Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
ACTIVE(sel(s(X), cons(Y, Z))) → SEL(X, Z)
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
ACTIVE(first(0, Z)) → MARK(nil)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
ACTIVE(first(s(X), cons(Y, Z))) → CONS(Y, first(X, Z))
ACTIVE(first(s(X), cons(Y, Z))) → FIRST(X, Z)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
ACTIVE(sel1(s(X), cons(Y, Z))) → SEL1(X, Z)
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
ACTIVE(sel1(0, cons(X, Z))) → QUOTE(X)
ACTIVE(first1(0, Z)) → MARK(nil1)
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
ACTIVE(first1(s(X), cons(Y, Z))) → CONS1(quote(Y), first1(X, Z))
ACTIVE(first1(s(X), cons(Y, Z))) → QUOTE(Y)
ACTIVE(first1(s(X), cons(Y, Z))) → FIRST1(X, Z)
ACTIVE(quote(0)) → MARK(01)
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
ACTIVE(quote1(cons(X, Z))) → CONS1(quote(X), quote1(Z))
ACTIVE(quote1(cons(X, Z))) → QUOTE(X)
ACTIVE(quote1(cons(X, Z))) → QUOTE1(Z)
ACTIVE(quote1(nil)) → MARK(nil1)
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
ACTIVE(quote(s(X))) → S1(quote(X))
ACTIVE(quote(s(X))) → QUOTE(X)
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
ACTIVE(quote(sel(X, Z))) → SEL1(X, Z)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
ACTIVE(quote1(first(X, Z))) → FIRST1(X, Z)
ACTIVE(unquote(01)) → MARK(0)
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
ACTIVE(unquote(s1(X))) → S(unquote(X))
ACTIVE(unquote(s1(X))) → UNQUOTE(X)
ACTIVE(unquote1(nil1)) → MARK(nil)
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(unquote1(cons1(X, Z))) → FCONS(unquote(X), unquote1(Z))
ACTIVE(unquote1(cons1(X, Z))) → UNQUOTE(X)
ACTIVE(unquote1(cons1(X, Z))) → UNQUOTE1(Z)
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
ACTIVE(fcons(X, Z)) → CONS(X, Z)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(0) → ACTIVE(0)
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
MARK(first(X1, X2)) → FIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(nil) → ACTIVE(nil)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(from(X)) → FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → SEL1(mark(X1), mark(X2))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(quote(X)) → ACTIVE(quote(X))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(first1(X1, X2)) → FIRST1(mark(X1), mark(X2))
MARK(first1(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(nil1) → ACTIVE(nil1)
MARK(cons1(X1, X2)) → ACTIVE(cons1(mark(X1), mark(X2)))
MARK(cons1(X1, X2)) → CONS1(mark(X1), mark(X2))
MARK(cons1(X1, X2)) → MARK(X1)
MARK(cons1(X1, X2)) → MARK(X2)
MARK(01) → ACTIVE(01)
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(s1(X)) → ACTIVE(s1(mark(X)))
MARK(s1(X)) → S1(mark(X))
MARK(s1(X)) → MARK(X)
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
MARK(unquote(X)) → UNQUOTE(mark(X))
MARK(unquote(X)) → MARK(X)
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
MARK(unquote1(X)) → UNQUOTE1(mark(X))
MARK(unquote1(X)) → MARK(X)
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(fcons(X1, X2)) → FCONS(mark(X1), mark(X2))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(fcons(X1, X2)) → MARK(X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(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)
FIRST(mark(X1), X2) → FIRST(X1, X2)
FIRST(X1, mark(X2)) → FIRST(X1, X2)
FIRST(active(X1), X2) → FIRST(X1, X2)
FIRST(X1, active(X2)) → FIRST(X1, X2)
FROM(mark(X)) → FROM(X)
FROM(active(X)) → FROM(X)
SEL1(mark(X1), X2) → SEL1(X1, X2)
SEL1(X1, mark(X2)) → SEL1(X1, X2)
SEL1(active(X1), X2) → SEL1(X1, X2)
SEL1(X1, active(X2)) → SEL1(X1, X2)
QUOTE(mark(X)) → QUOTE(X)
QUOTE(active(X)) → QUOTE(X)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST1(active(X1), X2) → FIRST1(X1, X2)
FIRST1(X1, active(X2)) → FIRST1(X1, X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)
CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(active(X1), X2) → CONS1(X1, X2)
CONS1(X1, active(X2)) → CONS1(X1, X2)
QUOTE1(mark(X)) → QUOTE1(X)
QUOTE1(active(X)) → QUOTE1(X)
S1(mark(X)) → S1(X)
S1(active(X)) → S1(X)
UNQUOTE(mark(X)) → UNQUOTE(X)
UNQUOTE(active(X)) → UNQUOTE(X)
UNQUOTE1(mark(X)) → UNQUOTE1(X)
UNQUOTE1(active(X)) → UNQUOTE1(X)
FCONS(mark(X1), X2) → FCONS(X1, X2)
FCONS(X1, mark(X2)) → FCONS(X1, X2)
FCONS(active(X1), X2) → FCONS(X1, X2)
FCONS(X1, active(X2)) → FCONS(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 15 SCCs with 46 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(X1, active(X2)) → FCONS(X1, X2)

