Termination w.r.t. Q proof of /tmp/tmp4-zX4H/Ex6_15_AEL02

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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.

↳ QTRS

  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ACTIVE(quote(s(X))) → QUOTE(X)
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
CONS(X1, mark(X2)) → CONS(X1, X2)
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(nil1) → ACTIVE(nil1)
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
MARK(sel1(X1, X2)) → SEL1(mark(X1), mark(X2))
ACTIVE(first(s(X), cons(Y, Z))) → FIRST(X, Z)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(first(X1, X2)) → FIRST(mark(X1), mark(X2))
ACTIVE(sel1(s(X), cons(Y, Z))) → SEL1(X, Z)
QUOTE(mark(X)) → QUOTE(X)
ACTIVE(unquote1(cons1(X, Z))) → UNQUOTE1(Z)
ACTIVE(unquote1(nil1)) → MARK(nil)
ACTIVE(unquote(01)) → MARK(0)
MARK(first(X1, X2)) → MARK(X2)
S1(mark(X)) → S1(X)
ACTIVE(quote(0)) → MARK(01)
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
FIRST1(active(X1), X2) → FIRST1(X1, X2)
QUOTE(active(X)) → QUOTE(X)
CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(X1, active(X2)) → CONS1(X1, X2)
SEL1(X1, active(X2)) → SEL1(X1, X2)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
MARK(s1(X)) → S1(mark(X))
FIRST(X1, mark(X2)) → FIRST(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS1(active(X1), X2) → CONS1(X1, X2)
MARK(s(X)) → MARK(X)
MARK(from(X)) → FROM(mark(X))
MARK(unquote(X)) → MARK(X)
MARK(fcons(X1, X2)) → MARK(X2)
QUOTE1(mark(X)) → QUOTE1(X)
QUOTE1(active(X)) → QUOTE1(X)
ACTIVE(quote1(cons(X, Z))) → QUOTE1(Z)
UNQUOTE1(active(X)) → UNQUOTE1(X)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(unquote1(cons1(X, Z))) → FCONS(unquote(X), unquote1(Z))
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
UNQUOTE1(mark(X)) → UNQUOTE1(X)
ACTIVE(first1(0, Z)) → MARK(nil1)
MARK(cons1(X1, X2)) → CONS1(mark(X1), mark(X2))
ACTIVE(first(0, Z)) → MARK(nil)
ACTIVE(first1(s(X), cons(Y, Z))) → QUOTE(Y)
ACTIVE(unquote(s1(X))) → UNQUOTE(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(quote(sel(X, Z))) → SEL1(X, Z)
ACTIVE(from(X)) → S(X)
S1(active(X)) → S1(X)
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote1(nil)) → MARK(nil1)
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
ACTIVE(first(s(X), cons(Y, Z))) → CONS(Y, first(X, Z))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(quote1(cons(X, Z))) → QUOTE(X)
SEL1(mark(X1), X2) → SEL1(X1, X2)
UNQUOTE(mark(X)) → UNQUOTE(X)
FROM(mark(X)) → FROM(X)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
FROM(active(X)) → FROM(X)
MARK(fcons(X1, X2)) → FCONS(mark(X1), mark(X2))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(01) → ACTIVE(01)
FIRST(active(X1), X2) → FIRST(X1, X2)
SEL1(active(X1), X2) → SEL1(X1, X2)
ACTIVE(unquote1(cons1(X, Z))) → UNQUOTE(X)
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(unquote1(X)) → UNQUOTE1(mark(X))
ACTIVE(fcons(X, Z)) → CONS(X, Z)
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
S(active(X)) → S(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(sel1(0, cons(X, Z))) → QUOTE(X)
ACTIVE(quote(s(X))) → S1(quote(X))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)
FCONS(X1, mark(X2)) → FCONS(X1, X2)
MARK(cons1(X1, X2)) → ACTIVE(cons1(mark(X1), mark(X2)))
MARK(s1(X)) → MARK(X)
FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST(mark(X1), X2) → FIRST(X1, X2)
MARK(sel1(X1, X2)) → MARK(X1)
FCONS(X1, active(X2)) → FCONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
FCONS(active(X1), X2) → FCONS(X1, X2)
MARK(first1(X1, X2)) → MARK(X2)
MARK(fcons(X1, X2)) → MARK(X1)
ACTIVE(from(X)) → FROM(s(X))
SEL(mark(X1), X2) → SEL(X1, X2)
MARK(unquote(X)) → UNQUOTE(mark(X))
MARK(quote(X)) → ACTIVE(quote(X))
SEL(X1, active(X2)) → SEL(X1, X2)
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
ACTIVE(unquote(s1(X))) → S(unquote(X))
MARK(s1(X)) → ACTIVE(s1(mark(X)))
MARK(cons1(X1, X2)) → MARK(X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
ACTIVE(quote1(first(X, Z))) → FIRST1(X, Z)
SEL(active(X1), X2) → SEL(X1, X2)
MARK(cons1(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → FIRST1(mark(X1), mark(X2))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
FCONS(mark(X1), X2) → FCONS(X1, X2)
MARK(first(X1, X2)) → MARK(X1)
MARK(s(X)) → S(mark(X))
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
FIRST1(X1, active(X2)) → FIRST1(X1, X2)
SEL1(X1, mark(X2)) → SEL1(X1, X2)
MARK(unquote1(X)) → MARK(X)
FIRST(X1, active(X2)) → FIRST(X1, X2)
ACTIVE(first1(s(X), cons(Y, Z))) → FIRST1(X, Z)
ACTIVE(first1(s(X), cons(Y, Z))) → CONS1(quote(Y), first1(X, Z))
ACTIVE(quote1(cons(X, Z))) → CONS1(quote(X), quote1(Z))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(0) → ACTIVE(0)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
UNQUOTE(active(X)) → UNQUOTE(X)
ACTIVE(sel(s(X), cons(Y, Z))) → SEL(X, Z)
MARK(nil) → ACTIVE(nil)
ACTIVE(from(X)) → CONS(X, from(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.
The approximation of the Dependency Graph [15,17,22] contains 15 SCCs with 46 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FCONS(X1, mark(X2)) → FCONS(X1, X2)
FCONS(X1, active(X2)) → FCONS(X1, X2)
FCONS(active(X1), 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

FCONS(X1, mark(X2)) → FCONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
FCONS(X1, active(X2)) → FCONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
FCONS(active(X1), X2) → FCONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
FCONS(mark(X1), X2) → FCONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

UNQUOTE1(active(X)) → UNQUOTE1(X)
The graph contains the following edges 1 > 1
UNQUOTE1(mark(X)) → UNQUOTE1(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

UNQUOTE(mark(X)) → UNQUOTE(X)
The graph contains the following edges 1 > 1
UNQUOTE(active(X)) → UNQUOTE(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

S1(active(X)) → S1(X)
The graph contains the following edges 1 > 1
S1(mark(X)) → S1(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

QUOTE1(mark(X)) → QUOTE1(X)
The graph contains the following edges 1 > 1
QUOTE1(active(X)) → QUOTE1(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

CONS1(active(X1), X2) → CONS1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS1(X1, mark(X2)) → CONS1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS1(X1, active(X2)) → CONS1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS1(mark(X1), X2) → CONS1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

FIRST1(X1, active(X2)) → FIRST1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
FIRST1(active(X1), X2) → FIRST1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

QUOTE(mark(X)) → QUOTE(X)
The graph contains the following edges 1 > 1
QUOTE(active(X)) → QUOTE(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

SEL1(X1, mark(X2)) → SEL1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL1(X1, active(X2)) → SEL1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL1(active(X1), X2) → SEL1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
SEL1(mark(X1), X2) → SEL1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FROM(mark(X)) → FROM(X)
FROM(active(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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

FROM(mark(X)) → FROM(X)
The graph contains the following edges 1 > 1
FROM(active(X)) → FROM(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

FIRST(active(X1), X2) → FIRST(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
FIRST(X1, active(X2)) → FIRST(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
FIRST(mark(X1), X2) → FIRST(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
FIRST(X1, mark(X2)) → FIRST(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

CONS(X1, active(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(active(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(X1, mark(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

S(mark(X)) → S(X)
S(active(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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

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

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

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

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

SEL(mark(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
SEL(X1, active(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(X1, mark(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(active(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(s(X)) → MARK(X)
MARK(quote(X)) → ACTIVE(quote(X))
MARK(unquote(X)) → MARK(X)
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quote1(X)) → ACTIVE(quote1(X))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(s1(X)) → ACTIVE(s1(mark(X)))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(cons1(X1, X2)) → MARK(X1)
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(first(X1, X2)) → MARK(X1)
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons1(X1, X2)) → ACTIVE(cons1(mark(X1), mark(X2)))
MARK(first(X1, X2)) → MARK(X2)
MARK(s1(X)) → MARK(X)
MARK(unquote1(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → MARK(X1)
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(first1(X1, X2)) → MARK(X2)
MARK(sel(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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(s1(X)) → ACTIVE(s1(mark(X)))
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.

