation)

(42) FALSE