Termination w.r.t. Q proof of /tmp/tmpoimYte/Ex1_GL02a

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 QDPOrderProof (⇔)

    ↳18 QDP

    ↳19 QDPOrderProof (⇔)

    ↳20 QDP

    ↳21 PisEmptyProof (⇔)

    ↳22 TRUE

  ↳23 QDP

    ↳24 QDPOrderProof (⇔)

    ↳25 QDP

    ↳26 QDPOrderProof (⇔)

    ↳27 QDP

    ↳28 QDPOrderProof (⇔)

    ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 PisEmptyProof (⇔)

    ↳33 TRUE

  ↳34 QDP

    ↳35 QDPOrderProof (⇔)

    ↳36 QDP

    ↳37 QDPOrderProof (⇔)

    ↳38 QDP

    ↳39 PisEmptyProof (⇔)

    ↳40 TRUE

  ↳41 QDP

    ↳42 QDPOrderProof (⇔)

    ↳43 QDP

    ↳44 QDPOrderProof (⇔)

    ↳45 QDP

    ↳46 PisEmptyProof (⇔)

    ↳47 TRUE

  ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 QDPOrderProof (⇔)

    ↳52 QDP

    ↳53 QDPOrderProof (⇔)

    ↳54 QDP

    ↳55 QDPOrderProof (⇔)

    ↳56 QDP

    ↳57 PisEmptyProof (⇔)

    ↳58 TRUE

  ↳59 QDP

    ↳60 QDPOrderProof (⇔)

    ↳61 QDP

    ↳62 QDPOrderProof (⇔)

    ↳63 QDP

    ↳64 QDPOrderProof (⇔)

    ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 QDPOrderProof (⇔)

    ↳69 QDP

    ↳70 DependencyGraphProof (⇔)

    ↳71 TRUE

(0) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(eq(0, 0)) → MARK(true)
ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))
ACTIVE(eq(s(X), s(Y))) → EQ(X, Y)
ACTIVE(eq(X, Y)) → MARK(false)
ACTIVE(inf(X)) → MARK(cons(X, inf(s(X))))
ACTIVE(inf(X)) → CONS(X, inf(s(X)))
ACTIVE(inf(X)) → INF(s(X))
ACTIVE(inf(X)) → S(X)
ACTIVE(take(0, X)) → MARK(nil)
ACTIVE(take(s(X), cons(Y, L))) → MARK(cons(Y, take(X, L)))
ACTIVE(take(s(X), cons(Y, L))) → CONS(Y, take(X, L))
ACTIVE(take(s(X), cons(Y, L))) → TAKE(X, L)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(X, L))) → MARK(s(length(L)))
ACTIVE(length(cons(X, L))) → S(length(L))
ACTIVE(length(cons(X, L))) → LENGTH(L)
MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))
MARK(0) → ACTIVE(0)
MARK(true) → ACTIVE(true)
MARK(s(X)) → ACTIVE(s(X))
MARK(false) → ACTIVE(false)
MARK(inf(X)) → ACTIVE(inf(mark(X)))
MARK(inf(X)) → INF(mark(X))
MARK(inf(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(X1, X2))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(nil) → ACTIVE(nil)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
EQ(mark(X1), X2) → EQ(X1, X2)
EQ(X1, mark(X2)) → EQ(X1, X2)
EQ(active(X1), X2) → EQ(X1, X2)
EQ(X1, active(X2)) → EQ(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
INF(mark(X)) → INF(X)
INF(active(X)) → INF(X)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 19 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(active(X)) → LENGTH(X)

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

Recursive Path Order [RPO].
Precedence:

[LENGTH₁, active₁]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > LENGTH₁

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[TAKE₁, mark₁]

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

Recursive Path Order [RPO].
Precedence:

mark₁ > TAKE₂

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(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.

TAKE(active(X1), X2) → TAKE(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[TAKE₁, active₁]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, active(X2)) → TAKE(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(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.

CONS(X1, mark(X2)) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[CONS₁, mark₁]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

mark₁ > CONS₂

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[CONS₁, active₁]

The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

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

CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) Obligation:

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

INF(active(X)) → INF(X)
INF(mark(X)) → INF(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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.