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

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(01) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(cons1(x₁, x₂)) = 0
POL(fcons(x₁, x₂)) = 1
POL(first(x₁, x₂)) = 1
POL(first1(x₁, x₂)) = 1
POL(from(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(nil1) = 0
POL(quote(x₁)) = 1
POL(quote1(x₁)) = 1
POL(s(x₁)) = 0
POL(s1(x₁)) = 0
POL(sel(x₁, x₂)) = 1
POL(sel1(x₁, x₂)) = 1
POL(unquote(x₁)) = 1
POL(unquote1(x₁)) = 1

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

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

ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(quote(X)) → ACTIVE(quote(X))
MARK(s(X)) → MARK(X)
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(unquote(X)) → MARK(X)
MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2)))
MARK(fcons(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(quote1(X)) → ACTIVE(quote1(X))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(cons1(X1, X2)) → MARK(X1)
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(first(X1, X2)) → MARK(X1)
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(first(X1, X2)) → MARK(X2)
MARK(s1(X)) → MARK(X)
MARK(unquote1(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → MARK(X1)
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(first1(X1, X2)) → MARK(X2)
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(sel(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.
By narrowing [15] the rule MARK(sel1(X1, X2)) → ACTIVE(sel1(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quote1(X)) → ACTIVE(quote1(X))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(unquote1(X)) → ACTIVE(unquote1(mark(X)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(first(X1, X2)) → MARK(X2)
MARK(s1(X)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(X1, X2)) → MARK(X1)
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(first1(X1, X2)) → MARK(X2)
MARK(fcons(X1, X2)) → MARK(X1)
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(quote(X)) → ACTIVE(quote(X))
MARK(s(X)) → MARK(X)
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(unquote(X)) → MARK(X)
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(cons1(X1, X2)) → MARK(X1)
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(unquote1(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

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.
By narrowing [15] the rule MARK(unquote1(X)) → ACTIVE(unquote1(mark(X))) at position [0] we obtained the following new rules:

MARK(unquote1(s1(x0))) → ACTIVE(unquote1(active(s1(mark(x0)))))
MARK(unquote1(quote(x0))) → ACTIVE(unquote1(active(quote(x0))))
MARK(unquote1(x0)) → ACTIVE(unquote1(x0))
MARK(unquote1(first(x0, x1))) → ACTIVE(unquote1(active(first(mark(x0), mark(x1)))))
MARK(unquote1(sel1(x0, x1))) → ACTIVE(unquote1(active(sel1(mark(x0), mark(x1)))))
MARK(unquote1(nil)) → ACTIVE(unquote1(active(nil)))
MARK(unquote1(quote1(x0))) → ACTIVE(unquote1(active(quote1(x0))))
MARK(unquote1(first1(x0, x1))) → ACTIVE(unquote1(active(first1(mark(x0), mark(x1)))))
MARK(unquote1(cons1(x0, x1))) → ACTIVE(unquote1(active(cons1(mark(x0), mark(x1)))))
MARK(unquote1(s(x0))) → ACTIVE(unquote1(active(s(mark(x0)))))
MARK(unquote1(unquote(x0))) → ACTIVE(unquote1(active(unquote(mark(x0)))))
MARK(unquote1(0)) → ACTIVE(unquote1(active(0)))
MARK(unquote1(01)) → ACTIVE(unquote1(active(01)))
MARK(unquote1(nil1)) → ACTIVE(unquote1(active(nil1)))
MARK(unquote1(fcons(x0, x1))) → ACTIVE(unquote1(active(fcons(mark(x0), mark(x1)))))
MARK(unquote1(cons(x0, x1))) → ACTIVE(unquote1(active(cons(mark(x0), x1))))
MARK(unquote1(from(x0))) → ACTIVE(unquote1(active(from(mark(x0)))))
MARK(unquote1(sel(x0, x1))) → ACTIVE(unquote1(active(sel(mark(x0), mark(x1)))))
MARK(unquote1(unquote1(x0))) → ACTIVE(unquote1(active(unquote1(mark(x0)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

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

MARK(unquote1(x0)) → ACTIVE(unquote1(x0))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(unquote1(from(x0))) → ACTIVE(unquote1(active(from(mark(x0)))))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
MARK(unquote1(s1(x0))) → ACTIVE(unquote1(active(s1(mark(x0)))))
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(first(X1, X2)) → MARK(X2)
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(unquote1(cons1(x0, x1))) → ACTIVE(unquote1(active(cons1(mark(x0), mark(x1)))))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(unquote1(fcons(x0, x1))) → ACTIVE(unquote1(active(fcons(mark(x0), mark(x1)))))
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(s(X)) → MARK(X)
MARK(unquote(X)) → MARK(X)
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(unquote1(nil)) → ACTIVE(unquote1(active(nil)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(unquote1(sel(x0, x1))) → ACTIVE(unquote1(active(sel(mark(x0), mark(x1)))))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(unquote1(01)) → ACTIVE(unquote1(active(01)))
MARK(unquote1(nil1)) → ACTIVE(unquote1(active(nil1)))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(unquote1(unquote1(x0))) → ACTIVE(unquote1(active(unquote1(mark(x0)))))
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(quote(x0))) → ACTIVE(unquote1(active(quote(x0))))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(s1(X)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(unquote1(s(x0))) → ACTIVE(unquote1(active(s(mark(x0)))))
MARK(unquote1(unquote(x0))) → ACTIVE(unquote1(active(unquote(mark(x0)))))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(unquote1(cons(x0, x1))) → ACTIVE(unquote1(active(cons(mark(x0), x1))))
MARK(first1(X1, X2)) → MARK(X2)
MARK(fcons(X1, X2)) → MARK(X1)
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(quote(X)) → ACTIVE(quote(X))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(unquote1(quote1(x0))) → ACTIVE(unquote1(active(quote1(x0))))
MARK(unquote1(first1(x0, x1))) → ACTIVE(unquote1(active(first1(mark(x0), mark(x1)))))
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(cons1(X1, X2)) → MARK(X1)
MARK(unquote(X)) → ACTIVE(unquote(mark(X)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(unquote1(first(x0, x1))) → ACTIVE(unquote1(active(first(mark(x0), mark(x1)))))
MARK(unquote1(sel1(x0, x1))) → ACTIVE(unquote1(active(sel1(mark(x0), mark(x1)))))
MARK(unquote1(X)) → MARK(X)
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(unquote1(0)) → ACTIVE(unquote1(active(0)))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

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.
By narrowing [15] the rule MARK(unquote(X)) → ACTIVE(unquote(mark(X))) at position [0] we obtained the following new rules:

MARK(unquote(sel(x0, x1))) → ACTIVE(unquote(active(sel(mark(x0), mark(x1)))))
MARK(unquote(fcons(x0, x1))) → ACTIVE(unquote(active(fcons(mark(x0), mark(x1)))))
MARK(unquote(unquote(x0))) → ACTIVE(unquote(active(unquote(mark(x0)))))
MARK(unquote(0)) → ACTIVE(unquote(active(0)))
MARK(unquote(first1(x0, x1))) → ACTIVE(unquote(active(first1(mark(x0), mark(x1)))))
MARK(unquote(01)) → ACTIVE(unquote(active(01)))
MARK(unquote(cons(x0, x1))) → ACTIVE(unquote(active(cons(mark(x0), x1))))
MARK(unquote(x0)) → ACTIVE(unquote(x0))
MARK(unquote(quote1(x0))) → ACTIVE(unquote(active(quote1(x0))))
MARK(unquote(cons1(x0, x1))) → ACTIVE(unquote(active(cons1(mark(x0), mark(x1)))))
MARK(unquote(sel1(x0, x1))) → ACTIVE(unquote(active(sel1(mark(x0), mark(x1)))))
MARK(unquote(unquote1(x0))) → ACTIVE(unquote(active(unquote1(mark(x0)))))
MARK(unquote(from(x0))) → ACTIVE(unquote(active(from(mark(x0)))))
MARK(unquote(first(x0, x1))) → ACTIVE(unquote(active(first(mark(x0), mark(x1)))))
MARK(unquote(s1(x0))) → ACTIVE(unquote(active(s1(mark(x0)))))
MARK(unquote(nil1)) → ACTIVE(unquote(active(nil1)))
MARK(unquote(s(x0))) → ACTIVE(unquote(active(s(mark(x0)))))
MARK(unquote(quote(x0))) → ACTIVE(unquote(active(quote(x0))))
MARK(unquote(nil)) → ACTIVE(unquote(active(nil)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