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

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

trivial

Status:

FCONS₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(active(X1), X2) → FCONS(X1, X2)

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

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

trivial

Status:

FCONS₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

FCONS(X1, mark(X2)) → FCONS(X1, X2)
FCONS(mark(X1), X2) → FCONS(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(X1, mark(X2)) → FCONS(X1, X2)

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

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

mark₁ > FCONS₁

Status:

FCONS₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

FCONS(mark(X1), X2) → FCONS(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(mark(X1), X2) → FCONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[FCONS₂, mark₁]

Status:

FCONS₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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

UNQUOTE1(active(X)) → UNQUOTE1(X)
UNQUOTE1(mark(X)) → UNQUOTE1(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE1(active(X)) → UNQUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UNQUOTE1(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

UNQUOTE1(mark(X)) → UNQUOTE1(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE1(mark(X)) → UNQUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UNQUOTE1(x1)  =  x1
mark(x1)  =  mark(x1)

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

trivial

Status:

mark₁: multiset

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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

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

UNQUOTE(active(X)) → UNQUOTE(X)
UNQUOTE(mark(X)) → UNQUOTE(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE(active(X)) → UNQUOTE(X)

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

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

UNQUOTE(mark(X)) → UNQUOTE(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE(mark(X)) → UNQUOTE(X)

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

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

S1(active(X)) → S1(X)
S1(mark(X)) → S1(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(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.

S1(active(X)) → S1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S1(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

S1(mark(X)) → S1(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(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.

S1(mark(X)) → S1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S1(x1)  =  x1
mark(x1)  =  mark(x1)

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

QUOTE1(active(X)) → QUOTE1(X)
QUOTE1(mark(X)) → QUOTE1(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE1(active(X)) → QUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOTE1(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

QUOTE1(mark(X)) → QUOTE1(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE1(mark(X)) → QUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOTE1(x1)  =  x1
mark(x1)  =  mark(x1)

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)
CONS1(active(X1), X2) → CONS1(X1, X2)
CONS1(X1, active(X2)) → CONS1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS1(X1, active(X2)) → CONS1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS1(x1, x2)  =  CONS1(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

CONS1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)
CONS1(active(X1), X2) → CONS1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS1(active(X1), X2) → CONS1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS1(x1, x2)  =  CONS1(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

CONS1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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.

CONS1(X1, mark(X2)) → CONS1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS1(x1, x2)  =  CONS1(x2)
mark(x1)  =  mark(x1)

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

mark₁ > CONS1₁

Status:

CONS1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

CONS1(mark(X1), X2) → CONS1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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.

CONS1(mark(X1), X2) → CONS1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[CONS1₂, mark₁]

Status:

CONS1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) Obligation:

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

FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
FIRST1(active(X1), X2) → FIRST1(X1, X2)
FIRST1(X1, active(X2)) → FIRST1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST1(X1, active(X2)) → FIRST1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST1(x1, x2)  =  FIRST1(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

FIRST1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
FIRST1(active(X1), X2) → FIRST1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST1(active(X1), X2) → FIRST1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST1(x1, x2)  =  FIRST1(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

FIRST1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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.

FIRST1(X1, mark(X2)) → FIRST1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST1(x1, x2)  =  FIRST1(x2)
mark(x1)  =  mark(x1)

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

mark₁ > FIRST1₁

Status:

FIRST1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

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

FIRST1(mark(X1), X2) → FIRST1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST1(mark(X1), X2) → FIRST1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[FIRST1₂, mark₁]

Status:

FIRST1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(63) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) Obligation:

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

QUOTE(active(X)) → QUOTE(X)
QUOTE(mark(X)) → QUOTE(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE(active(X)) → QUOTE(X)

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

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

QUOTE(mark(X)) → QUOTE(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE(mark(X)) → QUOTE(X)

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

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) Obligation:

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

SEL1(X1, mark(X2)) → SEL1(X1, X2)
SEL1(mark(X1), X2) → SEL1(X1, X2)
SEL1(active(X1), X2) → SEL1(X1, X2)
SEL1(X1, active(X2)) → SEL1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(X1, active(X2)) → SEL1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL1(x1, x2)  =  SEL1(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

SEL1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

SEL1(X1, mark(X2)) → SEL1(X1, X2)
SEL1(mark(X1), X2) → SEL1(X1, X2)
SEL1(active(X1), X2) → SEL1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(active(X1), X2) → SEL1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL1(x1, x2)  =  SEL1(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

SEL1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

SEL1(X1, mark(X2)) → SEL1(X1, X2)
SEL1(mark(X1), X2) → SEL1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(X1, mark(X2)) → SEL1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL1(x1, x2)  =  SEL1(x2)
mark(x1)  =  mark(x1)

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

mark₁ > SEL1₁

Status:

SEL1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(79) Obligation:

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

SEL1(mark(X1), X2) → SEL1(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(mark(X1), X2) → SEL1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[SEL1₂, mark₁]

Status:

SEL1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(82) PisEmptyProof (EQUIVALENT transformation)

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

(83) TRUE

(84) Obligation:

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

FROM(active(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(active(X)) → FROM(X)

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

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(86) Obligation:

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

FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(mark(X)) → FROM(X)

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

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(89) PisEmptyProof (EQUIVALENT transformation)

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

(90) TRUE

(91) Obligation:

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

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

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(X1, active(X2)) → FIRST(X1, X2)

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

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

trivial

Status:

FIRST₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(93) Obligation:

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

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

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(active(X1), X2) → FIRST(X1, X2)

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

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

trivial

Status:

FIRST₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

FIRST(X1, mark(X2)) → FIRST(X1, X2)
FIRST(mark(X1), X2) → FIRST(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(X1, mark(X2)) → FIRST(X1, X2)

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

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

mark₁ > FIRST₁

Status:

FIRST₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(97) Obligation:

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

FIRST(mark(X1), X2) → FIRST(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(mark(X1), X2) → FIRST(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[FIRST₂, mark₁]

Status:

FIRST₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(99) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(100) PisEmptyProof (EQUIVALENT transformation)

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

(101) TRUE

(102) 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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(103) 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)  =  CONS(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

CONS₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(104) 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)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(105) 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)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

CONS₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(106) 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)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(107) 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)

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

mark₁ > CONS₁

Status:

CONS₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

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

[CONS₂, mark₁]

Status:

CONS₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(110) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(111) PisEmptyProof (EQUIVALENT transformation)

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

(112) TRUE

(113) 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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(114) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(115) 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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(117) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(118) PisEmptyProof (EQUIVALENT transformation)

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

(119) TRUE

(120) Obligation:

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

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

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, active(X2)) → SEL(X1, X2)

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

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

trivial

Status:

SEL₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

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

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(active(X1), X2) → SEL(X1, X2)

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

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

trivial

Status:

SEL₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(125) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

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

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

mark₁ > SEL₁

Status:

SEL₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(126) Obligation:

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

SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[SEL₂, mark₁]

Status:

SEL₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

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

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(129) PisEmptyProof (EQUIVALENT transformation)

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

(130) TRUE

(131) Obligation:

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

MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(from(X)) → MARK(X)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(quote(X)) → ACTIVE(quote(X))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(first1(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(cons1(X1, X2)) → ACTIVE(cons1(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(cons1(X1, X2)) → MARK(X1)
MARK(cons1(X1, X2)) → MARK(X2)
MARK(quote1(X)) → ACTIVE(quote1(X))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(s1(X)) → ACTIVE(s1(mark(X)))
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(s1(X)) → MARK(X)
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(unquote(X)) → MARK(X)
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
MARK(unquote1(X)) → MARK(X)
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(fcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s1(X)) → ACTIVE(s1(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
sel(x1, x2)  =  sel
ACTIVE(x1)  =  x1
mark(x1)  =  mark
s(x1)  =  s
cons(x1, x2)  =  cons
0  =  0
first(x1, x2)  =  first
from(x1)  =  from
sel1(x1, x2)  =  sel1
quote(x1)  =  quote
first1(x1, x2)  =  first1
cons1(x1, x2)  =  cons1
quote1(x1)  =  quote1
s1(x1)  =  s1
unquote(x1)  =  unquote
unquote1(x1)  =  unquote1
fcons(x1, x2)  =  fcons
active(x1)  =  x1
nil  =  nil
nil1  =  nil1
01  =  01