INF(active(X)) → INF(X)

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

Recursive Path Order [RPO].
Precedence:

[INF₁, active₁]

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

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

INF(mark(X)) → INF(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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.

INF(mark(X)) → INF(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > INF₁

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

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

Recursive Path Order [RPO].
Precedence:

[S₁, active₁]

The following usable rules [FROCOS05] were oriented: none

(43) 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(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(44) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > S₁

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

EQ(X1, mark(X2)) → EQ(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[EQ₁, mark₁]

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

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

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

EQ(mark(X1), X2) → EQ(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

mark₁ > EQ₂

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

EQ(active(X1), X2) → EQ(X1, X2)
EQ(X1, active(X2)) → EQ(X1, X2)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

EQ(active(X1), X2) → EQ(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[EQ₁, active₁]

The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

EQ(X1, active(X2)) → EQ(X1, X2)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

EQ(X1, active(X2)) → EQ(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(56) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(57) PisEmptyProof (EQUIVALENT transformation)

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

(58) TRUE

(59) Obligation:

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

ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))
MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))
ACTIVE(inf(X)) → MARK(cons(X, inf(s(X))))
MARK(s(X)) → ACTIVE(s(X))
ACTIVE(take(s(X), cons(Y, L))) → MARK(cons(Y, take(X, L)))
MARK(inf(X)) → ACTIVE(inf(mark(X)))
ACTIVE(length(cons(X, L))) → MARK(s(length(L)))
MARK(inf(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(X1, X2))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(inf(X)) → MARK(cons(X, inf(s(X))))
ACTIVE(take(s(X), cons(Y, L))) → MARK(cons(Y, take(X, L)))
ACTIVE(length(cons(X, L))) → MARK(s(length(L)))
MARK(inf(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
eq(x1, x2) = eq
s(x1) = s
MARK(x1) = x1
inf(x1) = inf(x1)
cons(x1, x2) = cons(x1)
take(x1, x2) = take(x1, x2)
mark(x1) = x1
length(x1) = x1
false = false
active(x1) = x1
true = true
0 = 0
nil = nil

Recursive Path Order [RPO].
Precedence:

eq > false
eq > true
inf₁ > cons₁ > s
take₂ > cons₁ > s
take₂ > nil > 0

The following usable rules [FROCOS05] were oriented:

mark(false) → active(false)
mark(true) → active(true)
mark(0) → active(0)
active(take(0, X)) → mark(nil)
active(length(nil)) → mark(0)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
active(eq(0, 0)) → mark(true)
active(eq(X, Y)) → mark(false)
length(active(X)) → length(X)
length(mark(X)) → length(X)
mark(length(X)) → active(length(mark(X)))
mark(s(X)) → active(s(X))
active(inf(X)) → mark(cons(X, inf(s(X))))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(X1, X2))
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
active(eq(s(X), s(Y))) → mark(eq(X, Y))
mark(inf(X)) → active(inf(mark(X)))
active(length(cons(X, L))) → mark(s(length(L)))
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
inf(active(X)) → inf(X)
inf(mark(X)) → inf(X)
eq(active(X1), X2) → eq(X1, X2)
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
mark(nil) → active(nil)

(61) Obligation:

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

ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))
MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))
MARK(s(X)) → ACTIVE(s(X))
MARK(inf(X)) → ACTIVE(inf(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(X1, X2))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE
eq(x1, x2) = eq
s(x1) = s(x1)
MARK(x1) = x1
inf(x1) = x1
mark(x1) = x1
cons(x1, x2) = cons
take(x1, x2) = x2
length(x1) = x1
false = false
active(x1) = active(x1)
true = true
0 = 0
nil = nil

Recursive Path Order [RPO].
Precedence:

nil > 0 > active₁ > s₁ > [ACTIVE, eq, cons]
nil > 0 > active₁ > false > [ACTIVE, eq, cons]
nil > 0 > active₁ > true > [ACTIVE, eq, cons]

The following usable rules [FROCOS05] were oriented:

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

(63) Obligation:

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

ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))
MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))
MARK(inf(X)) → ACTIVE(inf(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(X1, X2))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(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.