MARK(unquote1(x0)) → ACTIVE(unquote1(x0))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(unquote1(from(x0))) → ACTIVE(unquote1(active(from(mark(x0)))))
MARK(unquote(s(x0))) → ACTIVE(unquote(active(s(mark(x0)))))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
MARK(unquote(sel(x0, x1))) → ACTIVE(unquote(active(sel(mark(x0), mark(x1)))))
MARK(unquote1(s1(x0))) → ACTIVE(unquote1(active(s1(mark(x0)))))
MARK(unquote(unquote(x0))) → ACTIVE(unquote(active(unquote(mark(x0)))))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(unquote(01)) → ACTIVE(unquote(active(01)))
MARK(unquote(cons(x0, x1))) → ACTIVE(unquote(active(cons(mark(x0), x1))))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(first(X1, X2)) → MARK(X2)
MARK(unquote(quote1(x0))) → ACTIVE(unquote(active(quote1(x0))))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
MARK(unquote1(cons1(x0, x1))) → ACTIVE(unquote1(active(cons1(mark(x0), mark(x1)))))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(unquote1(fcons(x0, x1))) → ACTIVE(unquote1(active(fcons(mark(x0), mark(x1)))))
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(first1(x0, x1))) → ACTIVE(unquote(active(first1(mark(x0), mark(x1)))))
MARK(unquote(0)) → ACTIVE(unquote(active(0)))
MARK(s(X)) → MARK(X)
MARK(unquote(X)) → MARK(X)
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(unquote(from(x0))) → ACTIVE(unquote(active(from(mark(x0)))))
MARK(unquote(nil1)) → ACTIVE(unquote(active(nil1)))
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(nil)) → ACTIVE(unquote1(active(nil)))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(unquote1(sel(x0, x1))) → ACTIVE(unquote1(active(sel(mark(x0), mark(x1)))))
MARK(unquote(nil)) → ACTIVE(unquote(active(nil)))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
MARK(unquote(fcons(x0, x1))) → ACTIVE(unquote(active(fcons(mark(x0), mark(x1)))))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(unquote(sel1(x0, x1))) → ACTIVE(unquote(active(sel1(mark(x0), mark(x1)))))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(unquote1(01)) → ACTIVE(unquote1(active(01)))
MARK(unquote1(nil1)) → ACTIVE(unquote1(active(nil1)))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
MARK(unquote(quote(x0))) → ACTIVE(unquote(active(quote(x0))))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(unquote1(x0))) → ACTIVE(unquote1(active(unquote1(mark(x0)))))
MARK(unquote1(quote(x0))) → ACTIVE(unquote1(active(quote(x0))))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(unquote(x0)) → ACTIVE(unquote(x0))
MARK(s1(X)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(unquote1(s(x0))) → ACTIVE(unquote1(active(s(mark(x0)))))
MARK(unquote1(unquote(x0))) → ACTIVE(unquote1(active(unquote(mark(x0)))))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(unquote1(cons(x0, x1))) → ACTIVE(unquote1(active(cons(mark(x0), x1))))
MARK(unquote(s1(x0))) → ACTIVE(unquote(active(s1(mark(x0)))))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(quote(X)) → ACTIVE(quote(X))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(unquote(cons1(x0, x1))) → ACTIVE(unquote(active(cons1(mark(x0), mark(x1)))))
MARK(unquote1(quote1(x0))) → ACTIVE(unquote1(active(quote1(x0))))
MARK(unquote1(first1(x0, x1))) → ACTIVE(unquote1(active(first1(mark(x0), mark(x1)))))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(first(x0, x1))) → ACTIVE(unquote(active(first(mark(x0), mark(x1)))))
MARK(cons1(X1, X2)) → MARK(X1)
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(unquote1(first(x0, x1))) → ACTIVE(unquote1(active(first(mark(x0), mark(x1)))))
MARK(unquote1(sel1(x0, x1))) → ACTIVE(unquote1(active(sel1(mark(x0), mark(x1)))))
MARK(unquote1(X)) → MARK(X)
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(unquote1(0)) → ACTIVE(unquote1(active(0)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(unquote(unquote1(x0))) → ACTIVE(unquote(active(unquote1(mark(x0)))))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

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.
By narrowing [15] the rule MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(sel(quote1(x0), y1)) → ACTIVE(sel(active(quote1(x0)), mark(y1)))
MARK(sel(y0, first1(x0, x1))) → ACTIVE(sel(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(sel(y0, fcons(x0, x1))) → ACTIVE(sel(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(y0, sel1(x0, x1))) → ACTIVE(sel(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel(y0, quote1(x0))) → ACTIVE(sel(mark(y0), active(quote1(x0))))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(unquote(x0), y1)) → ACTIVE(sel(active(unquote(mark(x0))), mark(y1)))
MARK(sel(s1(x0), y1)) → ACTIVE(sel(active(s1(mark(x0))), mark(y1)))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(y0, unquote(x0))) → ACTIVE(sel(mark(y0), active(unquote(mark(x0)))))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(mark(x0)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(sel(y0, cons1(x0, x1))) → ACTIVE(sel(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel(unquote1(x0), y1)) → ACTIVE(sel(active(unquote1(mark(x0))), mark(y1)))
MARK(sel(quote(x0), y1)) → ACTIVE(sel(active(quote(x0)), mark(y1)))
MARK(sel(y0, s1(x0))) → ACTIVE(sel(mark(y0), active(s1(mark(x0)))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(fcons(x0, x1), y1)) → ACTIVE(sel(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(sel(sel1(x0, x1), y1)) → ACTIVE(sel(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(first1(x0, x1), y1)) → ACTIVE(sel(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(mark(x0))), mark(y1)))
MARK(sel(y0, nil1)) → ACTIVE(sel(mark(y0), active(nil1)))
MARK(sel(nil1, y1)) → ACTIVE(sel(active(nil1), mark(y1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(y0, first(x0, x1))) → ACTIVE(sel(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(sel(y0, unquote1(x0))) → ACTIVE(sel(mark(y0), active(unquote1(mark(x0)))))
MARK(sel(cons1(x0, x1), y1)) → ACTIVE(sel(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, 01)) → ACTIVE(sel(mark(y0), active(01)))
MARK(sel(y0, quote(x0))) → ACTIVE(sel(mark(y0), active(quote(x0))))
MARK(sel(01, y1)) → ACTIVE(sel(active(01), mark(y1)))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(first(x0, x1), y1)) → ACTIVE(sel(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