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

0 > [MARK, sel, s, cons, first, from, sel1, quote, first1, cons1, quote1, unquote, unquote1, fcons] > [mark, s1, nil, nil1, 01]

Status:

MARK: multiset
sel: multiset
mark: multiset
s: multiset
cons: multiset
0: multiset
first: multiset
from: multiset
sel1: multiset
quote: multiset
first1: multiset
cons1: multiset
quote1: multiset
s1: multiset
unquote: multiset
unquote1: multiset
fcons: multiset
nil: multiset
nil1: multiset
01: multiset

The following usable rules [FROCOS05] were oriented:

sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
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)
first(X1, mark(X2)) → first(X1, X2)
first(mark(X1), X2) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(active(X)) → from(X)
from(mark(X)) → from(X)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(active(X)) → quote(X)
quote(mark(X)) → quote(X)
first1(X1, mark(X2)) → first1(X1, X2)
first1(mark(X1), X2) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(active(X)) → quote1(X)
quote1(mark(X)) → quote1(X)
s1(active(X)) → s1(X)
s1(mark(X)) → s1(X)
unquote(active(X)) → unquote(X)
unquote(mark(X)) → unquote(X)
unquote1(active(X)) → unquote1(X)
unquote1(mark(X)) → unquote1(X)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

(133) Obligation:

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

MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(from(X)) → MARK(X)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(quote(X)) → ACTIVE(quote(X))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(first1(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(cons1(X1, X2)) → ACTIVE(cons1(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(cons1(X1, X2)) → MARK(X1)
MARK(cons1(X1, X2)) → MARK(X2)
MARK(quote1(X)) → ACTIVE(quote1(X))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(s1(X)) → MARK(X)
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(unquote(X)) → MARK(X)
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
MARK(unquote1(X)) → MARK(X)
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(fcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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

(134) 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(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons1(X1, X2)) → ACTIVE(cons1(mark(X1), mark(X2)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
sel(x1, x2)  =  sel
ACTIVE(x1)  =  x1
mark(x1)  =  mark
s(x1)  =  s
cons(x1, x2)  =  cons
0  =  0
first(x1, x2)  =  first
from(x1)  =  from
sel1(x1, x2)  =  sel1
quote(x1)  =  quote
first1(x1, x2)  =  first1
cons1(x1, x2)  =  cons1
quote1(x1)  =  quote1
s1(x1)  =  s1
unquote(x1)  =  unquote
unquote1(x1)  =  unquote1
fcons(x1, x2)  =  fcons
active(x1)  =  active(x1)
nil  =  nil
nil1  =  nil1
01  =  01

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

s1 > mark > active₁ > [MARK, sel, first, from, sel1, quote, first1, quote1, unquote, unquote1, fcons] > s > cons1 > [nil, nil1]
s1 > mark > active₁ > [MARK, sel, first, from, sel1, quote, first1, quote1, unquote, unquote1, fcons] > cons > cons1 > [nil, nil1]
s1 > mark > active₁ > [MARK, sel, first, from, sel1, quote, first1, quote1, unquote, unquote1, fcons] > 0 > 01 > [nil, nil1]

Status:

MARK: []
sel: []
mark: multiset
s: []
cons: multiset
0: multiset
first: []
from: []
sel1: []
quote: []
first1: []
cons1: multiset
quote1: []
s1: multiset
unquote: []
unquote1: []
fcons: []
active₁: multiset
nil: multiset
nil1: multiset
01: multiset

The following usable rules [FROCOS05] were oriented:

sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
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)
first(X1, mark(X2)) → first(X1, X2)
first(mark(X1), X2) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(active(X)) → from(X)
from(mark(X)) → from(X)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(active(X)) → quote(X)
quote(mark(X)) → quote(X)
first1(X1, mark(X2)) → first1(X1, X2)
first1(mark(X1), X2) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(active(X)) → quote1(X)
quote1(mark(X)) → quote1(X)
unquote(active(X)) → unquote(X)
unquote(mark(X)) → unquote(X)
unquote1(active(X)) → unquote1(X)
unquote1(mark(X)) → unquote1(X)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

(135) Obligation:

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

MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(s(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(from(X)) → MARK(X)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(quote(X)) → ACTIVE(quote(X))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(first1(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(cons1(X1, X2)) → MARK(X1)
MARK(cons1(X1, X2)) → MARK(X2)
MARK(quote1(X)) → ACTIVE(quote1(X))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(s1(X)) → MARK(X)
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(unquote(X)) → MARK(X)
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
MARK(unquote1(X)) → MARK(X)
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(fcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(X2)))
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)

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