MARK(inf(X)) → ACTIVE(inf(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(X1, X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE
eq(x1, x2) = eq
s(x1) = s
MARK(x1) = x1
inf(x1) = inf
mark(x1) = x1
cons(x1, x2) = cons(x2)
take(x1, x2) = take
length(x1) = length(x1)
false = false
active(x1) = active(x1)
true = true
0 = 0
nil = nil

Recursive Path Order [RPO].
Precedence:

inf > active₁ > [0, nil] > [ACTIVE, eq, cons₁, take, true] > false > s
length₁ > active₁ > [0, nil] > [ACTIVE, eq, cons₁, take, true] > false > s

The following usable rules [FROCOS05] were oriented:

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

(65) Obligation:

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

ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))
MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE
eq(x1, x2) = eq
s(x1) = s
MARK(x1) = x1
take(x1, x2) = take
mark(x1) = mark(x1)
false = false
active(x1) = active(x1)
true = true
0 = 0
nil = nil
length(x1) = length
inf(x1) = inf
cons(x1, x2) = cons(x1)

Recursive Path Order [RPO].
Precedence:

take > [ACTIVE, eq] > false > [mark₁, active₁, 0, nil, length] > s
take > [ACTIVE, eq] > false > [mark₁, active₁, 0, nil, length] > true
take > cons₁ > [mark₁, active₁, 0, nil, length] > s
take > cons₁ > [mark₁, active₁, 0, nil, length] > true
inf > cons₁ > [mark₁, active₁, 0, nil, length] > s
inf > cons₁ > [mark₁, active₁, 0, nil, length] > true

The following usable rules [FROCOS05] were oriented:

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

(67) Obligation:

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

ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))
MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(eq(s(X), s(Y))) → MARK(eq(X, Y))

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

Recursive Path Order [RPO].
Precedence:

s₁ > eq₂

The following usable rules [FROCOS05] were oriented:

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

(69) Obligation:

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

MARK(eq(X1, X2)) → ACTIVE(eq(X1, X2))

The TRS R consists of the following rules:

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
mark(eq(X1, X2)) → active(eq(X1, X2))
mark(0) → active(0)
mark(true) → active(true)
mark(s(X)) → active(s(X))
mark(false) → active(false)
mark(inf(X)) → active(inf(mark(X)))
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(length(X)) → active(length(mark(X)))
eq(mark(X1), X2) → eq(X1, X2)
eq(X1, mark(X2)) → eq(X1, X2)
eq(active(X1), X2) → eq(X1, X2)
eq(X1, active(X2)) → eq(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
inf(mark(X)) → inf(X)
inf(active(X)) → inf(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

(70) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) QDPOrderProof (EQUIVALENT transformation)

(9) Obligation:

(10) PisEmptyProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) PisEmptyProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) QDPOrderProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPOrderProof (EQUIVALENT transformation)

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) PisEmptyProof (EQUIVALENT transformation)

(33) TRUE

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) PisEmptyProof (EQUIVALENT transformation)

(40) TRUE

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) PisEmptyProof (EQUIVALENT transformation)

(47) TRUE

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) QDPOrderProof (EQUIVALENT transformation)

(52) Obligation:

(53) QDPOrderProof (EQUIVALENT transformation)

(54) Obligation:

(55) QDPOrderProof (EQUIVALENT transformation)

(56) Obligation:

(57) PisEmptyProof (EQUIVALENT transformation)

(58) TRUE

(59) Obligation:

(60) QDPOrderProof (EQUIVALENT transformation)

(61) Obligation:

(62) QDPOrderProof (EQUIVALENT transformation)

(63) Obligation:

(64) QDPOrderProof (EQUIVALENT transformation)

(65) Obligation:

(66) QDPOrderProof (EQUIVALENT transformation)

(67) Obligation:

(68) QDPOrderProof (EQUIVALENT transformation)

(69) Obligation:

(70) DependencyGraphProof (EQUIVALENT transformation)

(71) TRUE