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

MARK(unquote1(x0)) → ACTIVE(unquote1(x0))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(sel(y0, fcons(x0, x1))) → ACTIVE(sel(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(y0, sel1(x0, x1))) → ACTIVE(sel(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel(unquote(x0), y1)) → ACTIVE(sel(active(unquote(mark(x0))), mark(y1)))
MARK(sel(s1(x0), y1)) → ACTIVE(sel(active(s1(mark(x0))), mark(y1)))
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(sel(y0, unquote(x0))) → ACTIVE(sel(mark(y0), active(unquote(mark(x0)))))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(unquote1(from(x0))) → ACTIVE(unquote1(active(from(mark(x0)))))
MARK(unquote(s(x0))) → ACTIVE(unquote(active(s(mark(x0)))))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel(y0, cons1(x0, x1))) → ACTIVE(sel(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
MARK(unquote1(s1(x0))) → ACTIVE(unquote1(active(s1(mark(x0)))))
MARK(unquote(sel(x0, x1))) → ACTIVE(unquote(active(sel(mark(x0), mark(x1)))))
MARK(unquote(unquote(x0))) → ACTIVE(unquote(active(unquote(mark(x0)))))
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(unquote(01)) → ACTIVE(unquote(active(01)))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(unquote(cons(x0, x1))) → ACTIVE(unquote(active(cons(mark(x0), x1))))
MARK(sel(sel1(x0, x1), y1)) → ACTIVE(sel(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(first1(x0, x1), y1)) → ACTIVE(sel(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(mark(x0))), mark(y1)))
MARK(unquote(quote1(x0))) → ACTIVE(unquote(active(quote1(x0))))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(unquote1(cons1(x0, x1))) → ACTIVE(unquote1(active(cons1(mark(x0), mark(x1)))))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(sel(y0, unquote1(x0))) → ACTIVE(sel(mark(y0), active(unquote1(mark(x0)))))
MARK(unquote1(fcons(x0, x1))) → ACTIVE(unquote1(active(fcons(mark(x0), mark(x1)))))
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(0)) → ACTIVE(unquote(active(0)))
MARK(unquote(first1(x0, x1))) → ACTIVE(unquote(active(first1(mark(x0), mark(x1)))))
MARK(s(X)) → MARK(X)
MARK(unquote(X)) → MARK(X)
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(y0, quote1(x0))) → ACTIVE(sel(mark(y0), active(quote1(x0))))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(unquote(from(x0))) → ACTIVE(unquote(active(from(mark(x0)))))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(unquote(nil1)) → ACTIVE(unquote(active(nil1)))
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel(quote(x0), y1)) → ACTIVE(sel(active(quote(x0)), mark(y1)))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(unquote1(nil)) → ACTIVE(unquote1(active(nil)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel(nil1, y1)) → ACTIVE(sel(active(nil1), mark(y1)))
MARK(sel(y0, nil1)) → ACTIVE(sel(mark(y0), active(nil1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
MARK(sel(01, y1)) → ACTIVE(sel(active(01), mark(y1)))
MARK(sel(y0, 01)) → ACTIVE(sel(mark(y0), active(01)))
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(unquote1(sel(x0, x1))) → ACTIVE(unquote1(active(sel(mark(x0), mark(x1)))))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(unquote(nil)) → ACTIVE(unquote(active(nil)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))
MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
MARK(unquote(fcons(x0, x1))) → ACTIVE(unquote(active(fcons(mark(x0), mark(x1)))))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(unquote(sel1(x0, x1))) → ACTIVE(unquote(active(sel1(mark(x0), mark(x1)))))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(unquote1(01)) → ACTIVE(unquote1(active(01)))
MARK(unquote1(nil1)) → ACTIVE(unquote1(active(nil1)))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(unquote(quote(x0))) → ACTIVE(unquote(active(quote(x0))))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(unquote1(unquote1(x0))) → ACTIVE(unquote1(active(unquote1(mark(x0)))))
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(quote(x0))) → ACTIVE(unquote1(active(quote(x0))))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(fcons(x0, x1), y1)) → ACTIVE(sel(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(x0)) → ACTIVE(unquote(x0))
MARK(s1(X)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel(y0, first(x0, x1))) → ACTIVE(sel(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(unquote1(s(x0))) → ACTIVE(unquote1(active(s(mark(x0)))))
MARK(unquote1(unquote(x0))) → ACTIVE(unquote1(active(unquote(mark(x0)))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(unquote1(cons(x0, x1))) → ACTIVE(unquote1(active(cons(mark(x0), x1))))
MARK(unquote(s1(x0))) → ACTIVE(unquote(active(s1(mark(x0)))))
MARK(first1(X1, X2)) → MARK(X2)
MARK(fcons(X1, X2)) → MARK(X1)
MARK(sel(first(x0, x1), y1)) → ACTIVE(sel(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel(quote1(x0), y1)) → ACTIVE(sel(active(quote1(x0)), mark(y1)))
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(sel(y0, first1(x0, x1))) → ACTIVE(sel(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(quote(X)) → ACTIVE(quote(X))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(unquote(cons1(x0, x1))) → ACTIVE(unquote(active(cons1(mark(x0), mark(x1)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(unquote1(quote1(x0))) → ACTIVE(unquote1(active(quote1(x0))))
MARK(unquote1(first1(x0, x1))) → ACTIVE(unquote1(active(first1(mark(x0), mark(x1)))))
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(unquote(first(x0, x1))) → ACTIVE(unquote(active(first(mark(x0), mark(x1)))))
MARK(cons1(X1, X2)) → MARK(X1)
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(sel(unquote1(x0), y1)) → ACTIVE(sel(active(unquote1(mark(x0))), mark(y1)))
MARK(first(X1, X2)) → MARK(X1)
MARK(sel(y0, s1(x0))) → ACTIVE(sel(mark(y0), active(s1(mark(x0)))))
MARK(from(X)) → MARK(X)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(unquote1(first(x0, x1))) → ACTIVE(unquote1(active(first(mark(x0), mark(x1)))))
MARK(unquote1(sel1(x0, x1))) → ACTIVE(unquote1(active(sel1(mark(x0), mark(x1)))))
MARK(unquote1(X)) → MARK(X)
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(unquote1(0)) → ACTIVE(unquote1(active(0)))
MARK(sel(cons1(x0, x1), y1)) → ACTIVE(sel(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(unquote1(x0))) → ACTIVE(unquote(active(unquote1(mark(x0)))))
MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2)))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(sel(y0, quote(x0))) → ACTIVE(sel(mark(y0), active(quote(x0))))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

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.
By narrowing [15] the rule MARK(first1(X1, X2)) → ACTIVE(first1(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(first1(y0, cons1(x0, x1))) → ACTIVE(first1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(first1(s1(x0), y1)) → ACTIVE(first1(active(s1(mark(x0))), mark(y1)))
MARK(first1(y0, fcons(x0, x1))) → ACTIVE(first1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(first1(quote1(x0), y1)) → ACTIVE(first1(active(quote1(x0)), mark(y1)))
MARK(first1(sel1(x0, x1), y1)) → ACTIVE(first1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(first1(y0, quote(x0))) → ACTIVE(first1(mark(y0), active(quote(x0))))
MARK(first1(y0, from(x0))) → ACTIVE(first1(mark(y0), active(from(mark(x0)))))
MARK(first1(unquote(x0), y1)) → ACTIVE(first1(active(unquote(mark(x0))), mark(y1)))
MARK(first1(y0, 01)) → ACTIVE(first1(mark(y0), active(01)))
MARK(first1(0, y1)) → ACTIVE(first1(active(0), mark(y1)))
MARK(first1(01, y1)) → ACTIVE(first1(active(01), mark(y1)))
MARK(first1(y0, 0)) → ACTIVE(first1(mark(y0), active(0)))
MARK(first1(first1(x0, x1), y1)) → ACTIVE(first1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(first1(unquote1(x0), y1)) → ACTIVE(first1(active(unquote1(mark(x0))), mark(y1)))
MARK(first1(y0, quote1(x0))) → ACTIVE(first1(mark(y0), active(quote1(x0))))
MARK(first1(y0, sel1(x0, x1))) → ACTIVE(first1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(first1(cons(x0, x1), y1)) → ACTIVE(first1(active(cons(mark(x0), x1)), mark(y1)))
MARK(first1(y0, nil1)) → ACTIVE(first1(mark(y0), active(nil1)))
MARK(first1(nil1, y1)) → ACTIVE(first1(active(nil1), mark(y1)))
MARK(first1(s(x0), y1)) → ACTIVE(first1(active(s(mark(x0))), mark(y1)))
MARK(first1(first(x0, x1), y1)) → ACTIVE(first1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(first1(y0, nil)) → ACTIVE(first1(mark(y0), active(nil)))
MARK(first1(nil, y1)) → ACTIVE(first1(active(nil), mark(y1)))
MARK(first1(fcons(x0, x1), y1)) → ACTIVE(first1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(first1(y0, first(x0, x1))) → ACTIVE(first1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(first1(y0, unquote(x0))) → ACTIVE(first1(mark(y0), active(unquote(mark(x0)))))
MARK(first1(cons1(x0, x1), y1)) → ACTIVE(first1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(first1(y0, x1)) → ACTIVE(first1(mark(y0), x1))
MARK(first1(y0, sel(x0, x1))) → ACTIVE(first1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(first1(x0, y1)) → ACTIVE(first1(x0, mark(y1)))
MARK(first1(sel(x0, x1), y1)) → ACTIVE(first1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first1(quote(x0), y1)) → ACTIVE(first1(active(quote(x0)), mark(y1)))
MARK(first1(from(x0), y1)) → ACTIVE(first1(active(from(mark(x0))), mark(y1)))
MARK(first1(y0, unquote1(x0))) → ACTIVE(first1(mark(y0), active(unquote1(mark(x0)))))
MARK(first1(y0, first1(x0, x1))) → ACTIVE(first1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(first1(y0, s1(x0))) → ACTIVE(first1(mark(y0), active(s1(mark(x0)))))
MARK(first1(y0, s(x0))) → ACTIVE(first1(mark(y0), active(s(mark(x0)))))
MARK(first1(y0, cons(x0, x1))) → ACTIVE(first1(mark(y0), active(cons(mark(x0), x1))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

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

MARK(unquote1(x0)) → ACTIVE(unquote1(x0))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(first1(quote1(x0), y1)) → ACTIVE(first1(active(quote1(x0)), mark(y1)))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(sel(y0, fcons(x0, x1))) → ACTIVE(sel(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(first1(y0, quote(x0))) → ACTIVE(first1(mark(y0), active(quote(x0))))
MARK(sel(y0, sel1(x0, x1))) → ACTIVE(sel(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(first1(unquote(x0), y1)) → ACTIVE(first1(active(unquote(mark(x0))), mark(y1)))
MARK(sel(unquote(x0), y1)) → ACTIVE(sel(active(unquote(mark(x0))), mark(y1)))
MARK(sel(s1(x0), y1)) → ACTIVE(sel(active(s1(mark(x0))), mark(y1)))
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(first1(unquote1(x0), y1)) → ACTIVE(first1(active(unquote1(mark(x0))), mark(y1)))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
MARK(sel(y0, unquote(x0))) → ACTIVE(sel(mark(y0), active(unquote(mark(x0)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(first1(nil1, y1)) → ACTIVE(first1(active(nil1), mark(y1)))
MARK(first1(y0, nil1)) → ACTIVE(first1(mark(y0), active(nil1)))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(first1(s(x0), y1)) → ACTIVE(first1(active(s(mark(x0))), mark(y1)))
MARK(unquote1(from(x0))) → ACTIVE(unquote1(active(from(mark(x0)))))
MARK(unquote(s(x0))) → ACTIVE(unquote(active(s(mark(x0)))))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel(y0, cons1(x0, x1))) → ACTIVE(sel(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
MARK(unquote(sel(x0, x1))) → ACTIVE(unquote(active(sel(mark(x0), mark(x1)))))
MARK(unquote1(s1(x0))) → ACTIVE(unquote1(active(s1(mark(x0)))))
MARK(unquote(unquote(x0))) → ACTIVE(unquote(active(unquote(mark(x0)))))
MARK(first1(y0, first(x0, x1))) → ACTIVE(first1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(first1(fcons(x0, x1), y1)) → ACTIVE(first1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(unquote(01)) → ACTIVE(unquote(active(01)))
MARK(unquote(cons(x0, x1))) → ACTIVE(unquote(active(cons(mark(x0), x1))))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(sel1(x0, x1), y1)) → ACTIVE(sel(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(first1(x0, x1), y1)) → ACTIVE(sel(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(mark(x0))), mark(y1)))
MARK(unquote(quote1(x0))) → ACTIVE(unquote(active(quote1(x0))))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
MARK(unquote1(cons1(x0, x1))) → ACTIVE(unquote1(active(cons1(mark(x0), mark(x1)))))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(first1(y0, unquote1(x0))) → ACTIVE(first1(mark(y0), active(unquote1(mark(x0)))))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(sel(y0, unquote1(x0))) → ACTIVE(sel(mark(y0), active(unquote1(mark(x0)))))
MARK(unquote1(fcons(x0, x1))) → ACTIVE(unquote1(active(fcons(mark(x0), mark(x1)))))
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(first1(x0, x1))) → ACTIVE(unquote(active(first1(mark(x0), mark(x1)))))
MARK(unquote(0)) → ACTIVE(unquote(active(0)))
MARK(s(X)) → MARK(X)
MARK(unquote(X)) → MARK(X)
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel(y0, quote1(x0))) → ACTIVE(sel(mark(y0), active(quote1(x0))))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(first1(X1, X2)) → MARK(X1)
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(unquote(from(x0))) → ACTIVE(unquote(active(from(mark(x0)))))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(unquote(nil1)) → ACTIVE(unquote(active(nil1)))
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel(quote(x0), y1)) → ACTIVE(sel(active(quote(x0)), mark(y1)))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first1(cons1(x0, x1), y1)) → ACTIVE(first1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(nil)) → ACTIVE(unquote1(active(nil)))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(first1(x0, y1)) → ACTIVE(first1(x0, mark(y1)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(sel(y0, nil1)) → ACTIVE(sel(mark(y0), active(nil1)))
MARK(sel(nil1, y1)) → ACTIVE(sel(active(nil1), mark(y1)))
MARK(first1(from(x0), y1)) → ACTIVE(first1(active(from(mark(x0))), mark(y1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2)))
MARK(sel(y0, 01)) → ACTIVE(sel(mark(y0), active(01)))
MARK(sel(01, y1)) → ACTIVE(sel(active(01), mark(y1)))
MARK(first1(y0, s1(x0))) → ACTIVE(first1(mark(y0), active(s1(mark(x0)))))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(unquote1(sel(x0, x1))) → ACTIVE(unquote1(active(sel(mark(x0), mark(x1)))))
MARK(unquote(nil)) → ACTIVE(unquote(active(nil)))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))
MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
MARK(unquote(fcons(x0, x1))) → ACTIVE(unquote(active(fcons(mark(x0), mark(x1)))))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first1(y0, from(x0))) → ACTIVE(first1(mark(y0), active(from(mark(x0)))))
MARK(first1(y0, 0)) → ACTIVE(first1(mark(y0), active(0)))
MARK(first1(0, y1)) → ACTIVE(first1(active(0), mark(y1)))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(first1(y0, quote1(x0))) → ACTIVE(first1(mark(y0), active(quote1(x0))))
MARK(unquote(sel1(x0, x1))) → ACTIVE(unquote(active(sel1(mark(x0), mark(x1)))))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(mark(x0)))))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(unquote1(01)) → ACTIVE(unquote1(active(01)))
MARK(unquote1(nil1)) → ACTIVE(unquote1(active(nil1)))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
MARK(unquote(quote(x0))) → ACTIVE(unquote(active(quote(x0))))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(unquote1(x0))) → ACTIVE(unquote1(active(unquote1(mark(x0)))))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(first1(nil, y1)) → ACTIVE(first1(active(nil), mark(y1)))
MARK(first1(y0, nil)) → ACTIVE(first1(mark(y0), active(nil)))
MARK(unquote1(quote(x0))) → ACTIVE(unquote1(active(quote(x0))))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(first1(y0, unquote(x0))) → ACTIVE(first1(mark(y0), active(unquote(mark(x0)))))
MARK(sel(fcons(x0, x1), y1)) → ACTIVE(sel(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(x0)) → ACTIVE(unquote(x0))
MARK(s1(X)) → MARK(X)
MARK(first1(y0, sel(x0, x1))) → ACTIVE(first1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel(y0, first(x0, x1))) → ACTIVE(sel(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(unquote1(s(x0))) → ACTIVE(unquote1(active(s(mark(x0)))))
MARK(unquote1(unquote(x0))) → ACTIVE(unquote1(active(unquote(mark(x0)))))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(unquote1(cons(x0, x1))) → ACTIVE(unquote1(active(cons(mark(x0), x1))))
MARK(unquote(s1(x0))) → ACTIVE(unquote(active(s1(mark(x0)))))
MARK(first1(y0, s(x0))) → ACTIVE(first1(mark(y0), active(s(mark(x0)))))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(sel(first(x0, x1), y1)) → ACTIVE(sel(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel(quote1(x0), y1)) → ACTIVE(sel(active(quote1(x0)), mark(y1)))
MARK(first1(s1(x0), y1)) → ACTIVE(first1(active(s1(mark(x0))), mark(y1)))
MARK(first1(y0, cons1(x0, x1))) → ACTIVE(first1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(first1(y0, fcons(x0, x1))) → ACTIVE(first1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(first1(sel1(x0, x1), y1)) → ACTIVE(first1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(quote(X)) → ACTIVE(quote(X))
MARK(sel(y0, first1(x0, x1))) → ACTIVE(sel(mark(y0), active(first1(mark(x0), mark(x1)))))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(first1(01, y1)) → ACTIVE(first1(active(01), mark(y1)))
MARK(first1(y0, 01)) → ACTIVE(first1(mark(y0), active(01)))
MARK(first1(first1(x0, x1), y1)) → ACTIVE(first1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(cons1(x0, x1))) → ACTIVE(unquote(active(cons1(mark(x0), mark(x1)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(unquote1(quote1(x0))) → ACTIVE(unquote1(active(quote1(x0))))
MARK(unquote1(first1(x0, x1))) → ACTIVE(unquote1(active(first1(mark(x0), mark(x1)))))
MARK(first1(y0, sel1(x0, x1))) → ACTIVE(first1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(first1(cons(x0, x1), y1)) → ACTIVE(first1(active(cons(mark(x0), x1)), mark(y1)))
MARK(unquote(first(x0, x1))) → ACTIVE(unquote(active(first(mark(x0), mark(x1)))))
MARK(cons1(X1, X2)) → MARK(X1)
MARK(first1(first(x0, x1), y1)) → ACTIVE(first1(active(first(mark(x0), mark(x1))), mark(y1)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(sel(unquote1(x0), y1)) → ACTIVE(sel(active(unquote1(mark(x0))), mark(y1)))
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(y0, s1(x0))) → ACTIVE(sel(mark(y0), active(s1(mark(x0)))))
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(first1(y0, x1)) → ACTIVE(first1(mark(y0), x1))
MARK(unquote1(first(x0, x1))) → ACTIVE(unquote1(active(first(mark(x0), mark(x1)))))
MARK(unquote1(sel1(x0, x1))) → ACTIVE(unquote1(active(sel1(mark(x0), mark(x1)))))
MARK(unquote1(X)) → MARK(X)
MARK(first1(sel(x0, x1), y1)) → ACTIVE(first1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first1(quote(x0), y1)) → ACTIVE(first1(active(quote(x0)), mark(y1)))
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(unquote1(0)) → ACTIVE(unquote1(active(0)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(unquote(unquote1(x0))) → ACTIVE(unquote(active(unquote1(mark(x0)))))
MARK(sel(cons1(x0, x1), y1)) → ACTIVE(sel(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(first1(y0, first1(x0, x1))) → ACTIVE(first1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(sel(y0, quote(x0))) → ACTIVE(sel(mark(y0), active(quote(x0))))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(first1(y0, cons(x0, x1))) → ACTIVE(first1(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

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.
By narrowing [15] the rule MARK(first(X1, X2)) → ACTIVE(first(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(first(unquote1(x0), y1)) → ACTIVE(first(active(unquote1(mark(x0))), mark(y1)))
MARK(first(y0, quote(x0))) → ACTIVE(first(mark(y0), active(quote(x0))))
MARK(first(from(x0), y1)) → ACTIVE(first(active(from(mark(x0))), mark(y1)))
MARK(first(y0, unquote(x0))) → ACTIVE(first(mark(y0), active(unquote(mark(x0)))))
MARK(first(s1(x0), y1)) → ACTIVE(first(active(s1(mark(x0))), mark(y1)))
MARK(first(y0, first1(x0, x1))) → ACTIVE(first(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(first(sel1(x0, x1), y1)) → ACTIVE(first(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(first(unquote(x0), y1)) → ACTIVE(first(active(unquote(mark(x0))), mark(y1)))
MARK(first(fcons(x0, x1), y1)) → ACTIVE(first(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(first(y0, cons1(x0, x1))) → ACTIVE(first(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(first(y0, nil)) → ACTIVE(first(mark(y0), active(nil)))
MARK(first(nil, y1)) → ACTIVE(first(active(nil), mark(y1)))
MARK(first(y0, x1)) → ACTIVE(first(mark(y0), x1))
MARK(first(first1(x0, x1), y1)) → ACTIVE(first(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(first(y0, s(x0))) → ACTIVE(first(mark(y0), active(s(mark(x0)))))
MARK(first(y0, from(x0))) → ACTIVE(first(mark(y0), active(from(mark(x0)))))
MARK(first(sel(x0, x1), y1)) → ACTIVE(first(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first(y0, 01)) → ACTIVE(first(mark(y0), active(01)))
MARK(first(quote(x0), y1)) → ACTIVE(first(active(quote(x0)), mark(y1)))
MARK(first(0, y1)) → ACTIVE(first(active(0), mark(y1)))
MARK(first(01, y1)) → ACTIVE(first(active(01), mark(y1)))
MARK(first(y0, 0)) → ACTIVE(first(mark(y0), active(0)))
MARK(first(y0, s1(x0))) → ACTIVE(first(mark(y0), active(s1(mark(x0)))))
MARK(first(cons1(x0, x1), y1)) → ACTIVE(first(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(first(y0, sel(x0, x1))) → ACTIVE(first(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(first(cons(x0, x1), y1)) → ACTIVE(first(active(cons(mark(x0), x1)), mark(y1)))
MARK(first(y0, unquote1(x0))) → ACTIVE(first(mark(y0), active(unquote1(mark(x0)))))
MARK(first(x0, y1)) → ACTIVE(first(x0, mark(y1)))
MARK(first(quote1(x0), y1)) → ACTIVE(first(active(quote1(x0)), mark(y1)))
MARK(first(y0, fcons(x0, x1))) → ACTIVE(first(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(first(first(x0, x1), y1)) → ACTIVE(first(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(first(y0, first(x0, x1))) → ACTIVE(first(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(first(s(x0), y1)) → ACTIVE(first(active(s(mark(x0))), mark(y1)))
MARK(first(y0, nil1)) → ACTIVE(first(mark(y0), active(nil1)))
MARK(first(nil1, y1)) → ACTIVE(first(active(nil1), mark(y1)))
MARK(first(y0, sel1(x0, x1))) → ACTIVE(first(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(first(y0, quote1(x0))) → ACTIVE(first(mark(y0), active(quote1(x0))))
MARK(first(y0, cons(x0, x1))) → ACTIVE(first(mark(y0), active(cons(mark(x0), x1))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

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

ACTIVE(fcons(X, Z)) → MARK(cons(X, Z))
MARK(first1(quote1(x0), y1)) → ACTIVE(first1(active(quote1(x0)), mark(y1)))
MARK(sel(y0, fcons(x0, x1))) → ACTIVE(sel(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(first(y0, unquote(x0))) → ACTIVE(first(mark(y0), active(unquote(mark(x0)))))
MARK(first1(y0, quote(x0))) → ACTIVE(first1(mark(y0), active(quote(x0))))
MARK(first1(unquote(x0), y1)) → ACTIVE(first1(active(unquote(mark(x0))), mark(y1)))
MARK(sel(unquote(x0), y1)) → ACTIVE(sel(active(unquote(mark(x0))), mark(y1)))
MARK(first1(unquote1(x0), y1)) → ACTIVE(first1(active(unquote1(mark(x0))), mark(y1)))
MARK(sel1(y0, s1(x0))) → ACTIVE(sel1(mark(y0), active(s1(mark(x0)))))
MARK(first(y0, s(x0))) → ACTIVE(first(mark(y0), active(s(mark(x0)))))
MARK(first1(nil1, y1)) → ACTIVE(first1(active(nil1), mark(y1)))
MARK(first1(y0, nil1)) → ACTIVE(first1(mark(y0), active(nil1)))
MARK(unquote(s(x0))) → ACTIVE(unquote(active(s(mark(x0)))))
MARK(sel(y0, cons1(x0, x1))) → ACTIVE(sel(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(unquote(sel(x0, x1))) → ACTIVE(unquote(active(sel(mark(x0), mark(x1)))))
MARK(first(cons1(x0, x1), y1)) → ACTIVE(first(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(unquote(x0))) → ACTIVE(unquote(active(unquote(mark(x0)))))
MARK(first1(y0, first(x0, x1))) → ACTIVE(first1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(sel1(y0, nil)) → ACTIVE(sel1(mark(y0), active(nil)))
MARK(sel1(nil, y1)) → ACTIVE(sel1(active(nil), mark(y1)))
MARK(unquote(cons(x0, x1))) → ACTIVE(unquote(active(cons(mark(x0), x1))))
MARK(sel1(y0, sel(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(sel1(x0, x1), y1)) → ACTIVE(sel(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(first(X1, X2)) → MARK(X2)
MARK(unquote1(cons1(x0, x1))) → ACTIVE(unquote1(active(cons1(mark(x0), mark(x1)))))
ACTIVE(sel1(0, cons(X, Z))) → MARK(quote(X))
MARK(sel1(y0, unquote1(x0))) → ACTIVE(sel1(mark(y0), active(unquote1(mark(x0)))))
MARK(unquote(0)) → ACTIVE(unquote(active(0)))
MARK(first(from(x0), y1)) → ACTIVE(first(active(from(mark(x0))), mark(y1)))
MARK(first(s1(x0), y1)) → ACTIVE(first(active(s1(mark(x0))), mark(y1)))
MARK(fcons(X1, X2)) → MARK(X2)
MARK(sel(y0, quote1(x0))) → ACTIVE(sel(mark(y0), active(quote1(x0))))
MARK(first(nil, y1)) → ACTIVE(first(active(nil), mark(y1)))
MARK(first(y0, nil)) → ACTIVE(first(mark(y0), active(nil)))
MARK(first1(X1, X2)) → MARK(X1)
MARK(first(first1(x0, x1), y1)) → ACTIVE(first(active(first1(mark(x0), mark(x1))), mark(y1)))
ACTIVE(quote1(first(X, Z))) → MARK(first1(X, Z))
MARK(unquote(nil1)) → ACTIVE(unquote(active(nil1)))
MARK(sel1(cons(x0, x1), y1)) → ACTIVE(sel1(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(quote(x0), y1)) → ACTIVE(sel(active(quote(x0)), mark(y1)))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first1(cons1(x0, x1), y1)) → ACTIVE(first1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote1(nil)) → ACTIVE(unquote1(active(nil)))
MARK(sel1(y0, cons1(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(first1(x0, y1)) → ACTIVE(first1(x0, mark(y1)))
MARK(sel(y0, nil1)) → ACTIVE(sel(mark(y0), active(nil1)))
MARK(sel(nil1, y1)) → ACTIVE(sel(active(nil1), mark(y1)))
MARK(first(first(x0, x1), y1)) → ACTIVE(first(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, nil1)) → ACTIVE(sel1(mark(y0), active(nil1)))
MARK(sel1(nil1, y1)) → ACTIVE(sel1(active(nil1), mark(y1)))
MARK(sel(y0, 01)) → ACTIVE(sel(mark(y0), active(01)))
MARK(sel(01, y1)) → ACTIVE(sel(active(01), mark(y1)))
MARK(first(nil1, y1)) → ACTIVE(first(active(nil1), mark(y1)))
MARK(first(y0, nil1)) → ACTIVE(first(mark(y0), active(nil1)))
MARK(first1(y0, s1(x0))) → ACTIVE(first1(mark(y0), active(s1(mark(x0)))))
MARK(unquote(nil)) → ACTIVE(unquote(active(nil)))
MARK(sel1(y0, fcons(x0, x1))) → ACTIVE(sel1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(unquote(fcons(x0, x1))) → ACTIVE(unquote(active(fcons(mark(x0), mark(x1)))))
MARK(first1(y0, from(x0))) → ACTIVE(first1(mark(y0), active(from(mark(x0)))))
ACTIVE(unquote(s1(X))) → MARK(s(unquote(X)))
MARK(sel1(quote(x0), y1)) → ACTIVE(sel1(active(quote(x0)), mark(y1)))
MARK(first(y0, x1)) → ACTIVE(first(mark(y0), x1))
MARK(unquote(sel1(x0, x1))) → ACTIVE(unquote(active(sel1(mark(x0), mark(x1)))))
MARK(sel1(y0, sel1(x0, x1))) → ACTIVE(sel1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel1(s1(x0), y1)) → ACTIVE(sel1(active(s1(mark(x0))), mark(y1)))
MARK(first(y0, from(x0))) → ACTIVE(first(mark(y0), active(from(mark(x0)))))
MARK(unquote1(nil1)) → ACTIVE(unquote1(active(nil1)))
MARK(sel1(y0, x1)) → ACTIVE(sel1(mark(y0), x1))
MARK(sel1(cons1(x0, x1), y1)) → ACTIVE(sel1(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(first1(nil, y1)) → ACTIVE(first1(active(nil), mark(y1)))
MARK(first1(y0, nil)) → ACTIVE(first1(mark(y0), active(nil)))
ACTIVE(quote(sel(X, Z))) → MARK(sel1(X, Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(first1(y0, unquote(x0))) → ACTIVE(first1(mark(y0), active(unquote(mark(x0)))))
MARK(first(y0, unquote1(x0))) → ACTIVE(first(mark(y0), active(unquote1(mark(x0)))))
MARK(sel(fcons(x0, x1), y1)) → ACTIVE(sel(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(sel1(x0, x1), y1)) → ACTIVE(sel1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(sel1(y0, quote1(x0))) → ACTIVE(sel1(mark(y0), active(quote1(x0))))
MARK(unquote1(cons(x0, x1))) → ACTIVE(unquote1(active(cons(mark(x0), x1))))
MARK(fcons(X1, X2)) → MARK(X1)
MARK(first1(X1, X2)) → MARK(X2)
MARK(first(unquote1(x0), y1)) → ACTIVE(first(active(unquote1(mark(x0))), mark(y1)))
MARK(first1(y0, cons1(x0, x1))) → ACTIVE(first1(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel1(unquote(x0), y1)) → ACTIVE(sel1(active(unquote(mark(x0))), mark(y1)))
MARK(first1(sel1(x0, x1), y1)) → ACTIVE(first1(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(quote(X)) → ACTIVE(quote(X))
MARK(first1(first1(x0, x1), y1)) → ACTIVE(first1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(cons1(x0, x1))) → ACTIVE(unquote(active(cons1(mark(x0), mark(x1)))))
MARK(first1(cons(x0, x1), y1)) → ACTIVE(first1(active(cons(mark(x0), x1)), mark(y1)))
MARK(first(y0, 0)) → ACTIVE(first(mark(y0), active(0)))
MARK(first(0, y1)) → ACTIVE(first(active(0), mark(y1)))
MARK(cons1(X1, X2)) → MARK(X1)
MARK(first(y0, s1(x0))) → ACTIVE(first(mark(y0), active(s1(mark(x0)))))
MARK(from(X)) → MARK(X)
MARK(sel(y0, s1(x0))) → ACTIVE(sel(mark(y0), active(s1(mark(x0)))))
MARK(sel1(y0, first1(x0, x1))) → ACTIVE(sel1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(unquote1(sel1(x0, x1))) → ACTIVE(unquote1(active(sel1(mark(x0), mark(x1)))))
MARK(unquote1(X)) → MARK(X)
MARK(first(quote1(x0), y1)) → ACTIVE(first(active(quote1(x0)), mark(y1)))
MARK(first(y0, first(x0, x1))) → ACTIVE(first(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(first(s(x0), y1)) → ACTIVE(first(active(s(mark(x0))), mark(y1)))
MARK(unquote1(0)) → ACTIVE(unquote1(active(0)))
MARK(sel1(y0, s(x0))) → ACTIVE(sel1(mark(y0), active(s(mark(x0)))))
MARK(unquote(unquote1(x0))) → ACTIVE(unquote(active(unquote1(mark(x0)))))
MARK(sel(cons1(x0, x1), y1)) → ACTIVE(sel(active(cons1(mark(x0), mark(x1))), mark(y1)))
MARK(first1(y0, first1(x0, x1))) → ACTIVE(first1(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(sel1(X1, X2)) → MARK(X2)
MARK(first1(y0, cons(x0, x1))) → ACTIVE(first1(mark(y0), active(cons(mark(x0), x1))))
MARK(unquote1(x0)) → ACTIVE(unquote1(x0))
MARK(sel1(y0, quote(x0))) → ACTIVE(sel1(mark(y0), active(quote(x0))))
MARK(sel1(s(x0), y1)) → ACTIVE(sel1(active(s(mark(x0))), mark(y1)))
MARK(sel(y0, sel1(x0, x1))) → ACTIVE(sel(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel(s1(x0), y1)) → ACTIVE(sel(active(s1(mark(x0))), mark(y1)))
MARK(quote1(X)) → ACTIVE(quote1(X))
MARK(first(y0, cons1(x0, x1))) → ACTIVE(first(mark(y0), active(cons1(mark(x0), mark(x1)))))
MARK(sel(y0, unquote(x0))) → ACTIVE(sel(mark(y0), active(unquote(mark(x0)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(sel1(y0, first(x0, x1))) → ACTIVE(sel1(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(first1(s(x0), y1)) → ACTIVE(first1(active(s(mark(x0))), mark(y1)))
MARK(unquote1(from(x0))) → ACTIVE(unquote1(active(from(mark(x0)))))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel1(x0, y1)) → ACTIVE(sel1(x0, mark(y1)))
MARK(unquote1(s1(x0))) → ACTIVE(unquote1(active(s1(mark(x0)))))
MARK(first1(fcons(x0, x1), y1)) → ACTIVE(first1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(01)) → ACTIVE(unquote(active(01)))
MARK(first(x0, y1)) → ACTIVE(first(x0, mark(y1)))
MARK(sel(first1(x0, x1), y1)) → ACTIVE(sel(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(mark(x0))), mark(y1)))
MARK(unquote(quote1(x0))) → ACTIVE(unquote(active(quote1(x0))))
MARK(first(y0, fcons(x0, x1))) → ACTIVE(first(mark(y0), active(fcons(mark(x0), mark(x1)))))
ACTIVE(unquote1(cons1(X, Z))) → MARK(fcons(unquote(X), unquote1(Z)))
MARK(first1(y0, unquote1(x0))) → ACTIVE(first1(mark(y0), active(unquote1(mark(x0)))))
MARK(sel(y0, unquote1(x0))) → ACTIVE(sel(mark(y0), active(unquote1(mark(x0)))))
MARK(unquote1(fcons(x0, x1))) → ACTIVE(unquote1(active(fcons(mark(x0), mark(x1)))))
MARK(sel1(first1(x0, x1), y1)) → ACTIVE(sel1(active(first1(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(first1(x0, x1))) → ACTIVE(unquote(active(first1(mark(x0), mark(x1)))))
MARK(s(X)) → MARK(X)
MARK(unquote(X)) → MARK(X)
MARK(sel1(y0, cons(x0, x1))) → ACTIVE(sel1(mark(y0), active(cons(mark(x0), x1))))
MARK(first(unquote(x0), y1)) → ACTIVE(first(active(unquote(mark(x0))), mark(y1)))
MARK(first(y0, first1(x0, x1))) → ACTIVE(first(mark(y0), active(first1(mark(x0), mark(x1)))))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(y0, from(x0))) → ACTIVE(sel1(mark(y0), active(from(mark(x0)))))
MARK(sel1(y0, unquote(x0))) → ACTIVE(sel1(mark(y0), active(unquote(mark(x0)))))
MARK(sel1(0, y1)) → ACTIVE(sel1(active(0), mark(y1)))
MARK(sel1(y0, 0)) → ACTIVE(sel1(mark(y0), active(0)))
MARK(unquote(from(x0))) → ACTIVE(unquote(active(from(mark(x0)))))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(first(quote(x0), y1)) → ACTIVE(first(active(quote(x0)), mark(y1)))
MARK(sel1(first(x0, x1), y1)) → ACTIVE(sel1(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(first(y0, sel(x0, x1))) → ACTIVE(first(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(first(cons(x0, x1), y1)) → ACTIVE(first(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel1(y0, 01)) → ACTIVE(sel1(mark(y0), active(01)))
MARK(sel1(01, y1)) → ACTIVE(sel1(active(01), mark(y1)))
MARK(first1(from(x0), y1)) → ACTIVE(first1(active(from(mark(x0))), mark(y1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel1(unquote1(x0), y1)) → ACTIVE(sel1(active(unquote1(mark(x0))), mark(y1)))
ACTIVE(quote(s(X))) → MARK(s1(quote(X)))
MARK(unquote1(sel(x0, x1))) → ACTIVE(unquote1(active(sel(mark(x0), mark(x1)))))
MARK(fcons(X1, X2)) → ACTIVE(fcons(mark(X1), mark(X2)))
MARK(first(y0, sel1(x0, x1))) → ACTIVE(first(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))
MARK(sel1(from(x0), y1)) → ACTIVE(sel1(active(from(mark(x0))), mark(y1)))
MARK(first(sel1(x0, x1), y1)) → ACTIVE(first(active(sel1(mark(x0), mark(x1))), mark(y1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first1(y0, 0)) → ACTIVE(first1(mark(y0), active(0)))
MARK(first1(0, y1)) → ACTIVE(first1(active(0), mark(y1)))
ACTIVE(first1(s(X), cons(Y, Z))) → MARK(cons1(quote(Y), first1(X, Z)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(first1(y0, quote1(x0))) → ACTIVE(first1(mark(y0), active(quote1(x0))))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(mark(x0)))))
MARK(unquote1(01)) → ACTIVE(unquote1(active(01)))
MARK(first(01, y1)) → ACTIVE(first(active(01), mark(y1)))
MARK(first(y0, 01)) → ACTIVE(first(mark(y0), active(01)))
MARK(unquote(quote(x0))) → ACTIVE(unquote(active(quote(x0))))
ACTIVE(sel(0, cons(X, Z))) → MARK(X)
MARK(unquote1(unquote1(x0))) → ACTIVE(unquote1(active(unquote1(mark(x0)))))
MARK(unquote1(quote(x0))) → ACTIVE(unquote1(active(quote(x0))))
MARK(unquote(x0)) → ACTIVE(unquote(x0))
MARK(s1(X)) → MARK(X)
MARK(first1(y0, sel(x0, x1))) → ACTIVE(first1(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel(y0, first(x0, x1))) → ACTIVE(sel(mark(y0), active(first(mark(x0), mark(x1)))))
MARK(unquote1(s(x0))) → ACTIVE(unquote1(active(s(mark(x0)))))
MARK(unquote1(unquote(x0))) → ACTIVE(unquote1(active(unquote(mark(x0)))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(unquote(s1(x0))) → ACTIVE(unquote(active(s1(mark(x0)))))
MARK(first1(y0, s(x0))) → ACTIVE(first1(mark(y0), active(s(mark(x0)))))
MARK(first(y0, cons(x0, x1))) → ACTIVE(first(mark(y0), active(cons(mark(x0), x1))))
MARK(first(y0, quote1(x0))) → ACTIVE(first(mark(y0), active(quote1(x0))))
MARK(sel(first(x0, x1), y1)) → ACTIVE(sel(active(first(mark(x0), mark(x1))), mark(y1)))
MARK(sel(quote1(x0), y1)) → ACTIVE(sel(active(quote1(x0)), mark(y1)))
MARK(first1(s1(x0), y1)) → ACTIVE(first1(active(s1(mark(x0))), mark(y1)))
MARK(first(y0, quote(x0))) → ACTIVE(first(mark(y0), active(quote(x0))))
MARK(first1(y0, fcons(x0, x1))) → ACTIVE(first1(mark(y0), active(fcons(mark(x0), mark(x1)))))
MARK(sel(y0, first1(x0, x1))) → ACTIVE(sel(mark(y0), active(first1(mark(x0), mark(x1)))))
ACTIVE(quote1(cons(X, Z))) → MARK(cons1(quote(X), quote1(Z)))
MARK(first(fcons(x0, x1), y1)) → ACTIVE(first(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(first1(01, y1)) → ACTIVE(first1(active(01), mark(y1)))
MARK(first1(y0, 01)) → ACTIVE(first1(mark(y0), active(01)))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(unquote1(quote1(x0))) → ACTIVE(unquote1(active(quote1(x0))))
MARK(unquote1(first1(x0, x1))) → ACTIVE(unquote1(active(first1(mark(x0), mark(x1)))))
MARK(first1(y0, sel1(x0, x1))) → ACTIVE(first1(mark(y0), active(sel1(mark(x0), mark(x1)))))
MARK(cons1(X1, X2)) → MARK(X2)
MARK(sel1(fcons(x0, x1), y1)) → ACTIVE(sel1(active(fcons(mark(x0), mark(x1))), mark(y1)))
MARK(unquote(first(x0, x1))) → ACTIVE(unquote(active(first(mark(x0), mark(x1)))))
MARK(first(sel(x0, x1), y1)) → ACTIVE(first(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first1(first(x0, x1), y1)) → ACTIVE(first1(active(first(mark(x0), mark(x1))), mark(y1)))
ACTIVE(sel1(s(X), cons(Y, Z))) → MARK(sel1(X, Z))
MARK(sel(unquote1(x0), y1)) → ACTIVE(sel(active(unquote1(mark(x0))), mark(y1)))
MARK(first(X1, X2)) → MARK(X1)
ACTIVE(first(s(X), cons(Y, Z))) → MARK(cons(Y, first(X, Z)))
MARK(first1(y0, x1)) → ACTIVE(first1(mark(y0), x1))
MARK(unquote1(first(x0, x1))) → ACTIVE(unquote1(active(first(mark(x0), mark(x1)))))
MARK(first1(sel(x0, x1), y1)) → ACTIVE(first1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(first1(quote(x0), y1)) → ACTIVE(first1(active(quote(x0)), mark(y1)))
MARK(sel1(sel(x0, x1), y1)) → ACTIVE(sel1(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel(y0, quote(x0))) → ACTIVE(sel(mark(y0), active(quote(x0))))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel1(quote1(x0), y1)) → ACTIVE(sel1(active(quote1(x0)), mark(y1)))

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.